Algebra & Number Theory
C.F. Gauss E. Noether D. Hilbert F.J. MacWilliams
Algebra and Number Theory Seminar Schedule, since Summer 2009
Tentative schedule of current semester
Dec 8, 2021 Kathryn Hechtel (University of Kentucky)
Roos Bound for Skew Cyclic Codes
In this talk, a Roos bound for the minimum distance of skew-cyclic codes is presented. Skew-cyclic codes are generated by a skew polynomial. With certain assumptions on the roots of this polynomial, we can guarantee a lower bound on the minimum distance in the Hamming and rank metrics. For the rank case, we obtain a construction of skew-cyclic MRD codes based on arithmetic progressions. We end with an open question about whether the arithmetic progression is a necessary condition.
Dec 1, 2021 Michael Morrow (University of Kentucky)
Finite Computation of Gröbner Bases for OI-Modules
Computing finite Gröbner bases for submodules of free modules over polynomial rings is a classical problem in commutative algebra that is solved via Buchberger's Algorithm. In this talk, we generalize these ideas to the setting of free OI-modules over a Noetherian polynomial OI-algebra. In particular, we develop an OI-Buchberger's Criterion and Algorithm whose finiteness depends on a combinatorial result about factoring OI-maps.
Oct 20, 2021 Erika Ordog (Texas A&M University)
Minimal resolutions of monomial ideals
There are two general constructions for minimal free resolutions of arbitrary monomial ideals. The first construction by Eagon is obtained via spectral sequences associated to certain Koszul complexes, and the second construction by Yuzvinsky is obtained from the Taylor resolution and utilizes the lcm lattice of monomial generators. In order to obtain a canonical, minimal free resolution for an arbitrary monomial ideal with a closed-form, combinatorial description of the differential, a combinatorial formula for the Moore-Penrose pseudoinverse is combined with these two constructions. The result is two differentials given as sums over lattice paths, one in the lcm lattice and one in N^n, of weights associated to higher-dimensional analogues of spanning trees. This is based on joint work with John Eagon and Ezra Miller.
Oct 20, 2021 Aida Maraj (University of Michigan)
Staged Tree Models with Toric Structure
The star of the talk will be some parametrised algebraic varieties motivated by the discrete statistical models called staged tree models. These models encode relationships between events. They are realised by directed trees with coloured vertices and include other well-known statistical models, such as Bayesian networks and hierarchical models. In algebro-geometric terms, a staged tree model consists of points inside a toric variety. For certain trees, called balanced, the model is in fact the intersection of the toric variety and the probability simplex. This gives the model a straightforward description, and has computational advantages. We will see that the class of staged tree models with a toric structure extends far outside of the balanced case, if we allow a change of coordinates. The talk is based on joint work with Lisa Nicklasson and Christiane Görgen.
Oct 20, 2021 Chris Manon (University of Kentucky)
Searching for Mori Dream Spaces
Mori dream spaces were defined by Hu and Keel to connect the minimal model program in birational geometry to geometric invariant theory. Along the lines of Hilbert's 14th problem, a space is a Mori dream space if a certain associated ring is finitely generated. This finiteness property makes Mori dream spaces an interesting and useful class of spaces in algebraic and arithmetic geometry. Many familiar spaces from algebraic geometry (flag varieties, complete symmetric spaces, toric varieties) end up being Mori dream spaces. On the other hand, results of Castravet, Tevelev, Mukai, Karu, Gonzalez, and others show that it's possible to do "nice" things to "nice" spaces and break the Mori dream space property. Determining just what goes wrong in a given case can be a subtle mixture of commutative algebra, algebraic geometry, and combinatorics. I'll give a general introduction to Mori dream spaces, and then describe some recent progress in determining this property for a special class of spaces using methods from tropical geometry.
Oct 13, 2021 Fernando Shao(University of Kentucky)
Singmaster's conjecture in the interior of Pascal's triangle
Singmaster's conjecture asserts that every natural number greater than one occurs at most a bounded number of times in Pascal's triangle. In this talk I will survey some results on this conjecture, and present a new result which establishes this conjecture in the interior region of Pascal's triangle. Our proof methods combine an Archimedean argument (due to Kand and reminiscent of the Bombieri-Pila determinant method) and a non-Archimedean argument based on Vinogradov's exponential sum estimates over primes. This is joint work with Kaisa Matomaki, Maksym Radziwill, Terence Tao, and Joni Teravainen.
Sep 29, 2021 Angela Hanson (University of Kentucky)
Combinatorial Approach to Surjectivity of the Wahl Map
The Wahl map defines a notion of the derivative on smooth curves but also has a nice structure when simplified to singular curves. The paper "The Gauss map for trivalent graphs� by Ciliberto and Franchetta shows that the Wahl map is surjective for trivalent, three-connected graphs with some particular structure which we will discuss. In particular, this proof breaks the Wahl map into a map on vertices and a map on edges of the simplicial curves which allows us to use graph theoretic techniques. We will define these maps and discuss the graphic structure which provides surjectivity of each. Depending on time we will delve deeper into enumeration of these particular graphs.
Sep 15, 2021 Dave Jensen (University of Kentucky)
Tropical Linear Series
In the past decade, many researchers have become interested in geometric objects defined over semirings, rather than rings. We will discuss some of the pathologies that arise when one attempts algebraic constructions in this setting. We will provide a definition of tropical linear series, analogous to linear series on an algebraic curve, and pose some of the many open questions that have been vexing us for years.
Sep 8, 2021 Joseph Cummings (University of Kentucky)
Phylogenetic Networks
Phylogenetic networks can model more complicated evolutionary phenomena that trees fail to capture such as horizontal gene transfer and hybridization. Using the same Markov processes used on trees, these models can be extended to networks. In particular, we will look at the Cavendar-Farris-Neyman (CFN) model. A discrete Fourier transform can be applied to the CFN model which greatly simplifies the parametrization. In this talk, we will review how to find phylogenetic invariants for trees, and then we will give a description of all quadratic invariants for level-1 networks.
Jun 22, 2021 Hunter Lehmann (University of Kentucky)
Weight Distributions and Automorphisms of Cyclic Orbit Codes
Cyclic orbit codes are subspace codes generated by the action of the Singer subgroup F_{q^n}^* on an F_q-subspace U of F_{q^n}. The weight distribution of a code is the vector whose i-th entry is the number of codewords with distance i to a fixed reference generator of the code. We will investigate the weight distribution for a few categories of cyclic orbit codes, including optimal codes. Further, we want to know when two cyclic orbit codes with the same weight distribution are isometric. To answer this question, we determine the possible automorphism groups for cyclic orbit codes.
May 5, 2021 Alex McDonough (Brown University)
A Family of Sandpile Multijections
Traditionally, the sandpile group is defined on a graph and the Matrix-Tree Theorem says that this group's size is equal to the number of spanning trees. An extension of the Matrix-Tree Theorem gives a relationship between the sandpile group and bases of an orientable arithmetic matroid. I provide a family of combinatorially meaningful maps between these two sets which I call "Sandpile Multijections". This generalizes a bijection given by Backman, Baker, and Yuen and extends work by Duval, Klivans, and Martin. I will not assume any background beyond undergraduate linear algebra.
Apr 28, 2021 Michael Morrow (University of Kentucky)
Equivariant Gröbner Bases
In the theory of computational commutative algebra, an algorithm due to Buchberger allows one to compute "good" generating sets for ideals of polynomial rings in finitely many variables. These generating sets are known as "Gröbner bases", and have many nice computational properties, such as allowing one to solve the ideal membership problem. We will generalize this theory to the setting of infinitely many variables, where noetherianity in the classical sense no longer holds. To deal with this, we will introduce a particular monoid action which will allow us to talk about finiteness results that hold "up to symmetry". This will lead to the notion of an "equivariant Gröbner basis", and we will exploit a result about strictly increasing maps on the natural numbers to give an algorithm for computing equivariant Gröbner bases in finite time.
Apr 27, 2021 Jay White (University of Kentucky)
Maximums of Total Betti Numbers in Hilbert Families
Two important invariants of an ideal are its Hilbert function and its Betti numbers. Naturally, the relationship between the two is of interest. We will explore this connection and in particular, will look at maximums for total Betti numbers when considering ideals that satisfy some constraint on their Hilbert functions and/or depths. Our results extend some existing results and lead to some other questions. We will also demonstrate an algorithm that efficiently computes these maximums.
Apr 21, 2021 Tony Várilly-Alvarado (Rice University)
Rational surfaces and locally recoverable codes
Motivated by large-scale storage problems around data loss, a budding branch of coding theory has surfaced in the last decade or so, centered around locally recoverable codes. These codes have the property that individual symbols in a codeword are functions of other symbols in the same word. If a symbol is lost (as opposed to corrupted), it can be recomputed, and hence a code word can be repaired. Algebraic geometry has a role to play in the design of codes with locality properties. In this talk I will explain how to use algebraic surfaces birational to the projective plane to both reinterpret constructions of optimal codes already found in the literature, and to find new locally recoverable codes, many of which are optimal (in a suitable sense). This is joint work with Cecília Salgado and Felipe Voloch.
Apr 20, 2021 Kathryn Hechtel (University of Kentucky)
Properties of Skew-Polynomial Rings and Skew-Cyclic Codes
A skew-polynomial ring is a polynomial ring where one must apply an automorphism to commute coefficients with x. It was first introduced by Ore in 1933 and later studied by Lam and Leroy in the 1980s. In this talk we will discuss some properties of skew polynomial rings and evaluations. Skew polynomials can have more roots than the degree suggests and hence we can produce more skew-cyclic codes. Further, we will discuss conditions presented in various research articles which guarantee a minimum distance for a skew-cyclic code.
Apr 14, 2021 Alberto Ravagnani (Technical University Eindhoven)
Common complements of linear subspaces and the density of optimal codes
An open question in coding theory asks whether or not MRD codes with the rank metric are dense as the field size tends to infinity. In this talk, I will survey this problem and the methods that have been developed to study it. I will then describe a new combinatorial method to obtain upper and lower bounds for the number of codes of prescribed parameters, based on the interpretation of optimal codes as the common complements linear subspaces. In particular, I will answer the above question in the negative, showing that MRD codes are almost always (very) sparse as the field size grows. The approach offers an explanation for the strong divergence in the behaviour of MDS and MRD codes with respect to density properties. I will also present partial results on the sparseness of MRD codes as their column length tends to infinity. The new results in this talk are joint work with A. Gruica.
Mar 31, 2021 Ralph Morrison (Williams College)
Higher-distance commuting varieties
The commuting variety is a well-studied object in algebraic geometry whose points are pairs of matrices that commute with one another. In this talk I present a generalization of the commuting variety by using the notion of commuting distance of matrices, which counts how many nonscalar matrices are required to form a commuting chain between two given matrices. I will prove that over any algebraically closed field, the set of pairs of matrices with bounded commuting distance forms an affine variety; and that this may or may not be the case over other fields. I will also discuss many open problems about these varieties, and present preliminary results in these directions. This is based on joint work with Madeleine Elyze, Alexander Guterman, and Klemen Sivic.
Mar 24, 2021 Al Shapere (University of Kentucky - Physics)
Quantum Error-Correcting Codes, Lattices and Conformal Field Theories
There is a rich web of connections between classical error-correcting codes, Euclidean integral lattices, and vertex operator algebras, known to physicists as conformal field theories (CFTs). I will describe an analogous relationship between quantum error-correcting codes, indefinite lattices, and a class of CFTs arising from toroidal compactifications of string theory. Specifically, I will show how quantum stabilizer codes are embedded in the operator algebras of these "quantum code" CFTs. For code CFTs, the constraints of modular invariance reduce to simple algebraic equations. Solving these equations provides many examples of physically distinct CFTs with the same spectrum.
Mar 2, 2021 Angela Hanson (University of Kentucky)
Surjectivity of the Wahl Map
The Wahl map defines a notion of the derivative on smooth curves. The paper `On the surjectivity of the Wahl map' by Ciliberto, Harris, and Miranda shows that the Wahl map is surjective for general curves of sufficiently high genus using techniques which resemble tropical geometry with a specific family of graphs. We look to generalize their argument and determine the necessary graph criteria which make the Wahl map surjective. We will explore motivation for examining surjectivity of the Wahl map and our generalized results.
Feb 24, 2021 David Stapleton (UC San Diego)
Studying Fano hypersurface with holomorphic forms
In characteristic 0 Fano varieties never admit holomorphic forms. In characteristic p, Kollár showed that Fano varieties that are p-cyclic covers can admit differential forms. This is a powerful tool for studying these varieties. By specializing to characteristic p one can use the positivity of these forms to show that the birational geometry complex Fano hypersurfaces can have quite different behavior from the birational geometry of projective space. For example, Kollár used these forms to prove that there are Fano hypersurfaces which are not rational (or even ruled), and Totaro showed that these Fano hypersurfaces are not even stably rational. In this talk we give new applications of these degeneration techniques. We explain that Fano hypersurfaces can have arbitrarily large degrees of irrationality and we show that the degrees of rational endomorphisms of Fano hypersurfaces must satisfy certain congruence constraints modulo p. This is joint work with Nathan Chen.
Oct 28, 2020 Courtney George (University of Kentucky)
Sagbi Bases of Cox-Nagata Rings
Cox-Nagata Rings can be described both in terms of blow-ups of projective space and as the collection of polynomials fixed under a group action - the very group action used to disprove Hilbert's 14th problem. We wish to examine the combinatorics of these rings. Under the right circumstances, we can extract desirable combinatorial properties, and the minimal generators of the Cox-Nagata Ring form a Khovanskii basis. We'll discuss the impact of tropical geometry on these structures, finishing with an application of Khovanskii bases in phylogenetic algebraic geometry. This talk is primarily attributed to a paper by Sturmfels and Xu by the same name.
Oct 21, 2020 Benjamin Jany (University of Kentucky)
Independent Space of q-Polymatroids
q-Matroids and q-polymatroids, a generalization of the former, were introduced and studied to understand algebraic and combinatorial invariants of rank metric codes. These two objects were initially defined via a rank function which is a monotone, non-negative, bounded, submodular function on the collection of subspaces of F_q^n that takes respectively integer or rational values. R. Jurrius and R. Pellikaan defined a notion of independent spaces for q-matroids and showed that these spaces fully determine a matroid and its rank function. For this talk I will focus on q-polymatroids and define a notion of independent spaces. Furthermore I will discuss some of the properties of the independent spaces of a q-polymatroid and show when a function defined on a collection of spaces which satisfy these same properties determine a q-polymatroid.
Oct 14, 2020 Milena Wrobel (University of Oldenburg)
On (singular) arrangement varieties
Arrangement varieties are T-varieties for which the torus action gives rise to a specific rational quotient to a projective space having a hyperplane arrangement as critical values. Important example classes are the toric varieties and the rational T-varieties of complexity one. We present an explicit description of their Cox rings, which gives us access to the geometry of these varieties. Using this description, we present i.a. a combinatorial tool for the classification of singular arrangement varieties.
Oct 7, 2020 Laura Escobar (Washington University in St. Louis)
Which Schubert varieties are Hessenberg varieties?
Schubert varieties are subvarieties of the flag variety parametrized by permutations; they induce an important basis for the cohomology of the flag variety. Hessenberg varieties are also subvarieties of the flag variety with connections to both algebraic combinatorics and representation theory. I will discuss joint work with Martha Precup and John Shareshian in which we investigate which Schubert varieties in the full flag variety are Hessenberg varieties.
Sep 30, 2020 Lara Bossinger (UNAM Oaxaca)
Positively well-poised embeddings
In this talk I aim to combine two subject: the tropicalization of an ideal and cluster algebras of finite type. To be more precise, the tropicalization of a homogeneous prime ideal J inside a polynomial ring is a subfan of the Gröbner fan of J. We think of the ideal as determining an embedding for the projective variety X=V(J). This embedding is called "well-poised" if all the initial ideals of maximal cones in the tropicalization of J are prime. For example, the Plücker embedding for the Grassmannian Gr(2,n) is well-poised. As this is a very strong condition, I suggest to focus only on initial ideals corresponding to maximal cones in the positive part of the tropicalization. If they are all prime, we say the embedding is "positively well-poised". It turns out that several finite type cluster algebras give rise to positively-well poised embeddings. This talk is based on work in progress and I will present partial results that I have obtained so far.
Sep 23, 2020 Alessandro Neri (Technical University of Munich)
Spaces of matrices with high rank using Galois theory
In this work we present an approach for studying linear spaces of n x n matrices with high rank. We do so for the case where the underlying field K admits a Galois extension L of degree n. In the special case where the Galois group is abelian, we derive the analogues of the celebrated Alon-Füredi theorem and of the Schwartz-Zippel lemma for endomorphisms, which produce nontrivial lower bounds on the rank of a linear endomorphism. The main motivation arises from algebraic coding theory, and in particular from the theory of rank-metric codes.
Well-Poised Embeddings of Arrangement Varieties
An affine variety V(I) is said to be well-poised if the initial ideal in_u(I) is prime for every u in Trop(V(I)). Arrangement varieties are a special class of T-varieties built from a hyperplane arrangement decorated by polyhedra. We will show that arrangement varieties always have a well-poised embedding and explore their toric degenerations coming from their tropicalizations. As a class of examples, we realize the Cox ring of any projectivized cotangent bundle on a smooth toric variety as the coordinate ring of an affine arrangement variety.
Sep 2, 2020 Thomas Tran (University of Kentucky)
Secondary terms in asymptotics for the number of zeros of quadratic forms
Let F be a non-degenerate quadratic form on an n-dimensional vector space over the rational numbers. One is interested in counting the number of integral zeros of the quadratic form whose coordinates are restricted in a smoothed box of size B. Heath-Brown gave an asymptotic of the form: c_1 B^{n-2} + O_{F,\epsilon}(B^{(n-1+\delta)/2+\epsilon}), for any \epsilon > 0 and n \geq 5, where c_1 is a complex number, and \delta=0 or 1, according as n is odd or even. For n = 3, 4, Heath-Brown also gave similar asymptotics. More recently, Getz gave an asymptotic of the form: c_1 B^{n-2} + c_2 B^{n/2}+O_{F,\epsilon}(B^{n/2+\epsilon-1}) when n is even, in which c_2 has a pleasant geometric interpretation. We consider the case where n is odd and give an analogous asymptotic of the form: c_1 B^{n-2} +c_2B^{(n-1)/2}+O_{F,\epsilon}(B^{n/2+\epsilon-1}). Notably it turns out that the geometric interpretation of the constant c_2 of the asymptotic in the odd degree and even degree cases is strikingly different.
Mar 25, 2020 Tefjol Pllaha (Aalto University)
Binary Subspace Chirps
One of the challenges of 5G wireless communications is to support massive Machine Type Communications (mMTC), where the network has a large number of connected devices but only small fractions are active at any given time interval. In this scenario, encoding and decoding complexity are of prime interest. In this talk we will introduce the Binary Subspace Chirps (BSSCs) as a set of complex Grassmannian lines in N = 2^m dimension, and use their algebraic and geometric features to construct a low complexity decoding algorithm.
Mar 11, 2020 Ian Le (Northwestern University)
Positive Tropicalization of Character Varieties
I will start by defining framed character varieties for a surface S and the group SL_n. Then I will introduce the notion of a cluster algebra and describe the cluster structure on the framed character variety. This allows us to define the positive tropicalization of the character variety. I will end by giving many examples of how to explicitly write down tropical points of the character variety using the affine Grassmannian.
Mar 4, 2020 Benjamin Jany (University of Kentucky)
Rank metric codes and their associated q-polymatroids
Rank metric codes are subspaces of the F_q vector space consisting of (m x n) matrices to which we associate the rank metric defined by d(M,N)=rk(M-N). The study of algebraic and combinatorial properties of codes has proven useful for linear network coding problems. One of the tools constructed to study those properties are q-polymatroids. We will go over the construction of q-polymatroids for rank metric codes and discuss the q-polymatroid associated to Maximum Rank Distance (MRD) codes.
Feb 26, 2020 Heide GLuesing-Luerssen (University of Kentucky)
On the Proportion of MRD codes and Semifields
Rank-metric codes are subspaces of a full matrix space over a finite field, where we endow the matrix space with the rank metric: d(A,B)=rk(A-B). The rank distance of such a code is defined as the minimum rank of all its nonzero elements. Codes with the maximum possible size for a given rank distance are called MRD codes (maximum rank-distance codes). In this talk I will discuss the proportion of MRD codes within the space of all rank-metric codes of the same dimension. More specifically, I will consider the asymptotic behavior of this proportion as the field size tends to infinity. I will report on the few answers that exist at this point. Special attention will be paid to the case of [3x3;3]-MRD codes, where a close relation to 3-dimensional semifields arises.
Feb 19, 2020 Yoav Len (Georgia Tech)
Brill-Noether Theory of Prym Varieties
The talk will revolve around combinatorial aspects Prym varieties, a class of Abelian varieties that occurs in the presence of double covers. Pryms have deep connections with torsion points of Jacobians, bi-tangent lines of curves, and spin structures. As I will explain, problems concerning Pryms may be reduced, via tropical geometry, to combinatorial games on graphs. As a consequence, we obtain new results concerning the geometry of special algebraic curves, and bounds on dimensions of certain Brill–Noether loci.
Feb 19, 2020 Ben Hollering (NC State)
Identifiability in Phylogenetics using Algebraic Matroids
Identifiability is a crucial property for a statistical model since it implies that distributions in the model uniquely determine the parameters that produce them. In phylogenetics, the identifiability of the tree parameter is of particular interest since it means that phylogenetic models can be used to infer evolutionary histories from data. Typical strategies for proving identifiability results require Gröbner basis computations which become untenable as the size of the model grows. In this talk I'll give some background on phylogenetic algebraic geometry and then discuss a new computational strategy for proving the identifiability of discrete parameters in algebraic statistical models that uses algebraic matroids naturally associated to the models. This algorithm allows us to avoid computing Gröbner bases and is also parallelizable.
Feb 12, 2020 Kaelin Cook-Powell (University of Kentucky)
Divisors on Curves
The study of algebraic curves has a rich and vibrant history in algebraic geometry. A divisor on a (relatively nice) curve is just a finite Z-linear combination of its points. Special families of divisors on a curve, known as Brill-Noether Loci, carry rich geometric information about that curve. This talk will start with a brief survey of the objects just mentioned and a summary of several classical results about them. Afterwards we will discuss recent developments in this area, including a generalization of the famous Brill-Noether Theorem. The main goal of this talk is to discuss a proof of this generalization that uses tools and techniques from tropical geometry, due to CP-Jensen. If time permits we will also discuss some work in progress and future directions. (Based on joint work with Dave Jensen)
Feb 5, 2020 Joseph Cummings (University of Kentucky)
Gorenstein Fan Compactifications of SL_2(C)-character varieties
We are interested in studying the character varieties, X(F_g, SL_2(C)), which are moduli spaces of representations \rho : F_g \to SL_2(C). In fact, these moduli are always affine. Chris Manon, Sean Lawton, and others have described compactifications using GIT quotients. In this talk, I will present a more canonical compactification, X_g, using Chow quotients. Along the way, we will describe a toric degeneration of X_g, and show that X_g and its toric degeneration are Gorenstein Fano.
Jan 22, 2020 Kalila Sawyer (University of Kentucky)
Scrollar Invariants of Tropical Curves
The scrollar invariants of a special divisor D on a k-gonal algebraic curve X are a family of invariants that provide insight into the behavior of D and the geometry of X. In the classical case, the ranks of multiples cD of D form a convex sequence completely determined by the scrollar invariants, which are extremely useful but hard to compute in this setting. We use degeneration techniques to investigate this question in the tropical setting, where combinatorial tools provide helpful insight. We examine the behavior of scrollar invariants under specialization, and compute these invariants for a much-studied family of tropical curves. Our examples highlight many parallels between the classical and tropical theories, but also point to some substantive distinctions.
Dec 11, 2019 Darleen Perez-Lavin (University of Kentucky)
Some additive combinatorial problem over finite fields
Let G be a finite abelian group, written additively. The Davenport constant D(G) is the smallest positive number s such that for any set {g1, g2, . . . , gs} of s elements in G, with repetition allowed, there exists a subset {gi1, gi2, . . . , git} such that gi1 + gi2 + · · · + git = 0. The plus-minus Davenport constant, D±(G), is defined similarly but instead we only require that gi1 ± gi2 ± · · · ± git = 0. In this talk, we provide current results for D±(G) for general finite abelian groups G. Then we present current results for groups such as C3r⊕C5s and C2q⊕C3r. In working on the plus-minus Davenport constant, we came upon an interesting problem with rectangular matrices with entries in finite fields. For a positive integer m, let hq(m) denote the largest integer n such that there exists an m×n matrix M with entries in Fq where every m × m submatrix W of M has rank m. We will present our current results on the constant hq(m).
Nov 20, 2019 Harry Richman (University of Michigan)
Weierstrass points on a tropical curve
The set of (higher) Weierstrass points on an algebraic curve of genus g > 1 is an analogue of the set of N-torsion points on an elliptic curve. As N grows, the torsion points "distribute evenly" over a complex elliptic curve. This makes it natural to ask how Weierstrass points distribute, as the degree of the corresponding divisor grows. We will explore how Weierstrass points behave on tropical curves (i.e. graphs with edge lengths), and explain how their distribution can be described in terms of electrical networks. Knowledge of tropical curves will not be assumed, but knowledge of how to compute resistances (e.g. in series and parallel) will be useful.
Nov 13, 2019 Hunter Lehmann (University of Kentucky)
Distance Distributions of Cyclic Orbit Codes
Subspace codes are collections of subspaces of the finite vector space F_q^n under the subspace metric. The distance distribution of such a code is the vector whose i-th entry counts the number of pairs of codewords with subspace distance i. Constant dimension cyclic orbit codes, which are contained in a Grassmannian G_q(n,k) and are the orbit of a subspace under an action of F_{q^n}^* on G_q(n,k), are of particular interest. We show that for optimal such codes, the distance distribution depends only on q, n, and k. For more general codes, we can relate the distance distribution to the number of orbits which contain intersections between different codewords.
Oct 30, 2019 Kaelin Cook-Powell (University of Kentucky)
Components of Brill-Noether Loci for Curves with Fixed Gonality
The geometry of an algebraic curve is intimately related to the theory of linear systems of special divisors defined on that curve, and the study of such divisors is known as Brill-Noether theory. A sequence of results in the 80's, the seminal being the Brill-Noether Theorem, concerns the geometry of Brill-Noether varieties for a curve C that is general in the moduli space M_g of smooth, projective curves of genus g. Recently, there has been a surge of interest when C is general in the Hurwitz space H_g,k parameterizing smooth, projective curves of genus g equipped with a prescribed degree k branched cover of the projective line. In this talk we will aim to present a generalization of the Brill-Noether Theorem to this more general setting that extends results of Pfleuger and Jensen-Ranganathan by extending their analysis of certain families of tableaux. This talk is based on joint work with Dave Jensen.
Oct 23, 2019 Michael Loper (University of Minnesota)
What Makes a Complex Virtual
Let S be the Cox ring of a smooth toric variety and B be the irrelevant ideal. In 2017, Berkesch, Erman, and Smith introduced virtual resolutions for toric varieties as an analogue of minimal free resolutions for projective varieties. Virtual resolutions are complexes of free S-modules that allow B-torsion homology. I will discuss virtual resolutions and name two algebraic conditions that determine whether a bounded chain complex of free S-modules is a virtual resolution. This theorem is similar to the depth criterion of exactness that Buchsbaum and Eisenbud published in 1973.
Oct 16, 2019 Adriana Solerno (Bates College)
Hasse-Witt matrices and mirror toric pencils
Mirror symmetry predicts unexpected relationships between arithmetic properties of distinct families of algebraic varieties. For example, Wan and others have shown that for some mirror pairs, the number of rational points over a finite field matches modulo the order of the field. In this talk, we obtain a similar result for certain mirror pairs of toric varieties. We use recent results by Huang, Lian, Yau and Yu describing the relationship between the Picard-Fuchs equations and the Hasse-Witt matrix of these varieties, which encapsulates information about the number of points. The result allows us to compute the number of points modulo the order of the field explicitly, and we illustrate this by computing K3 surface examples related to hypergeometric functions. This is joint work with Ursula Whitcher (AMS).
Oct 9, 2019 Kalina Mincheva (Yale University)
Prime tropical ideals
Tropical geometry provides a new set of purely combinatorial tools, which has been used to approach classical problems. In tropical geometry most algebraic computations are done on the classical side - using the algebra of the original variety. The theory developed so far has explored the geometric aspect of tropical varieties as opposed to the underlying (semiring) algebra and there are still many commutative algebra tools and notions without a tropical analogue. In the recent years, there has been a lot of effort dedicated to developing the necessary tools for commutative algebra using different frameworks, among which prime congruences, tropical ideals, tropical schemes. These approaches allows for the exploration of the properties of tropicalized spaces without tying them up to the original varieties and working with geometric structures inherently defined in characteristic one (that is, additively idempotent) semifields. In this talk we explore the relationship between tropical ideals and congruences to conclude that the variety of a non-zero prime tropical ideal is either empty or consists of a single point. This is joint work with D. Jo'o.
Oct 2, 2019 Angela Hanson (University of Kentucky)
Tropicalization of the Max Noether Theorem
The classical Max Noether Theorem determines the number of quadrics containing a canonical curve by examining the map from Sym^2 L(K_X) to L(2K_X). With tropical geometry, the algebraic geometry in this theorem can be translated into a problem in combinatorics. This tropicalization leads to a proof using metric graphs to prove the same result for algebraic curves with trivalent, three-edge-connected skeletons.
Sep 25, 2019 Hannah Larson (Stanford University)
A refined Brill-Noether theory over Hurwitz spaces
The Brill-Noether theorem describes the maps of general curves to projective space. Recently, the Brill-Noether theory of general k-gonal curves C has gathered much interest: Coppens-Martens exhibited components of the Brill-Noether loci W^r_d(C) with different dimensions; work of Pflueger and Jensen-Ranganathan determined the dimension of the largest component. In this talk, I will introduce a natural refinement of Brill-Noether loci for curves with a distinguished map C --> P^1, using the splitting type of push forwards of line bundles to P^1. In particular, studying this refinement determines the dimensions of all irreducible components of W^r_d(C) for general k-gonal C.
Sep 18, 2019 Courtney George (University of Kentucky)
Metric graphs and points in the tropical variety of the Coble quartic
For a given polynomial, we can think of the Groebner fan as the dual fan of the polynomial's Newton polytope. By utilizing techniques in tropical geometry and combinatorics, we are able to better understand the Groebner fan of a particular polynomial, the Coble quartic. After establishing a background in Groebner theory and algebraic geometry through definitions and examples, we will go through the procedure for using genus-3 directed graphs to determine points in the tropicalization of the Coble quartic.
Chip Firing and Algebraic Curves
The chip firing game is played with poker chips on the vertices of a graph. Though seemingly simple, this game has deep connections to various fields of mathematics. We will discuss one of these connections, to the theory of algebraic curves. We will see that the chip firing is a combinatorial analogue of Brill-Noether theory, the study of divisors on algebraic curves. This observation allows us to prove theorems in algebraic geometry using graph theory, and vice-versa.
Sep 4, 2019 Max Kutler (University of Kentucky)
The motivic zeta function of a matroid
We associate to any loop-free matroid a motivic zeta function. If the matroid is representable by a complex hyperplane arrangement, then this coincides with the motivic Igusa zeta function of the arrangement. We show that this zeta function is rational and satisfies a functional equation. Moreover, it specializes to the topological zeta function of Robin van der Veer. We answer two questions of van der Veer about this topological zeta function. This is joint work with Dave Jensen and Jeremy Usatine.
Aug 28, 2019 Jenny Kenkel (University of Kentucky)
Local cohomology of thickenings
Let R be a standard graded polynomial ring that is finitely generated over a field, and let I be a homogenous prime ideal of R. A recent paper of Bhatt, Blickle, Lyubeznik, Singh, and Zhang examined the local cohomology of the thickenings R/I^t in characteristic 0, and provided a stabilization result for these cohomology modules. I will explicitly construct isomorphisms between local cohomology modules of thickenings in several cases.
Jun 20, 2019 Jared Antrobus (University of Kentucky)
The State of Ferrers Diagram Rank-Metric Codes
In coding theory we wish to find as many codewords as possible, while simultaneously maintaining high distance between codewords to ease the detection and correction of errors. For linear codes, this translates to finding high-dimensional subspaces of a given metric space, where the induced distance between vectors stays above a specified minimum. Today we investigate the recent advances of this problem in the context Ferrers diagram rank-metric codes, a key ingredient in the construction of subspace codes. A well-known upper bound for dimension of these codes is conjectured to be sharp. In this talk we will discuss several solved cases of this conjecture, as well as the genericity of maximal codes.
Apr 24, 2019 Juliette Bruce (University of Wisconsin-Madison)
Asymptotic Syzygies for Products of Projective Space
I will discuss results describing the asymptotic syzygies of products of projective space, in the vein of the explicit methods of Ein, Erman, and LazarsfeldÂ's non-vanishing results on projective space.
Apr 17, 2019 Jeremy Usatine (Yale University)
Hyperplane Arrangements and Compactifying the Milnor Fiber
Milnor fibers are invariants that arise in the study of hypersurface singularities. A major open conjecture predicts that for hyperplane arrangements, the Betti numbers of the Milnor fiber depend only on the combinatorics of the arrangement. I will discuss how tropical geometry can be used to study related invariants, the virtual Hodge numbers of a hyperplane arrangement's Milnor fiber. This talk is based on joint work with Max Kutler.
Apr 3, 2019 Aida Maraj (University of Kentucky)
Quantitative Properties of Ideals arising from Hierarchical Models
We will discuss hierarchical models and certain toric ideals as a way of studying these objects in algebraic statistics. Some algebraic properties of these ideals will be described and a formula for the Krull dimension of the corresponding toric rings will be presented. One goal is to study the invariance properties of families of ideals arising from hierarchical models with varying parameters. We will present classes of examples where we have information about an equivariant Hilbert series. This is joint work with Uwe Nagel.
Mar 27, 2019 Kiumars Kaveh (University of Pittsburgh)
Tropical geometry and amoebas in matrix groups
We start with the basic and remarkable notions of amoeba and tropical variety of a subvariety Y in the algebraic torus (C \setminus {0})^n. We will demonstrate how these notions lead us to finding a minimal compactification of Y (usually referred to as "tropical compactification"). In the course of this we will introduce the notion of a toric variety as well. Next, I will discuss recent results about extending these notions from the algebraic torus to other matrix groups such as GL(n, C). Some interesting linear algebra, such as singular value decomposition and Smith normal form, pops up. For the most part, I assume only basic background from algebra and geometry and the talk should be understandable to a general math crowd. There will be a nonzero number of pictures!
Mar 27, 2019 Melody Chan (Brown University)
Tropical curves, graph complexes, and the cohomology of M_g
Joint with Søren Galatius and Sam Payne. The cohomology ring H^*(M_g,Q) of the moduli space of curves of genus g is not fully understood, even for g small. For example, in the 1980s, Harer-Zagier showed that the Euler characteristic (up to sign) grows super-exponentially with g --- yet most of this cohomology is not explicitly known. I will explain how we obtained new results on the rational cohomology of moduli spaces of curves of genus g, via Kontsevich's graph complexes and the moduli space of tropical curves.
Mar 26, 2019 Joseph Cummings (University of Kentucky)
Presenting T-varieties
A normal T-variety is a variety X with effective T \cong (C^*)^n-action. We define the complexity of X to be dim(X) - n. In the affine complexity-0 case, we get toric varieties which are completely determined by a rational cone. In a 2005 paper, Klaus Altmann and Jurgen Hausen showed that a normal affine T-variety is determined by the data of a base space Y and a polyhedral divisor. In 2017, Nathan Ilten and Chris Manon described a semi-canonical embedding of any affine rational complexity-1 T-variety. In this talk, we will show that when the base, Y \subseteq P^n, is projectively normal, and the polyhedral divisor, D, is good enough, the same construction works. The plan is to give generators for the ideal, describe the tropical variety of X, and work through some examples.
Mar 6, 2019 Robert Walker (University of Michigan)
Symbolic Generic Initial Systems and Matroid Configurations
We survey dissertation work of my academic sister Sarah Mayes-Tang (2013 PhD). As time allows, we aim towards two objectives. First, in terms of combinatorial algebraic geometry we weave a narrative from linear star configurations in projective spaces to matroid configurations therein, the latter being a recent development investigated by the quartet of Geramita -- Harbourne -- Migliore -- Nagel. Second, we pitch a prospectus for further work in follow-up to both Sarah's work and the matroid configuration investigation.
Feb 27, 2019 Jay White (University of Kentucky)
Betti Number Maximization and Bounds
Think about all ideals with your favorite Hilbert function. What are the largest possible Betti numbers for those ideals? Is there an ideal whose Betti numbers attain maximums? Now, think about your favorite Hilbert polynomial. Let's ask the same questions. These questions have been answered by Bigatti and Hulett (1993) and Caviglia and Murai (2010). We will talk about other, related, families of ideals that we could ask these questions about.
Feb 20, 2019 Oliver Pechenik (University of Michigan)
Degenerations of cohomology rings
An associative algebra is encoded by its structure constants, describing how to multiply elements in a distinguished basis and expand in that basis. Such algebras are rigid in the sense that you can't generally maintain associativity while modifying some of the structure constants. Motivated by analogues of Horn's problem on eigenvalues of sums of Hermitian matrices, Belkale and Kumar (2006) nonetheless obtained important new associative algebras from the cohomology of generalized flag varieties by setting various structure constants equal to zero. The existence of this degeneration was originally established via geometric invariant theory; another proof was supplied by Graham and Evens (2013) using relative Lie algebra cohomology. We give an elementary proof. This leads us to an additional degeneration, which we interpret geometrically. (Joint work with Dominic Searles.)
Feb 13, 2019 Alexandra Seceleanu (University of Nebraska-Lincoln)
Generalized minimum distance functions for schemes and linear codes
In coding theory, a linear code is a subspace of a finite dimensional vector space. To a linear code one can associate a set of points in projective space. If these points are distinct, then the Hamming distance of the code can be computed based on the geometry of the points.
Motivated by these considerations, we introduce commutative algebraic generalizations for the notion of Hamming distance, which are better suited for working with non-reduced schemes. We describe the properties of these generalized minimum distance functions, as well as bounding them in the spirit of the classical singleton bound and by means of a Cayley Bacharach-type conjecture.
This talk is based on joint work with Susan Cooper, Stefan Tohaneanu, Maria Vaz Pinto and Rafael Villarreal.
Jan 17, 2019 Julian Rosen (University of Maine)
Elements of an infinite product of finite fields coming from geometry
The ring A is defined to be the quotient of the product the prime finite fields, modulo those elements with only finitely many non-zero coordinates. In this talk, I will describe some arithmetically interesting elements of A coming from algebra and geometry, and I will explain how these elements are analogous to integrals of algebraic differential forms.
Jan 10, 2019 Gene Kopp (Bristol University)
From Hilbert's 12th problem to complex equiangular lines
We describe a connection between two superficially disparate open problems: Hilbert's 12th problem in number theory and Zauner's conjecture in quantum mechanics and design theory. Hilbert asked for a theory giving explicit generators of the abelian Galois extensions of a number field; such an "explicit class field theory" is known only for the rational numbers and imaginary quadratic fields. Zauner conjectured that a configuration of d^2 pairwise equiangular complex lines exists in d-dimensional Hilbert space (and additionally that it may be chosen to satisfy certain symmetry properties); such configurations are known only in a finite set of dimensions d.
We prove a conditional result toward Zauner's conjecture, refining insights of Appleby, Flammia, McConnell, and Yard gleaned from the numerical data on complex equiangular lines. We prove that, if there exists a set of real units in a certain abelian extension of a real quadratic field (depending on d) satisfying certain congruence conditions and algebraic properties, a set of d^2 equiangular lines may be constructed, when d is an odd prime congruent to 2 modulo 3. We give an explicit analytic formula that we expect to yield such a set of units. Our construction uses values of derivatives of zeta functions at s = 0 and is closely connected to the Stark conjectures over real quadratic fields. We will work through the example d = 5 in detail to help illustrate our results and conjectures.
Jan 9, 2019 Tefjol Pllaha (University of Kentucky)
Equivalence of Classical and Quantum Codes
In classical and quantum information theory there are different types of error-correcting codes being used. We study the equivalence of codes via a classification of their isometries. The isometries of various codes over Frobenius alphabets endowed with various weights typically have a rich and predictable structure. On the other hand, when the alphabet is not Frobenius the isometry group behaves unpredictably. We use character theory to develop a duality theory of partitions over Frobenius bimodules, which is then used to study the equivalence of codes. We also consider instances of codes over non-Frobenius alphabets and establish their isometry groups. Secondly, we focus on quantum stabilizer codes over local Frobenius rings. We estimate their minimum distance and conjecture that they do not underperform quantum stabilizer codes over fields. We introduce symplectic isometries. Isometry groups of binary quantum stabilizer codes are established and then applied to the LU-LC conjecture.
Dec 5, 2018 Kalila Sawyer (University of Kentucky)
Scrollar Invariants are *Always* Fun
In the quest to understand curves, we often look at their divisors, that is, how many ways we can map them into complex projective space. In particular, we like to study the spaces W^r_d(C) of such maps that have rank r and degree d. The scrollar invariants of a curve give us some notion of how the rank of each divisor changes as we repeatedly add it to itself, which in turn yields some insight into W^r_d(C). In this talk we'll introduce and motivate these ideas more carefully and give an overview of how we can use tropical tools to calculate scrollar invariants.
Sidon Spaces and Subspace Codes
Subspace codes are collections of subspaces of the finite vector space (F_q)^n under the subspace metric. Constant dimension cyclic orbit codes, which are contained in a Grassmannian G_q(n,k) and are the orbit of a subspace under an action of F_{q^n}^* on G_q(n,k), are of particular interest. We will see how to construct examples of these spaces with optimal parameters for some choices of q,n, and k using Sidon spaces based on the work of Roth, Raviv, & Tamo in 2018.
Nov 7, 2018 Kaelin Cook-Powell (University of Kentucky)
Combinatorial Brill-Noether Theory
For a given algebraic curve C the object W^r_d(C) can be thought of as a space of degree d maps from C to a projective space of dimension at least r. A landmark result, proven by Griffiths-Harris in 1980, was that for a general curve C of genus g that dim W^r_d(C) = g-(r+1)(g-d+r) [the Brill-Noether Number]. However, a general curve C of genus g comes equipped with a map of minimal degree k=[(g+3)/2] to 1 - dimensional projective space [known as the gonality of the curve] and little is known about curves with a specified gonality. In this talk we will discuss dim W^r_d(C) for general curves of genus g with a prescribed gonality using combinatorial objects such as k-uniform displacement tableaux which capture the same data as sets of divisor classes on a certain unexpected metric graph.
Oct 31, 2018 Christin Bibby (University of Michigan)
Combinatorics of orbit configuration spaces
From a group action on a space, define a variant of the configuration space by insisting that no two points inhabit the same orbit. When the action is almost free, this "orbit configuration space'' is the complement of an arrangement of subvarieties inside the cartesian product, and we use this structure to study its topology. We give an abstract combinatorial description of its poset of layers (connected components of intersections from the arrangement) which turns out to be of much independent interest as a generalization of partition and Dowling lattices. The close relationship to these classical posets is then exploited to give explicit cohomological calculations akin to those of (Totaro '96). Joint work with Nir Gadish.
Oct 24, 2018 Dan Corey (University of Wisconsin)
Initial degenerations of Grassmannians
Let Gr_0(d,n) denote the open subvariety of the Grassmannian Gr(d,n) consisting of d-1 dimensional subspaces of P^(n-1) meeting the toric boundary transversely. We prove that Gr_0(3,7) is schoen in the sense that all of its initial degenerations are smooth. We use this to show that the Chow quotient of Gr(3,7) by the maximal torus in GL(7) is the log canonical compactification of the moduli space of 7 lines in P^2 in linear general position. T his provides a positive answer to a conjecture of Hacking, Keel, and Tevelev from "Geometry of Chow quotients of Grassmannians."
Oct 11, 2018 Bill Robinson (Denison University)
Linear Structural Equation Models
Linear structural equation models (L-SEMs) are a class of multivariate statistical models which study possible causal dependencies among variables. These models are associated with a path diagram, a graph with a directed acyclic part and a bidirected part. When a model has been specified, it is of interest to determine whether the model parameters can be recovered from the covariance matrix which they define. In this talk we will introduce the topic of causal inference using L-SEMs and present recent progress on generic identifiability using algebraic methods.
Oct 10, 2018 Alberto Ravagnani (University College Dublin, Ireland)
Adversarial Network Coding
In the context of Network Coding, one or more sources of information attempt to transmit messages to multiple receivers through a network of intermediate nodes. In order to maximize the throughput, the nodes are allowed to recombine the received packets before forwarding them towards the sinks. In this talk, we present a mathematical model for adversarial network transmissions, studying the scenario where one or multiple (possibly coordinated) adversaries can maliciously corrupt some of the transmitted messages, according to certain restrictions. For example, the adversaries may be constrained to operate on a vulnerable region of the network. If noisy channels (traditionally studied in Information Theory) are described within a theory of "probability", adversarial channels are described within a theory of "possibility". Accordingly, in this talk we take a discrete combinatorial approach in defining and studying network adversaries and channels. We propose various notions of capacity region of an adversarial network, and illustrate a general technique that allows to port upper bounds for the capacities of point-to-point channels to the networking context. We then present some new upper bounds on the capacity regions of an adversarial network, and describe some new capacity-achieving communication schemes. The new results in this talk are joint work with Frank R. Kschischang (University of Toronto).
Oct 8, 2018 Linquan Ma (Purdue University)
Lech's inequality and a conjecture of Stuckrad-Vogel
Let (R, m) be a Noetherian local ring and let M be a finitely generated R-module of dimension d. We prove that the set {l(M/IM)/e(I,M)}, when I runs through all m-primary ideals, is bounded below by 1/d!e(R). Moreover, when the completion of M is equidimensional, this set is bounded above by a finite constant depending only on M. This extends a classical inequality of Lech and answers a question of Stuckrad-Vogel. Our main tool is to use Vasconcelos's homological degree. The talk is based on joint work with Patricia Klein, Pham Hung Quy, Ilya Smirnov, and Yongwei Yao.
Oct 3, 2018 Patricia Klein (University of Kentucky)
Characterizing Finite Length Local Cohomology in Terms of Bounds on Koszul Cohomology
We will define Koszul (co)homology and give some of its basic properties. We will then explain, in terms of work of Lech and Serre, why we might expect Koszul cohomology modules to be asymptotically small. Lastly, we will share the surprising result that this intuitive asymptotic smallness of Koszul cohomology obtains exactly when particular local cohomology modules are finite length.
Sep 26, 2018 Joseph Cummings (University of Kentucky)
Khovanskii bases and rational, complexity-1 T-varieties
Khovanskii bases (introduced by Kiumars Kaveh and Christopher Manon) are an important tool used to study K-domains relative to a valuation, and in fact, are a vast generalization of the notion of a SAGBI basis. We begin by introducing Khovanskii bases and provide several examples. We are particularly interested in the case when a pair (A, v) has a finite Khovanskii basis. Next, we discuss T-varieties, which are a natural generalization of toric varieties. We will see how affine T-varieties arise out of the quasi-combinatorial data of a polyhedral divisor via the construction of Jürgen Hausen and Klaus Altmann. Finally, we will see a result by Nathan Ilten and C. Manon: for any homogeneous valuation v on the coordinate ring of a rational, complexity-1 T-variety, there is an embedding whose coordinates are a Khovanskii basis for v.
Sep 21, 2018 Daniel Smolkin (University of Utah)
Symbolic powers via test ideals
An important problem in commutative algebra is studying the relationship between symbolic and ordinary ideals. One striking result in this direction was found by Ein-Lazarsfeld-Smith, who showed that for regular rings in characteristic 0, the dn-th symbolic power of any ideal is contained in the n-th ordinary power of that ideal, where d is the dimension of the ring. Their method proved to be quite powerful, and was adapted to the positive characteristic setting by Hara and the mixed characteristic setting by Ma and Schwede. However, all of this work was done in the regular setting. This is because the above method relies on the so-called subadditivity property of test ideals, which only holds for regular rings.
In this talk, we will discuss an approach to extending Ein-Lazarsfeld-Smith's result to the non-regular setting by using a new subadditivity formula for test ideals. Recent joint work with Carvajal-Rojas, Page, and Tucker shows that this approach works for a large class of rings, including all Segre products of polynomial rings. Time permitting, we will discuss how applying this approach to any toric variety reduces to solving a certain combinatorial problem.
Sep 21, 2018 Jennifer Kenkel (University of Utah)
Let R be a standard graded polynomial ring that is finitely generated over a field, and let I be a homogenous prime ideal of R. Bhatt, Blickle, Lyubeznik, Singh, and Zhang examined the local cohomology of R/I^t, as t goes to infinity, which led to the development of an asymptotic invariant by Dao and Montaño. I will discuss their results and give concrete examples of the calculation of this new invariant in the case of determinantal rings.
Sep 19, 2018 Tim Roemer (Universität Osnabrück, Germany)
Asymptotic Algebra and FI-modules
Symmetric ideals in increasingly larger polynomial rings that form an ascending chain arise in various contexts like algebraic statistics, commutative algebra, and representation theory. In this talk we discuss some recent results and open questions on the asymptotic behavior of algebraic/homological invariants of ideals in such chains. Our approach is based on FI-modules with varying coefficients and various related techniques. This talk is based on joint work with Dinh Van Le, Uwe Nagel, and Hop D. Nguyen.
Linear Systems on General Curves of Fixed Gonality
The geometry of an algebraic curve is governed by its linear systems. While many curves exhibit bizarre and pathological linear systems, the general curve does not. This is a consequence of the Brill-Noether theorem, which says that the space of linear systems of given degree and rank on a general curve has dimension equal to its expected dimension. In this talk, we will discuss a generalization of this theorem to general curves of fixed gonality. To prove this result, we use tropical and combinatorial methods. This is joint work with Dhruv Ranganathan, based on prior work of Nathan Pflueger.
Sep 5, 2018 Nathan Fieldsteel (University of Kentucky)
OI-algebras, strongly stable ideals, and cellular resolutions
A common occurrence, in commutative algebra and elsewhere, is a family of ideals I_n in k[x_1,...,x_n] in a family of polynomial rings in increasingly many variables, satisfying that f(I_n) is in I_{n+1} for any f in a certain family of ring homomorphisms. In the context where f is any order-preserving function of the indices of the variables, the theory of OI-algbras gives a categorical re-framing of this situation. In this framework one can study OI-ideals using resolutions by free OI-modules, in analogy with classical commutative algebra. After introducing and motivating the subject, we will exhibit a family of OI-ideals (coming from strongly-stable ideals) that have explicit free resolutions supported on OI-simplicial complexes. This talk is based on ongoing work with Uwe Nagel.
Aug 29, 2018 Max Kutler (University of Kentucky)
Motivic manifestations of matroids
A matroid is a combinatorial object which abstracts the notion of (linear or algebraic) independence. Recent work of Adiprasito-Huh-Katz demonstrates a surprising connection to algebraic geometry: every matroid possesses a ``cohomology ring'' which behaves like the Chow ring of a smooth projective variety. In this talk, I will define another ``algebro-geometric'' matroid invariant, the motivic zeta function of a matroid. I will assume no prior knowledge of matroids or motivic zeta functions. This is work in progress with Jeremy Usatine.
Apr 25, 2018 Robert Denomme (University of Kentucky)
Primality tests, elliptic curves and field theory
This talk would be of interest to anyone interested in algebraic varieties, field theory or number theory. We'll start by reviewing Pepin's primality test for Fermat numbers, which are numbers of the form F_n = 2^n+1. This test features a 'universal point' on an algebraic group (a multiplicative group) whose powers determine the primality of the given number. We will show how this is related to a field theory problem involving the splitting of rational primes for quadratic extensions. I'll also describe some directly analogous primality tests for other special types of numbers, which use elliptic curves. The field theory part of the problem is slightly elevated to the more general splitting of primes in abelian extensions. I will be working this summer on a project to develop a primality test that uses results about elliptic curves and non-abelian extensions, which would provide many more opportunities to develop tests for specific types of numbers.
Apr 18, 2018 Martina Juhnke-Kubitzke (Universität Osnabrück, Germany)
Balanced shellings on manifolds
A classical result by Pachner states that any two PL homeomorphic manifolds with boundary are related by a sequence of shellings and inverse shellings. We show that, for balanced manifolds, such a sequence can be chosen in such a way that in each step balancedness is preserved. This is joint work with Lorenzo Venturello.
Apr 11, 2018 Hunter Lehmann (University of Kentucky)
Subspace Polynomials and Cyclic Subspace Codes
Subspace codes are collections of subspaces of the finite vector space (F_q)^n under the subspace metric, where F is a finite field. Of particular interest due to their efficient encoding/decoding algorithms are constant-dimension cyclic codes where all codewords have the same dimension and where the code is invariant under an action of F_{q^n}^*. We will see how to represent subspace codes using particular polynomials and deduce properties of the codes from the structure of these polynomials based on the work of Ben-Sasson et. al. in 2016. Using these results, we will give a construction of a constant-dimension cyclic code with highest possible subspace distance containing multiple orbits under the F_{q^n}^* action.
Apr 4, 2018 Robert Krone (UC Davis)
FI-algebras
An FI-algebra encodes a family of algebras with symmetric group actions, and an ideal of an FI-algebra represents an infinite family of ideals with symmetry. I will give an overview of some results about when such ideals are finitely generated, and how to compute with them. Then I will explain how to compute Hilbert series of these ideals. Along the way we will see some surprising connections to combinatorics, such as well-partial orders and regular languages.
Mar 28, 2018 Anastassia Etropolski (Rice University)
Explicit rational point calculations for certain hyperelliptic curves
Given a curve of genus at least 2, it was proven in 1983 by Faltings that it has only finitely many rational points. Unfortunately, this result is ineffective, in that it gives no bound on the number of rational points. 40 years earlier, Chabauty proved the same result under the condition that the rank of the Jacobian of the curve is strictly smaller than the genus. While this is obviously a weaker result, the methods behind that proof could be made effective, and this was done by Coleman in 1985 using p-adic analysis. Coleman's work led to a procedure known as the Chabauty-Coleman method, which has shown to be extremely effective at determining the set of rational points exactly, particularly in the case of hyperelliptic curves. I n this talk I will discuss how we implement this method using Magma and Sage to provably determine the set of rational points on a large set of genus 3, rank 1 hyperelliptic curves, and how these calculations fit into the context of the state of the art conjectures in the field. The subject of this talk is joint work with Jennifer Balakrishnan, Francesca Bianchi, Victoria Cantoral-Farfan, and Mirela Ciperiani.
Feb 28, 2018 Brian Davis (University of Kentucky)
Rationality of the Poincare series for lattice simplices
We often study rings by resolving their defining ideal as a module over a polynomial ring. We may also study a ring by resolving the ground field as a module over the ring itself. Such resolutions do not generally have finite length, and so new interesting questions arise about the growth of Betti sequences. In particular, it is interesting to study the Poincare series, a generating function for the Betti numbers. In this talk we present a sketch of this phenomenon and a result about the Poincare series of rings associated to a particular family of lattice simplices. This is joint work with Ben Braun.
Feb 21, 2018 Kalila Sawyer (University of Kentucky)
The Maroni Invariant of Trigonal Chains of Loops
What is a Maroni invariant? What are trigonal chains of loops? Why should you care? We'll answer all these questions and more in our excursion into divisor theory on graphs, complete with an explicit computation of the Maroni invariant of a chain of loops.
Improvements to the Brill-Noether Theorem
In 1980 Griffiths and Harris proved what is known as the "Brill-Noether Theorem", which essentially says that for a general curve C of genus g the dimension of a variety of special linear series on C is precisely equal to the Brill-Noether number of that variety. However, it is also known that a general curve C of genus g must have a particular gonality k, so the next natural question to ask is, "Can we compute the dimension of a variety of special linear series on a general curve C of genus g with specified gonality k?" This week we will see a result from Nathan Pflueger that says we can, at least, bound the dimension above via a modification to the Brill-Noether number using recent results from Tropical Geometry.
Feb 7, 2018 Martin Ulirsch (University of Michigan)
Tropical geometry of the Hodge bundle
The Hodge bundle is a vector bundle over the moduli space of smooth curves (of genus g) whose fiber over a smooth curve is the space of abelian differentials on this curve. We may define a tropical analogue of its projectivization as the moduli space of pairs (\Gamma, D) consisting of a stable tropical curve \Gamma and an effective divisor D in the canonical linear system on \Gamma. This tropical Hodge bundle turns out to be of dimension 5g-5, while the classical projective Hodge bundle has dimension 4g-4. This means that not every pair (\Gamma, D) in the tropical Hodge bundle arises as the tropicalization of a suitable element in the algebraic Hodge bundle. In this talk I am going to outline a comprehensive (and completely combinatorial) solution to the realizability problem, which asks us to determine the locus of points in the tropical Hodge bundle that arise as tropicalizations. Our approach is based on recent work of Bainbridge-Chen-Gendron-Grushevsky-M\"oller on compactifcations of strata of abelian differentials. Along the way, I will also develop a moduli-theoretic framework to understand the specialization of divisors to tropical curves as a natural tropicalization map in the sense of Abramovich-Caporaso-Payne. This talk is based on joint work with Bo Lin as well as with Martin Moeller and Annette Werner.
Jan 31, 2018 Chris Manon (University of Kentucky)
The Combinatorics, Algebra, and Geometry of Conformal Blocks
For any choice of smooth, marked projective curve and some representation-theoretic data, the Wess-Zumino-Novikov-Witten (WZNW) model of conformal field theory produces a finite dimensional vector space called a space of conformal blocks. As the marked curve is varied, the conformal blocks form a vector bundle over the space of curves; this gives rise to the so-called "fusion rules" of the WZNW theory. I will explain how these rules allow us to count the dimensions of the space of conformal blocks using Ehrhart theory in a case associated to the Lie algebra sl_2. Hiding behind this surprising connection are deep algebraic properties of the total coordinate ring of a closely related moduli space associated to a curve: it's spaces of parabolic SL_2 principal bundles. An honorary appearance will be made by the chain of loops.
Jan 24, 2018 Dave Jensen (University of Kentucky)
The Kodaira dimension of the moduli space of curves
After introducing the moduli space of curves and some notions from birational geometry, we will describe recent progress on the Kodaira dimension of the moduli space of curves.
Dec 6, 2017 Tefjol Pllaha (University of Kentucky)
On quantum stabilizer codes derived from local Frobenius rings
Since their discovery in 1997, quantum stabilizer codes over fields have attracted a vast number of researchers. Recently, there has been a growing interest in stabilizer codes over different types of rings. In this talk I will discuss stabilizer codes over local Frobenius rings. We focus on isometries and minimum distance. We conjecture that stabilizer codes over local Frobenius rings do not underpeform stabilizer codes over fields, and present some interesting open problems. This is based on joint work with Heide Gluesing-Luerssen.
Nov 29, 2017 Aida Maraj (University of Kentucky)
Markov Bases of Hierarchical Models
We will start by discussing the significance of Markov Bases for investigating Hierarchical Models occurring in Algebraic Statistics. Markov Bases are often very large and hard to compute. This talk is going to introduce an alternative way of thinking about them using tools from Algebraic Geometry and present some of the latest computational results.
Nov 8, 2017 Uwe Nagel (University of Kentucky)
Sequences of Symmetric Ideals
Various problems arise in spaces of any dimension. As an attempt to investigate the problem in all dimensions simultaneously, one considers suitable ascending sequences of symmetric ideals in polynomial rings in more and more variables. We discuss recent results on such sequences and their limits. This is based on joint work with Tim Roemer.
Oct 25, 2017 James Maynard (Magdalen College, Oxford)
The strange consequences of Siegel zeros
If you believe the Generalised Riemann Hypothesis, then there are no zeros of L-functions with real part bigger than 1/2, but unfortunately we don't know how to show this. A `Siegel zero' is a putative strong counterexample to GRH, and if such exceptional zeros do exist, then there are many strange consequences for the distribution of prime numbers. However, prime numbers would also become very regular, and this allows us to prove things which go beyond even the consequences of GRH, if these exceptional zeros exist! I will survey some of these results, including recent joint work showing we can prove results towards the horizontal Sato-Tate conjecture for Kloosterman sums in this alternative world where Siegel zeros exist.
Oct 18, 2017 Martha Yip (University of Kentucky)
A minimaj-preserving crystal on ordered multiset partitions
One of the main objects of study in the Delta Conjecture is the polynomial Val_{n,k}(x;q,t). In this talk, we will give some background on the conjecture, and focus on two combinatorial aspects of the specialization of Val at q=0. We will give a proof that the polynomial is Schur-positive via the use of crystal bases, and we will show how the crystal structure leads to a bijective proof that the major index and the so-called minimaj statistic on multiset partitions are equidistributed.
Oct 9, 2017 Chris Manon (University of Kentucky)
The path semigroup of a graph with applications to moduli spaces of geometric structures
I'll introduce some open questions about an elementary object: a commutative, finitely generated semigroup formed by the supports of (unions of) paths in a finite graph. These semigroups are related to affine algebraic schemes called character varieties; also known as moduli of flat principal bundles and moduli of geometric structures. I'll explain how answers to my questions could say something about the symplectic and tropical geometry of these moduli spaces.
Oct 4, 2017 Nathan Fieldsteel (University of Kentucky)
Topological Complexity and Graphic Hyperplane Arrangements.
A motion planning algorithm for a topological space X is a (possibly not continuous) assignment of paths to ordered pairs of points in X. The topological complexity of X is an integer invariant which measures the extent to which any motion planning algorithm for X must be discontinuous. In practice it is difficult to compute, but it is bounded below by an integer invariant of the cohomology ring of X, called the zero-divisors-cup-length of X. When X is the complement of an arrangement of hyperplanes, this ring is the Orlik-Solomon algebra of A and the lower bound can be controlled by some combinatorial conditions. When X is the complement of a graphic arrangement, there is a connection between topological complexity and a classical result in graph theory, the Nash-Williams decomposition theorem.
Sep 27, 2017 Bill Trok (University of Kentucky)
Unexpected Curves and Hyperplane Arrangements
A recent paper by Cook, Harbourne, Migliore and Nagel, showed deep connections between objects known as unexpected curves in algebraic geometry and an open conjecture on Line Arrangements known as Terao's conjecture. After giving an exposition of this connection, I discuss some ongoing work in this area. We put restrictions on when unexpected curves may occur and discuss why we might expect these curves to have certain types of symmetry.
Sep 13, 2017 Fernando Shao (University of Kentucky)
Value sets of polynomials modulo primes
Let f be a polynomial with integer coefficients. For each prime p, we can reduce f modulo p and consider the (relative) size of the value set of f mod p. There is much to be desired concerning how small the relative sizes can be on average over p. In this talk I will discuss some results and some open problems.
Sep 6, 2017 Chris Manon (University of Kentucky)
An Introduction to Khovanskii Bases
Khovanskii bases are subsets of algebras which have nice computational properties. They are a generalization of so-called SAGBI bases or canonical bases in an algebra, and in this sense they are meant to do for algebras what Groebner bases do for ideals. I'll give an introduction to Khovanskii bases, describe a few theorems about their structure and their existence, and describe their relationship to other interesting topics in combinatorial algebraic geometry, like tropical geometry and toric geometry.
Aug 30, 2017 Nathan Fieldsteel (University of Kentucky)
Ideals of Geometrically Characterized Point Sets and Simplicial Complexes
Let X be a finite set of points in an affine space. A Lagrange polynomial for X is a polynomial which vanishes at all but one of the points of X. In a way which can be made precise, not every point set admits Lagrange polynomials, but the existence of Lagrange polynomials for X guarantees that a polynomial function is completely determined by its values on X. If in addition, these Lagrange polynomials fully factor into products of linear polynomials, the set X is called geometrically characterized. In this talk, we will discuss the geometry of geometrically characterized sets, the ideals of such point sets, and will make some connections to simplicial complexes.
Aug 25, 2017 Jared Antrobus (University of Kentucky)
Ferrers Diagram Rank-Metric Codes
Our codes of interest are subspaces of F_q^{m\times n} in which every nonzero matrix has rank at least \delta, and conforms to the shape of a given Ferrers diagram. In 2009, Etzion and Silberstein proved an upper bound for the dimension of such codes, and conjectured that it is achievable for any given parameters. In particular, the case for unrestricted matrices was solved in 1985 by Gabidulin, predating the complications brought on by nontrivial Ferrers diagram shapes. In this talk, we will prove the bound and discuss several known cases of the conjecture, including two new cases.
Jul 26, 2017 Aida Maraj (University of Kentucky)
Some Algebraic Properties of Hierarchical Models
We are going to introduce certain toric ideals that correspond to structures of Hierarchical Models in Algebraic Statistics and explore the properties of these ideals. One challenge will be to describe ideals that have infinitely many generators in a finite way, which we do using the symmetry group action on the set of indices. We will also show a formula for calculating the dimension of the variety defined by these ideals, and make progress in calculating the corresponding Hilbert function.
May 11, 2017 Jay White (University of Kentucky)
Upper Bounds for Betti numbers
The Hilbert function/polynomial and the Betti numbers each give fruitful information about an ideal, so relationships between the two are interesting. We will discuss the question of whether the Betti numbers have maximums for certain families of ideals.
Apr 27, 2017 Rachel Petrik (University of Kentucky)
The Chevalley-Warning Theorems
The Chevalley-Warning Theorems are a collection of results that give different lower bounds for the number of solutions to systems of equations over finite fields. In particular, for a system of equations over a finite field, F_q, if the number of variables (n) is strictly greater than the total degree of the system (d), then the number of solutions is greater than or equal to q^{n-d}. In 2011, D. R. Heath-Brown improved some of these results. In this talk, I will briefly state the Chevalley-Warning Theorems and then discuss Heath-Brown's results.
Apr 26, 2017 Ben Bakker (University of Georgia)
A global Torelli theorem for singular symplectic varieties
The local and global deformation theories of holomorphic symplectic manifolds enjoy many beautiful properties. By work of Namikawa, some of the local results generalize to singular symplectic varieties, but the moduli theory is badly behaved. In joint work with C. Lehn we show that for locally trivial deformations the entire picture is exactly analogous to the smooth case. In particular, we prove a global Torelli theorem and deduce some applications to birational contractions of moduli spaces of vector bundles on K3 surfaces. In place of twistor lines, the crucial global input is Verbitsky's work on ergodic complex structures using Ratner's theorems.
Apr 20, 2017 Isaiah Harney (University of Kentucky)
Colorings of Hamming-Distance Graphs
Error-correcting codes can be given a graphical interpretation via the Hamming-distance graphs, allowing for the application of graph theoretical approaches to what has traditionally been an algebra-based field. In a recent paper, El Rouayheb et al. used this graphical interpretation to produce new proofs for, and in some cases actually improve, known bounds on the size of error-correcting codes. In this talk, we study various standard graph properties of the Hamming-distance graphs with special emphasis placed on vertex colorings. In particular, we give a partial answer to an open problem of El Rouayheb et al. concerning the chromatic number of the Hamming-distance graphs. Furthermore, we define a notion of robustness for colorings which allows us to extend a classical result of Greenwell/Lovasz concerning the structure of minimal colorings of certain families of Hamming-distance graphs.
Apr 19, 2017 Kalila Sawyer (University of Kentucky)
Special divisors on marked chains of cycles
One area of current research in algebraic geometry is whether it is possible to classify all divisors on an algebraic curve. In general, this is a difficult question, but it turns out that the chain of loops has enough combinatorial structure that some progress can be made. In 2012, Cools, Draisma, Payne, and Robeva classified divisors on a generic chain of loops, and in 2016, Pflueger generalized their techniques to classify divisors on an arbitrary chain of loops. We will discuss Pflueger's techniques and results, focusing on this classification. No prior knowledge of algebraic geometry is assumed.
Apr 18, 2017 Luis Sordo Vieira (University of Kentucky)
Quadratic Forms Over p-adic Fields
Apr 17, 2017 Anupam Kumar (University of Kentucky)
Finite generation of symmetric ideals
The Hilbert Basis Theorem says that for a commutative Noetherian ring A and for a finite collection of variables X, every ideal of A[X] is finitely generated over A[X]. This certainly is not true if X is an infinite collection of variables. However, we can say that an ideal invariant under the action of the permutation group S_X is finitely generated over the group ring A[X][S_X]. In this talk, we will discuss a sketch of a proof of this result. It involves introducing a certain well-partial-ordering on monomials of X and developing a theory of Groebner bases and reduction in this setting.
Apr 5, 2017 Bill Trok (University of Kentucky)
Waring's Problem on Forms, Fat Points and the Macaulay Inverse Systems
Waring's Problem on forms is the question, given a homogeneous form F of degree d in n variables, what is the smallest integer K, so that there are linear forms L_1,...,L_K such that F = L_1^d + ... + L_K^d. For a given form F this integer is called the Waring rank of F. Ehrenborg and Rota showed that for generic forms this problem is equivalent to computing the Hilbert polynomial of generic double point ideals, a problem solved in 1995 by Alexander and Hirschowitz. In this talk, we discuss the connection between these problems and discuss results on forms whose Waring rank exceeds that of generic forms.
Mar 29, 2017 Alberto Ravagnani (University of Toronto)
An algebraic framework for end-to-end physical-layer network coding
In the framework of physical-layer network-coding (PLNC), multiple terminals attempt to exchange sets of messages through intermediate relay nodes. Recently, Feng, Silva and Kschischang developed an algebraic framework to study PLNC schemes, where messages can be represented as modules over a finite principal ideal ring. In this talk, in analogy with random linear network coding, we propose an algebraic framework for module transmission based on module length. We define a submodule code as a collection of submodules of a given ambient space module, and measure the distance between submodules via a function which we call the submodule distance. Both information loss and errors are captured by the submodule distance. Using the row-echelon form of a matrix over a principal ideal ring, we reduce the computation of the distance between submodules to the computation of the length of certain ideals in the base ring. We then present two bounds on the size of a submodule code of given minimum distance and whose codewords have fixed length. For certain classes of rings, we state our bounds explicitly in terms of the ring and code invariants. Finally, we construct classes of submodule codes with maximum error-correction capability. In particular, we construct asymptotically optimal codes over certain rings that are relevant from an applied viewpoint. Joint work with Elisa Gorla.
Mar 22, 2017 Tif Shen (Yale University)
Break divisors and compactified Jacobians
Let X be a genus g strictly semistable family of curves over C[[t]]. We show, using break divisors introduced by Mikhalkin and Zharkov, that all degree g Simpson compactified Jacobians of X are identical. As a consequence , we resolve a conjecture of Payne, and show that the unique degree g Simpson compactified Jacobian can be constructed from the break divisor decomposition introduced by An-Baker-Kuperberg-Shokrieh, using Mumford's non-Archimedean uniformization theory.
Mar 1, 2017 Juan Migliore (University of Notre Dame)
Lefschetz properties for ideals of powers of linear forms
If R/I is an artinian algebra over an infinite field K, and if L is a general linear form, then the Lefschetz question asks for which k does the multiplication map \times L^k : [R/I]_{j-k} \rightarrow [R/I]_j have maximal rank for all j? If this is true for k=1 then we say R/I has the Weak Lefschetz Property (WLP). If it is true for all k then we say that R/I has the Strong Lefschetz Property (SLP). But it can happen that this holds for some k but not others. At the end of a paper from 2010, MirÂ'o-Roig, Nagel and I gave some computer evidence suggesting that the question might be an interesting one for ideals generated by powers of general linear forms. This generated a number of papers by ourselves and others. Now the situation is largely understood for WLP and few variables, and largely wide open for more variables or higher k. IÂ'll give an overview of the current state of affairs and try to indicate some interesting open directions. IÂ'll also describe some of the methods that have been used in the study of this problem, especially a duality result of Emsalem and Iarrobino to translate the problem to a study of fat points in projective space.
Feb 22, 2017 Ashwin Deopurkar (Columbia University)
Theta characteristics and the Weil pairing on degenerate curves
Often degenerate curves play a crucial role in our understanding of a generic smooth curve. The classical theory of limit linear series developed by Griffiths and Harris studies deformation of a line bundle on a smooth curve to a degenerate curve of compact type. They used to this method to prove the Brill-Noether theorem. A totally different proof was given by Cools, Draisma, Payne, and Robeva using degenerate curves of non-compact type. The theory of limit linear series for such curves takes shape of metric graphs and divisors on them. After explaining this interplay of tropical geometry and algebraic geometry, I'd talk about specilizations of theta characteristics and the Weil pairing.
Feb 15, 2017 Martina Juhnke-Kubitzke (Universität Osnabrück, Germany)
Flawlessness of h-vectors of broken circuit complexes
One of the major open questions in matroid theory asks whether the h-vector (h_0,h_1,...,h_s) of the broken circuit complex of a matroid is weakly increasing in its first half and also satisfies that h_i is at most h_{s-i} whenever i is between zero and s/2. In this talk, we give an affirmative answer to this question for matroids that are representable over a field of characteristic zero. This is joint work with Din van Le.
Nov 30, 2016 Dave Jensen (University of Kentucky)
The geometry of an algebraic curve is governed by the maps it admits to various projective spaces. While many curves exhibit bizarre and pathological maps, the general curve does not. This is a consequence of the Brill-Noether theorem, which says that the space of maps of given degree and rank on a general curve has dimension equal to its expected dimension. In this talk, we will discuss a generalization of this theorem to general curves of fixed gonality. To prove this result, we use tropical and combinatorial methods. This is joint work with Dhruv Ranganathan.
Nov 16, 2016 Sam Payne (Yale University)
Top weight cohomology of moduli spaces of curves
The top weight cohomology of the moduli space of algebraic curves is naturally identified (with a degree shift) with the reduced rational homology of a moduli space of stable tropical curves. I will discuss the structure of this tropical moduli space and applications to computing new cohomology classes on M_g, based on recent joint work with M. Chan and S. Galatius.
Nov 9, 2016 Luis Sordo Vieira (University of Kentucky)
On the transfer map
Let G be a finite group and let H be a group of finite index. Then there is a nice way of defining a homomorphism from G to H/H'. This map is ubiquitous in group theory. A bit more surprisingly, it shows up in other areas of mathematics such as in Class Field Theory, Algebraic Topology and classical number theory. We will study the transfer map, and get some information on how the Sylow subgroups of a finite group G control the structure of G using transfer theory. We will also talk about a really neat application in number theory.
Nov 2, 2016 Jay White (University of Kentucky)
Maximum Betti Numbers for a Hilbert Function
The Hilbert function and the Betti numbers each give fruitful information about an ideal, so relationships between the two are interesting. We will answer the question of whether the Betti numbers have a maximum for a given Hilbert function. To do this, we will discuss strongly stable and lexicographic ideals.
Oct 19, 2016 Bill Trok (University of Kentucky)
Matroids and Regularity of Fat Points
Given a scheme of fat points Z in P^n, the regularity index can be defined as the point where the Hilbert function of the quotient ring, R/I(Z), is equal to the degree. We discuss new results which give upper bounds on this regularity index, based on the linear position of the points in space. A key technique is some results in the theory of Matroid Partitions.
Oct 19, 2016 David Zureick-Brown (Emory University)
Canonical rings of stacky curves
We give a generalization to stacks of the classical (1920's) theorem of Petri -- we give a presentation for the canonical ring of a stacky curve. This is motivated by the following application: we give an explicit presentation for the ring of modular forms for a Fuchsian group with cofinite area, which depends on the signature of the group. This is joint work with John Voight.
Oct 12, 2016 Anna-Lenna Horlemann-Trautmann (EPF Lausanne, Switzerland)
Network Coding and Schubert Varieties Over Finite Fields
Network coding deals with noisy transmission of data over a network, in our case from one sender to several receivers. It turns out that linear vector spaces over a finite field are a good tool for error correction in this setting. In this talk we give a quick introduction to network coding in general and show that many network coding theoretic problems translate into enumerative geometry problems in the Grassmann variety over the finite field. In particular, the Plücker embedding of the Grassmann variety, and Schubert varieties therein, are useful tools for code constructions and decoding algorithms. We show how to set up such an error correcting decoding algorithm and what its advantages and disadvantages are. Moreover, we briefly explain classical Schubert calculus over the complex numbers and show that the classical results do not hold (in general) over finite fields.
Oct 5, 2016 Tefjol Pllaha (University of Kentucky)
Equivalence of stabilizer codes
I will start this talk with a short discussion on equivalence of classical codes. Then, I will define stabilizer codes and equivalencies that naturally arise from quantum information theory. Using insights from the classical case, we introduce an equivalence notion for stabilizer codes. We show that this notion coincides with the natural notion of equivalence used widely in quantum information theory. No prior knowledge of coding theory and quantum computation is assumed.
Sep 28, 2016 McCabe Olsen (University of Kentucky)
Euler-Mahonian statistics and descent bases for semigroup algebras
We consider quotients of the unit cube semigroup algebra by particular $\mathbb{Z} \wr S_n$ -invariant ideals. Using Groebner basis methods, we can show that the resulting graded quotient algebra has a basis with each element indexed by colored permutation and each element encodes descent and major index statistics of the colored permutation. We can use this basis to recover combinatorial identities. This talk is based on joint work with Ben Braun.
Sep 14, 2016 Jared Antrobus (University of Kentucky)
Lexicodes Over Principal Ideal Rings
Let R be a finite principal left ideal ring. Place a lexicographic ordering on the vectors in the free left R-module R^n with respect to some basis B. The code produced by searching through this list, greedily collecting vectors satisfying some property P, is called a lexicode. For decades, several iterations of a greedy algorithm have appeared, to produce maximal lexicodes with nice properties. The most recent, in 2014, was generalized to work over finite chain rings. In this talk, I will present a new version of the greedy algorithm, generalized to produce lexicodes over a much larger class of rings.
Sep 7, 2016 Uwe Nagel (University of Kentucky)
Hilbert Series of Monomial Orbits
Hilbert functions or Hilbert series contain useful information about ideals. First, we will define and compute these invariants in the classical setting of a polynomial ring in finitely many variables if the ideal is a principal ideal. Then we discuss the analogous result in the spirit of commutative algebra up to symmetry. For this we consider the orbit of a monomial in a polynomial ring in infinitely many variables. The talk is based on joint work with Sema Gunturkun.
Aug 31, 2016 Tim Roemer (Universität Osnabrück, Germany)
Commutative Algebra up to Symmetry
Ideal theory over a polynomial ring in infinitely many variables is rather complicated which is (beside other things) due to the fact that this ring is not Noetherian. Since very recently one is interested in ideals in such a ring which are invariant under certain well-behaved monoid actions. We present some new results and open questions on algebraic properties of these ideals and associated objects of interest. The talk is based on joint work with Uwe Nagel.
Apr 27, 2016 Heide Gluesing-Luerssen (University of Kentucky)
Code-Based Cryptography and a Variant of the McEliece Cryptosystem
After a brief overview of public-key cryptography I will turn to a specific realization of a cryptosystem that relies on the hardness of decoding a random code. These cryptosystems were introduced by McEliece in 1978, but became popular only recently when it was discovered that RSA and elliptic-curve cryptography won't be secure in the presence of quantum computers. I will discuss the workings, advantages and drawbacks of the McEliece cryptosystem and also present a variant that aims at overcoming some of its drawbacks. No prior knowledge on public-key cryptography and coding theory is assumed.
Apr 20, 2016 Jonathan Constable (University of Kentucky); PhD defense
Kronecker's Theory of Binary Bilinear Forms with Applications to Representations of Integers as Sums of Three Squares
In 1883 Leopold Kronecker published a paper containing "a few explanatory remarks" to an earlier paper of his from 1866. His work loosely connected the theory of integral binary bilinear forms to the theory of integral binary quadratic forms. In this defense we shall discover the key statements within Kronecker's paper and offer insight into new, detailed arithmetic proofs. Further, I will present some additional results on the proper and complete class numbers for bilinear forms, before demonstrating their use in rigorously developing the connection between binary bilinear forms and binary quadratic forms. We conclude by giving an application of this material to the number of representations of an integer as a sum of three squares and show the resulting formula is equivalent to the well-known result due to Gauss.
Constructing the p-adic Integers
For a prime p, the p-adic integers Z_p is the valuation ring of the field of p-adic numbers Q_p. In this talk, we will explicitly construct Zp as a ring of coherent sequences and explore its algebraic and topological properties. We will then explore the multiplicative structure of Q_p using Hensel's Lemma.
Apr 18, 2016 Neville Fogarty (University of Kentucky); PhD Defense
On Skew-Constacyclic Codes
Cyclic codes are a well-known class of linear block codes with efficient decoding algorithms. In recent years they have been generalized to skew-constacyclic codes; such a generalization has previously been shown to be useful. After a brief introduction of skew-polynomial rings and their quotient modules, which we use to study skew-constacyclic codes algebraically, we motivate and define a notion of idempotent elements in these quotient modules. We are particularly concerned with the existence and uniqueness of idempotents that generate a given submodule; as such, we generalize relevant results from previous work on skew-constacyclic codes by Gao/Shen/Fu in 2013 and well-known results from the classical case.
Apr 6, 2016 Luis Sordo Vieira (University of Kentucky)
Elliptic Curves over finite fields and some of its applications
We will introduce elliptic curves and talk about (some) applications of elliptic curves, including factorizations of integers and elliptic curve protocols.
Mar 30, 2016 Bill Trok (University of Kentucky)
The Alexander Hirschowitz Theorem
A double point corresponds to the square of an ideal of a point. Geometrically we say a function vanishes at a double point, if its value and all derivatives vanish at the point. Given a collection of double points then, we can ask how many polynomials of a given degree vanish at all of them. This is equivalent to asking what the Hilbert function of the corresponding ideal is. In this talk we present a theorem of Alexander and Hirschowitz which classifies the Hilbert function of generic double points in projective space.
Mar 29, 2016 David Cook II (Eastern Illinois University)
The absence of the weak Lefschetz property
Mezetti, Miro-Roig and Ottaviani showed that in some cases the failure of the weak Lefschetz property can be used to produce a variety satisfying (nontrivial) Laplace equation. We define a graded algebra to have a Lefschetz defect of delta in degree d if the rank of the multiplication map of a general linear form between the degree d-1 and degree d components has rank delta less than the expected rank. Mezzetti and Miro-Roig recently explored the minimal and maximal number of generators of ideals that fail to have the weak Lefschetz property, i.e., to have a positive Lefschetz defect. In contrast to this, we will discuss constructions of ideals that have large Lefschetz defects and thus can be used to produce toric varieties satisfying many Laplace equations.
Mar 23, 2016 David Leep (University of Kentucky)
Levels and Pythagoras numbers of commutative rings
The level s(R) of a commutative ring R is the smallest integer n such that -1 is a sum of n squares of elements in R. Set s(R) = infinity if no such representation exists. The Pythagoras number p(R) is the smallest integer m such that every sum of squares of elements in R is already a sum of m squares in R. Set p(R) = infinity if no such bound exists. The study of levels and Pythagoras numbers of fields is a classical topic. Many results are known, but many open questions still remain. The study of levels and Pythagoras numbers of arbitrary commutative rings is more recent. I will survey known results and report on recent research with Detlev Hoffmann.
Feb 24, 2016 Tefjol Pllaha (University of Kentucky)
Quantum error-correcting codes II
This talk will be a continuation of the talk from last week. Definitions from quantum computing will be formalized and algebra will finally come into play to translate quantum questions to classical coding theory. Physics background is neither assumed nor required.
Quantum error-correcting codes
Quantum error correction is a necessity for eventual quantum computers and unfortunately much more difficult than the classical one. In this talk, we will explore these difficulties and how to fight them with good quantum codes. The focus will be on stabilizer formalism, as a compact description of almost every known quantum code. We will use this algebraic language to translate questions raising from quantum computation to classical error correction. Physics background is neither assumed nor required.
Feb 3, 2016 Uwe Nagel (University of Kentucky)
Dimensions of secant varieties
A variety is the set of solutions of a polynomial system of equations. Considering the union of all linear subspaces spanned by k points on a variety V, one obtains the k-th secant variety of V. Determining the dimension of a secant variety is an interesting and challenging problem. We illustrate this in two instances. The first one concerns the Waring rank. Any homogeneous polynomial f of degree d can be written as a sum of d-th powers of linear forms. The minimum number of summands in such a decomposition is the Waring rank of f. It admits a geometric interpretation using secant varieties. In the second instance we use linear algebra to solve the problem in some cases. The general problem (of decomposing tensors as sums of pure tensors) is open.
Jan 27, 2016 Uwe Nagel (University of Kentucky)
The Waldschmidt constant, II
Abstract: We discuss the Waldschmidt constant of ideals that are generated by products of two distinct variables. Each such ideal corresponds to a graph. It turns out that the Waldschmidt constant of the ideal is equal to the fractional chromatic number of the graph. This leads to the new bounds and computations of the Waldschmidt constant.
No prior knowledge of monomial ideals or graph theory is assumed. All concepts will be explained in the talk.
The Waldschmidt constant
Abstract: A (projective) variety V is a set of common zeros of the polynomials in an ideal I that is generated by homogenous polynomials. Given the generators of the ideal I, one would like to know the minimum degree of a polynomial F such that each point of V is a root of f of a given multiplicity, say k. As this is often a difficult problem, one studies first the corresponding question for large k. This leads to the Waldschmidt constant, which gives an asymptotic answer to the problem.
If I is an ideal that is generated by squarefree monomials, then the Waldschmidt constant can be expressed as the optimal solution to a linear program or as a fractional chromatic number. This leads to the new bounds and computations of the Waldschmidt constant.
Dec 9, 2015 Bill Trok (University of Kentucky)
Regularity of Fat Points
Abstract: Every point P in K^n, has an associated ideal I, which consists of the polynomials which vanish on it. Given P, we can define the fat point with weight m, as the ideal I^m. This ideal corresponds to polynomials which vanish on it, and whose m-1 partial derivatives vanish on it as well. The Hilbert function of intersections of these ideals is an interesting topic of study, as it has nice geometric meaning. I will discuss some results and conjectures about Hilbert functions of these ideals and there intersections, as well as discussing some work in progress in the area.
Dec 2, 2015 Luis Vieira (University of Kentucky)
ArtinÂ's Conjecture for Diagonal Forms
Abstract: One of ArtinÂ's famous conjectures states that a homogeneous polynomial of the type $a_1x_1^d+\cdots+a_sx_s^d$ over a $p$-adic field $K$ has a nontrivial zero in $K^s$ provided $s>d^2$. The conjecture is known to be true over$\mathbb{Q}_p$ and recently now by collaborative work with David Leep for all unramified extensions of $\mathbb{Q}_p$ with $p>2$. We will talk about the history of ArtinÂ's conjecture and explore some of the known results about ArtinÂ's conjecture on local fields and other fields of interest.
Nov 11, 2015 Tefjol Pllaha (University of Kentucky)
Isometries of Codes
Abstract: A code is endowed with the Hamming distance, which in turn determines its error-correcting capabilities. This suggests the study of isometries between codes. In 1961, MacWilliams with her famous Extension Theorem, classified Hamming isometries of linear codes over finite fields. This result was generalized for different weight functions, different alphabets and recently, for sublinear codes. We discuss the extension theorems for isometries of codes in their full generality and generalize some existing results.
Nov 4, 2015 Adam Chapman (Michigan State University)
The Word Problem for the Brauer Group
Abstract: By the renowned Merkurjev-Suslin Theorem, the Brauer group of a field is generated by symbol algebras. This gives rise to the word problem - given two ``words" (i.e. tensor products of symbol algebras), one should like to determine whether they represent the same element in the group or not. One way to approach this problem is to come up with a way of producing all the equivalent words to one given word. We shall discuss the special case of quaternion algebras and present some new results.
Oct 28, 2015 Jenn Park (University of Michigan)
A heuristic for boundedness of elliptic curves
Abstract: I will discuss a heuristic that predicts that the ranks of all but finitely many elliptic curves defined over Q are bounded above by 21. This is joint work with Bjorn Poonen, John Voight, and Melanie Matchett Wood.
Oct 22, 2015 Angie Cueto (Ohio State University)
Repairing tropical curves by means of tropical modifications
Abstract: Tropical geometry is a piecewise-linear shadow of algebraic geometry that preserves important geometric invariants. Often, we can derive classical statements from these (easier) combinatorial objects. One general difficulty in this approach is that tropicalization strongly depends on the embedding of the algebraic variety. Thus, the task of finding a suitable embedding or of repairing a given "bad" embedding to obtain a nicer tropicalization that better reflects the geometry of the input object becomes essential for many applications. In this talk, I will show how to use linear tropical modifications and Berkovich skeleta to achieve such goal in the curve case. Our motivating examples will be plane elliptic cubics and genus two hyperelliptic curves. Based on joint work with Hannah Markwig (arXiv:1409.7430) and ongoing work in progress with Hannah Markwig and Ralph Morrison.
Oct 21, 2015 Dhruv Ranganathan (Yale University)
Motivic Hilbert Zeta Functions of Curves
Abstract: The Grothendieck ring of varieties is a beautiful and intricate object which bears witness to an interplay between arithmetic, geometry, and topology. I will discuss in particular the behavior of Hilbert schemes of points on singular curves and the associated "motivic zeta function". After surveying work in this area, I will report on joint work in progress with Dori Bejleri and Ravi Vakil, in which we prove that the motivic Hilbert zeta function of an arbitrary curve is rational, in the spirit of the Weil conjectures.
Oct 14, 2015 Nathan Pflueger (Brown University)
Young tableaux and the geometry of algebraic curves
Abstract: A classical computation of Castelnuovo in enumerative geometry (made rigorous in the 1980s) shows that, for certain choices of numerical invariants, the number of linear series on a general curve of genus g is equal to the number of standard Young tableaux on a certain rectangular partition. Later proofs show that this equality becomes a bijection when the algebraic curve degenerates in a particular way. I will discuss joint work with Melody Chan, Alberto Lopez, and Montserrat Teixidor i Bigas, in which we prove that in the case where the variety of linear series is 1-dimensional rather than a finite set of points, then the holomorphic Euler characteristic of this variety can be computed by an analogous enumeration of tableaux. Time permitting, I will explain how similar methods translate other aspects of the geometry of algebraic curves to enumeration of tableaux.
Sep 30, 2015 Neville Fogarty (University of Kentucky)
A Circulant Approach to Skew-Constacyclic Codes
Abstract: We introduce a type of skew-generalized circulant matrices that captures the structure of a skew-polynomial ring F[x;theta] modulo the left ideal generated by a polynomial of the form x^n-a. This allows us to develop an approach to skewconstacyclic codes based on skew-generalized circulants. We show that for the code-relevant case, the transpose of a skew-generalized circulant is also a skew-generalized circulant. This recovers the well-known result that the dual of a skew-constacyclic code is also a skew-constacyclic code.
Sep 23, 2015 Anna Kazanova (University of Georgia)
Vector bundles on moduli space of stable curves with marked points
Abstract: Conformal block vector bundles are vector bundles on the moduli space of stable curves with marked points defined using certain Lie theoretic data. Over smooth curves, these vector bundles can be identified with generalized theta functions. In this talk we discuss extension of this identification over the stable curves. This is joint work with P. Belkale and A. Gibney.
Sep 16, 2015 Theodoros Kyriopoulos (University of Kentucky)
The Fundamental Theorem of Commutative Algebra (FTOCA)
Abstract: The Lecture is about the theorem of right invertibility of matrices over Commutative Rings (FTOCA) A proof of FTOCA in the context of linear systems and various applications of it (like the lemma of Nakayama and vanishing of the tensor product) are being presented in the lecture.
Sep 2, 2015 Eric Kaper (University of Kentucky)
Annihilators of Homogeneous Symmetric Polynomials Under the Action of Partial Differentiation
Abstract: Let E ans S be polynomial rings of degree n over a field k and let S act on E by partial differentiation. Given a homogeneous symmetric polynomial f, what can we say about Ann(f)? For a general polynomial the annihilator turns out to be as 'small' as possible. We will discuss what 'small' means in this context and exhibit a specific polynomial that satisfies these conditions (based on work of M. Boij, J. Migliore, R. Miro-Roig, U. Nagel).
Sep 2, 2015 Dave Jensen (University of Kentucky)
Matroids in Algebra and Geometry
Abstract: Matroids are combinatorial structures that generalize the notion of independence in linear algebra and graph theory. Rota conjectured that certain invariants of a matroid should always form a log concave sequence. We will report on recent work of Adripasato, Huh, and Katz, in which they use techniques from algebra and geometry to prove Rota's Conjecture.
Apr 29, 2015 Andrew Obus (University of Virginia)
The Oort conjecture and its generalizations
Abstract: A common mathematical problem is to be given a mathematical object in characteristic p, and to ask whether it is the reduction, in some sense, of an analogous structure in characteristic zero. If so, the structure in characteristic zero is called a "lift" of the structure in characteristic p.
We will consider a power series version of this problem, called the "local lifting problem." Given a characteristic p algebraically closed field k, and an action of a finite group G on k[[t]] by k-automorphisms, is there a DVR R in characteristic zero with residue field k such that the action of G lifts to R[[t]]? Oort conjectured that this is true when G is cyclic, and this conjecture was proven by the speaker, Stefan Wewers, and Florian Pop. We will discuss this conjecture and its generalizations. Examples will be given throughout.
We remark that the motivation for studying the local lifting problem comes from understanding lifts of branched covers of curves from characteristic p to characteristic zero.
Apr 23, 2015 Lars Christensen (Texas Tech University)
Tate homology over associative rings
Abstract: Tate homology and cohomology originated in the realm of group algebras and evolved through a series of generalizations to the setting of Iwanaga-Gorenstein rings. The cohomological theory has a more far-reaching generalization to the setting of associative rings; it is now called stable cohomology, and it agrees with Tate homology over Iwanaga-Gorenstein rings.
Apr 22, 2015 Bill Robinson (University of Kentucky) ; PhD defense
Tableau Ideals
Abstract: We study a class of determinantal ideals called skew tableau ideals, which are generated by (t x t) minors in a subset of a symmetric matrix of indeterminates. The initial ideals have been studied in the (2 x 2) case by Corso, Nagel, Petrovic and Yuen. Using liaison techniques, we have extended their results to include the original determinantal ideals in the (2 x 2) case, and obtained some partial results in the (t x t) case. A critical tool we use is an elementary biliaison, and producing these requires some technical determinantal calculations. We have uncovered in error a previous determinantal lemma that was applied in several papers, and have used the straightening law for minors of a matrix to establish a new determinantal relation. This new tool is quite versatile; it fixes the gaps in the previous papers and provides the main computational power in several of our own arguments. This is joint work with Uwe Nagel.
Apr 21, 2015 Nicholas Armenoff (University of Kentucky); PhD defense
Free Resolutions Associated to Representable Matroids
Abstract: As a matroid is naturally a simplicial complex, one can study its combinatorial properties via the associated Stanley-Reisner ideal and its corresponding free resolution. Using results by Johnsen and Verdure, we prove that a matroid is the dual to a perfect matroid design if and only if its corresponding Stanley-Reisner ideal has a pure free resolution, and, motivated by applications to their generalized Hamming weights, characterize free resolutions corresponding to the vector matroids of the parity check matrices of Reed-Solomon codes and certain BCH codes. Furthermore, using an inductive mapping cone argument, we construct a cellular resolution for the matroid duals to finite projective geometries and discuss consequences for finite affine geometries. Finally, we provide algorithms for computing such cellular resolutions explicitly.
Apr 17, 2015 Carolyn Troha (University of Kentucky); PhD defense
A Linkage Constructions for Subspace Codes
Abstract: In this thesis defense, we will begin by giving an overview of random network coding and how subspace codes are used in this context. In this talk I will focus on the linkage construction, which builds a code by linking previously constructed codes. We will explore the properties of codes created by this construction. In particular, we will explore how to utilize the linkage construction to create partial spread codes. Finally we will look at cases in which linkage codes are efficiently decodable.
Apr 15, 2015 Florian Kohl (University of Kentucky)
Ehrhart from Hilbert
Abstract: Many discrete problems in various mathematical areas arise from linear systems, thus they ask about integer points of polytopes in disguise. Ehrhart theory tries to develop tools to encode information about integer points of polytopes. One of the most important objects in Ehrhart theory is the so-called Ehrhart function. We will show that Ehrhart theory is closely related to commutative algebra. In particular, we will show how graded modules and the Hilbert function can be used to prove interesting results about the Ehrhart function of a lattice polytope.
Apr 8, 2015 Jeff Giansiracusa (Swansea University)
Semirings and schemes in tropical geometry
Abstract: The tropicalization of a variety is usually considered as a polyhedral set inside Euclidean space, but people often think of it heuristically as an algebraic set defined over the idempotent semiring of real numbers with the (max,+) structure. I'll explain how to give this heuristic picture teeth: it turns out that the tropicalization trop(X) is actually the solution set to a system of (max,+) polynomial equations canonically associated with X. This leads to tropical Hilbert polynomials and several other interesting things.
Apr 1, 2015 Ivan Soprunov (Cleveland State University)
On zero dimensional complete intersections in the torus
Abstract: Consider an n-variate system of n Laurent polynomials over an algebraically closed field K with prescribed Newton polytopes P_1, ..., P_n. If the coefficients of the system are generic, the solution set Z consists of isolated points in the torus (K^*)^n. ?We concentrate on the following two questions. Given a polytope P, let L(P) be the space of Laurent polynomials spanned by monomials corresponding to the lattice points in P. What is the dimension of the subspace of those h\in L(P) that vanish on Z? If h\in L(P) does not vanish identically on Z, what is the smallest number of points p in Z where h(p)\neq 0? These questions are related to the multigraded Hilbert function of ideals in the homogeneous coordinate ring of a toric variety and the Cayley-Bacharach theorem. Although we cannot answer these questions in full generality, we will see how much can be said in terms of geometry of the polytopes P_1, ..., P_n and P. Both questions have applications to algebraic coding theory.
Mar 25, 2015 Ralph Morrison (Berkeley)
Moduli of Tropical Plane Curves
Abstract: Smooth curves in the tropical plane come from unimodular triangulations of lattice polygons. The skeleton of such a curve is a metric graph whose genus is the number of lattice points in the interior of the polygon. In this talk we report on work concerning the following realizability problem: Characterize all metric graphs that admit a planar representation as a smooth tropical curve. For instance, about 29.5 percent of metric graphs of genus 3 have this property. (Joint work with Sarah Brodsky, Michael Joswig, and Bernd Sturmfels.)
Mar 4, 2015 Luis Sordo Vieira (University of Kentucky)
A brief introduction to quadratic spaces
Feb 25, 2015 Liam Solus (University of Kentucky)
Connecting Two Problems with a Spectrahedron
Abstract: Spectrahedra are natural generalizations of polyhedra that arise as parameterizations of slices of the cone of positive semidefinite matrices. To a graph G we can associate a spectrahedron known as an elliptope. We will use the elliptope to connect two well-studied problems via their underlying geometry. The first problem is the well-known max-cut problem from linear programming, the feasible region of which is the cut polytope of G. The second problem is the positive semidefinite matrix completion problem whose solution is characterized by extremal rays of the cone of concentration matrices for G. We will begin with a brief introduction to the geometry of positive semidefinite matrices and spectrahedra. Then we will define these two problems and their associated convex bodies. Finally, we will see that for some graphs the facets of the cut polytope correspond to extremal rays of the cone of concentration matrices, and that this correspondence is given by the elliptope and its dual. Moreover, we will see that the shape of the facet decides the rank of the corresponding extremal ray.
Feb 18, 2015 Carl Lee (University of Kentucky)
Stress and the Stanley-Reisner Ring, Part 2
Abstract: I will discuss some connections between classical stress on bar and joint frameworks, a generalization of stress to simplicial complexes, the Stanley-Reisner ring, and a consequent interpretation of the g-theorem for simplicial polytopes.
Feb 4, 2015 Alina Jacob (Georgia Southern University)
Gorenstein Projective Precovers
Abstract: We consider a right coherent and left n-perfect ring R. We prove that the class of Gorenstein projective complexes is special precovering in the category of unbounded complexes, Ch(R). As a corollary, we show that the class of Gorenstein projective modules is special precovering over such a ring. This is joint work with Sergio Estrada and Sinem Odabasi.
Dec 10, 2014 Carl Lee (University of Kentucky)
Stress and the Stanley-Reisner Ring
Dec 3, 2014 Luis Sordo Vieira (University of Kentucky)
On Artin's Conjecture for diagonal forms
Abstract: Emil Artin conjectured that any form of degree d in more than d^2 variables with coefficients in a p-adic field K has a nontrivial zero in K. Terjanian provided a counterexample to the conjecture, and many more have been found afterwards. However, in the case of diagonal forms, the result is known to hold for K=Q_p. The conjecture for diagonal forms over arbitrary p-adic fields remains unproved. We investigate partial results.
Nov 19, 2014 Sonja Petrovic (Illinois Institute of Technology)
Bouquet algebra of toric ideals
Abstract: To any integer matrix A one can associate a toric ideal I_A, whose sets of generators are basic objects in discrete linear optimization, statistics, and graph/hypergraph sampling algorithms. The basic algebraic problem is that of implicitization: given the matrix A, find a set of generators with some given property (minimal, Groebner, Graver, etc.). Then there is a related problem of complexity: how complicated can these generators be? In general, it is known that Graver bases are much more complicated than minimal generators. But there are some classical families of toric ideals where these sets actually agree, providing very nice results on complexity and sharp degree bounds.
This talk is about combinatorial signatures of generating sets of I_A. For the special case when A is a 0/1 matrix, bicolored hypergraphs give the answer. It turns out that such hypergraphs give an intuition for constructing basic building blocks for the general case too. Namely, we introduce the bouquet graph and bouquet ideal of the toric ideal I_A, whose structure determines the Graver basis. This, in turn, leads to a complete characterization of toric ideas for which the following sets are equal: the Graver basis, the universal Groebner basis, any reduced Groebner basis and any minimal generating set. This generalizes many of the classical examples.
Nov 12, 2014 Dustin Cartwright (University of Tennessee)
Crossing numbers for tropical curves
Abstract: In tropical geometry, curves have both an intrinic side, as metric graphs and an embedded representation in terms of so-called balanced polyhedral complexes. I will discuss the relationship between these two representations. Since most curves can't be embedded in the plane, it is often useful to relax the embedding condition by allowing transverse crossings. A tropical crossing number for a metric graph is defined to be the fewest number of crossings in a planar immersion, and I will give some results on this crossing number.
Nov 5, 2014 Yoav Len (Yale University)
Tropical Plane Quartics
Abstract: I will begin with a brief introduction to tropical geometry, and explain how algebraic curves give rise to tropical curves. I will then show that every tropical plane quartic admits 7 families of bitangent lines. This is analogous to the remarkable fact in classical geometry that a smooth plane quartic has exactly 28 bitangent lines. While the proof is purely combinatorial, I will discuss recent developments which suggest that the classical and tropical results are closely related. This is joint work with Matt Baker, Ralph Morrison, Nathan Pflueger, and Qingchun Ren.
Oct 29, 2014 Neville Fogarty (University of Kentucky)
Skew θ-Constacyclic Codes
Abstract: We generalize cyclic codes to skew θ-constacyclic codes using skew polynomial rings. We provide a useful tool for exploring these codes: the circulant. In addition to presenting some properties of the circulant, we use it to re-examine a theorem giving the dual code of a skew θ-constacyclic code first presented by Boucher/Ulmer (2011). This talk includes work with Dr. Heide Gluesing-Luerssen.
Oct 15, 2014 Bill Robinson (University of Kentucky)
Introduction to Linkage
Abstract: Liaison theory studies ideals by ``linking" them to nicer ideals that are well understood, and so gaining some interesting information about the original ideals. In this talk we will introduce some key ideas in liaison theory and apply them to the study of ideals generated by minors in a symmetric skew tableau. This will include some recent work with Uwe Nagel.
Oct 1, 2014 Luis Sordo Vieira (University of Kentucky)
A result by Davenport and Lewis on additive equations
Abstract: We will present a result by Davenport and Lewis which states that an additive form with coefficients in Q_p of degree d in s>d^2 variables has a non-trivial p-adic solution.
Chip-Firing and Tropical Independence II
Abstract: We continue discussing the basic theory of divisors on graphs, with a primary focus on concrete examples. If time permits, we will describe how these tools are used to provide new proofs of some well-known theorems in algebraic geometry.
Chip-Firing and Tropical Independence
Abstract: We will discuss the basic theory of divisors on graphs, with a primary focus on concrete examples. If time permits, we will describe how these tools are used to provide new proofs of some well-known theorems in algebraic geometry.
Classical and Tropical Brill-Noether Theory
Abstract: Classical Brill-Noether theory studies the existence and behavior of maps from a given algebraic curve to projective space. Recent years have witnessed the development of combinatorial techniques for studying such questions. After briefly surveying some of the major results in classical Brill-Noether theory, we will explain how these and related problems can be reduced to problems about combinatorial properties of graphs. This talk should be accessible to a broad audience - we will assume no familiarity with algebraic geometry.
Apr 23, 2014 Sema Gunturkun (University of Kentucky); PhD defense
Homogeneous Gorenstein Ideals and Boij Söderberg Decompositions
Abstract: This talk consists of two parts. Part one revolves around a construction for homogeneous Gorenstein ideals and properties of these ideals. Part two focuses on the behavior of the Boij-Söderberg decomposition of lex ideals. Gorenstein ideals are known for their nice duality properties. For codimension two and three, the structures of Gorenstein ideals have been established by Hilbert-Burch and Buchsbaum-Eisenbud, respectively. However, although some important results have been found about Gorenstein ideals of higher codimension, there is no structure theorem proven for higher codimension cases. Kustin and Miller showed how to construct a Gorenstein ideals in local Gorenstein rings starting from smaller such ideals. We discuss a modification of their construction in the case of graded rings. In a Noetherian ring, for a given two homogeneous Gorenstein ideals, we construct another homogeneous Gorenstein ideal and so we describe the resulting ideal in terms of the initial homogeneous Gorenstein ideals. Gorenstein liaison theory plays a central role in this construction. For the second part, we talk about Boij- Söderberg theory which is a very recent theory. It arose from two conjectures given by Boij and Söderberg and their proof by Eisenbud and Schreyer. It establishes a unique decomposition for Betti diagram of graded modules over polynomial rings. We focus on Betti diagrams of lex ideals which are the ideals having the largest Betti numbers among the ideals with the same Hilbert function. We describe Boij-Söderberg decomposition of a lex ideal in terms of Boij-Söderberg decomposition of some related lex ideals.
Apr 22, 2014 Casey Monday (University of Kentucky); PhD defense
A Characterization of Serre Classes of Reflexive Modules Over a Complete Local Noetherian Ring
Abstract: Serre classes of modules over a ring R are important because they describe relationships between certain classes of modules and sets of ideals of R. In this talk we characterize the Serre classes of three different types of modules. First we characterize all Serre classes of noetherian modules over a commutative noetherian ring. By relating noetherian modules to artinian modules via Matlis duality, we characterize the Serre classes of artinian modules. When R is complete local and noetherian, define E as the injective envelope of the residue field of R. Then denote M^\nu=Hom_R(M,E) as the dual of M. A module M is reflexive if the natural evaluation map from M to M^{\nu\nu} is an isomorphism. The main result provides a characterization of the Serre classes of reflexive modules over such a ring. This characterization depends on an ability to ``construct'' reflexive modules from noetherian modules and artinian modules. We find that Serre classes of reflexive modules over a complete local noetherian ring are in one-to-one correspondence with pairs of collections of prime ideals which are closed under specialization.
Apr 17, 2014 Furuzan Ozbek (University of Kentucky); PhD defense
Subfunctors of Extension Functors
Abstract: In this talk we examine subfunctors of Ext relative to covering (enveloping) classes and the theory of covering (enevloping) ideals. The notion of covers and envelopes by modules was introduced independently by Auslander-Smalo and Enochs and has proven to be beneficial for module theory as well as for representation theory. First we will focus on subfunctors of Ext and their properties. We show how the class of precoverings give us subfunctors of Ext. Later, we investigate the sunfunctor of Hom called ideals. The definition of cover and envelope carry over to the ideals naturally. Classical conditions for existence theorems for covers led to similar approaches in the ideal case. Even though some theorems such as Salce's Lemma were proven to extend to ideals, most of the theorems do not directly apply to the new case. We show how Eklof-Trlifaj's result can partially be extended to the ideals generated by a set. Moreover by relating the existence theorems for covering ideals of morphisms by identifying the morphisms with objects in A_2 we obtain a sufficient condition for the existence of covering ideals in a more general setting and finish with applying this result to the class of phantom morphisms.
Apr 17, 2014 Neville Fogarty (University of Kentucky)
Duals of Skew θ-Constacyclic Codes
Abstract: We generalize cyclic codes to skew θ-constacyclic codes using skew polynomial rings. We provide a useful tool for exploring these codes: the circulant. In addition to presenting some properties of the circulant, we use it to re-examine a theorem giving the dual code of a θ-constacyclic code first presented by Boucher/Ulmer (2011).
Apr 15, 2014 Ray Kremer (University of Kentucky); PhD defense
Homological Algebra with Filtered Modules
Abstract: Classical homological algebra begins with the study of projective and injective modules. In this talk I will discuss analogous notions of projectivity and injectivity in a category of filtered modules. In particular, projective and injective objects with respect to the restricted class of strict morphisms are defined and characterized. Additionally, an analogue to the injective envelope is discussed with examples showing how this differs from the usual notion of an injective envelope.
Mar 26, 2014 Alexandra Seceleanu (University of Nebraska)
Constructing ideals with large projective dimension
Abstract: Projective dimension is a homological measure of the complexity of algebraic objects. Motivated by computational considerations, M. Stillman asked for upper bounds on the projective dimension of homogeneous ideals in polynomial rings, based solely on invariants of the ideal, not of the ambient ring. In this talk, we discuss several constructions that shed some light on what ideal invariants can or cannot be used to bound projective dimension and we give lower bounds on any possible answer to Stillman's question. The talks is based on joint work with Huneke-Mantero-McCullough and Beder-McCullough-Nunez-Snapp-Stone.
Mar 13, 2014 Stephen Sturgeon (University of Kentucky); PhD defense
Polar Self-Dual Polytopes and the n-gon
Abstract: Given a monomial ideal we can sometimes interpret its free resolution as arising from a labeled cell complex. We seek to explore this connection in the case of some Gorenstein rings. The symmetry in Gorenstein rings can sometimes be modeled by polar self-dual polytopes. We will establish some basic theory concerning polar self-dual polytopes and discuss some motivating questions. In particular we construct the family of Ferrers polytopes and show that, given a proper labeling, they support a minimal free resolution of the Stanley-Reisner ring of the n-gon. Although applied in a particular case we will see that the techniques used are quite general and lead us to hope for results in greater generality.
Jan 22, 2014 Adam Boocher (University of California at Berkeley)
Closures of a Linear Space
Abstract: Let L be an affine linear space. Once we fix coordinates, it makes sense to discuss the closure of L inside a product of projective lines. In this talk I'll present joint work with Federico Ardila concerning the defining ideal of the closure. It turns out that the combinatorics of this ideal are completely determined by a matroid associated to L and we are able to explicitly compute its degree, universal Gr"obner basis, Betti numbers, and initial ideals. I'll include several examples along the way and discuss how this closure operation comes up naturally when one searches for ideals with "nice" behavior upon degeneration.
Jan 15, 2014 Mohanad Farhan Hamid Al Saidi (University of Kentucky, visiting from University of Mustansiriyah, Bagdad)
Classes of modules relative to torsion theories
Abstract: Our purpose is extend known results of some classes of modules to torsion theoretic setting in a way so that the former results are recovered when some torsion theory is chosen. For example we generalize the concept of relative pure injectivity to relative pure \tau-injectivity, where \tau is a given hereditary torsion theory. If \tau is the improper torsion theory then relative pure injectivity and relative pure \tau-injectivity are equivalent. This new concept retains some of the important properties of pure injectives. For instance, we show that the class of pure \tau-injective modules is enveloping.
Nov 20, 2013 Liam Solus (University of Kentucky)
The Hilbert Series of Algebras of the Veronese Type
Abstract: Algebras of the Veronese type are semigroup algebras that have attracted considerable attention from the algebra and algebraic combinatorics communities. In 1996, Sturmfels described Groebner bases for presentations of these algebras, and in 1997 De Negri and Hibi classified those that are Gorenstein. In his 2005 paper "The Hilbert Series of Algebras of the Veronese Type," Mordechai Katzman added to these results by providing an explicit formula for the Hilbert series of these algebras. In this talk, we will describe Katzman's formula and its connections to the combinatorics of a family of polytopes known as hypersimplices.
Nov 20, 2013 Luis Sordo Viera (University of Kentucky)
Artin's Conjecture on homogeneous forms over Q_p
Abstract: A field k is called a C_i field if any homogeneous form of degree i in more than d^i variables has a nontrivial zero in k. It is well known that finite fields are C_1. What about the p-adics? It was conjectured by Emil Artin that Q_p is C_2. The result turned out to be false. We will investigate some positive results.
Nov 13, 2013 Katherine Morrison (University of Northern Colorado)
Enumerating Equivalence Classes of Rank-Metric and Matrix Codes
Abstract: Due to their applications in network coding, public-key cryptography, and space-time coding, both rank-metric codes and matrix codes, also known as array codes and space-time codes over finite fields, have garnered significant attention. We focus on characterizing rank-metric and matrix codes that are both efficient, i.e. have high dimension, and effective at error correction, i.e. have high minimum distance. A number of researchers have contributed to the foundation of duality theory for rank-metric and matrix codes, which has demonstrated that the inherent trade-off between dimension and minimum distance for a code is reversed for its dual code; specifically, if a code has high dimension and low minimum distance, then its dual code will have low dimension and high minimum distance. Thus, with an aim towards finding codes with a perfectly balanced trade-off, we study self-dual matrix codes. In particular, we enumerate the equivalence classes of self-dual matrix codes of short lengths over small finite fields. To perform this classification, we also examine the notion of equivalence for rank-metric and matrix codes and use this to characterize the automorphism groups of these codes.
Non-negative polynomials and Gorenstein ideals
Abstract: A homogenous polynomial of degree d in n variables is called non-negative if it is at least zero when evaluated at any point with real coordinates. The cone of such non-negative polynomials contains the cone of the homogeneous polynomials that are sums of squares. Hilbert characterized the pairs (n,d) such that the two cones are the same. Recently, Blekherman strengthened Hilbert's results by describing the extremal rays of the cone that is dual to the cone of non-negative polynomials. These rays correspond to certain extremal Gorenstein ideals. We will discuss these results.
Oct 30, 2013 Mohanad Farhan Hamid Al Saidi (University of Kentucky, visiting from University of Mustansiriyah, Bagdad)
(Flat) modules that are fully invariant in their pure-injective (cotorsion) envelopes
Abstract: We introduce two concepts. A module M is said to be purely quasi-injective (resp. quasi-cotorsion) if it is fully invariant in its pure-injective envelope (resp. if it is flat and fully invariant in its cotorsion envelope). Endomorphism rings of both of the above types of modules are proved to be regular and self injective modulo their Jacobson radicals. If M is a purely quasiinjective (resp. quasi-cotorsion) module, then so is any finite direct sum of copies of M. Each of the above concepts is stronger than the well-known concept of quasi-pure-injectivity, but not equivalent. This solves, negatively, a problem of Mao and Ding's of whether every flat quasi pure-injective module is fully invariant in its cotorsion envelope. Certain types of rings are characterized in terms of purely quasi-injective modules. For example, a ring R is regular if and only if every purely quasiinjective R-module is quasi-injective, and is pure-semisimple if and only if every R-module is purely quasi-injective.
Oct 16/23, 2013 Heide Gluesing-Luerssen (University of Kentucky)
Codes over Frobenius rings and an extension theorem
Abstract: An important result in algebraic coding theory tells us that every Hamming weight-preserving isomorphism between subspaces in F^n, where F is a finite field, extends to a Hamming weight-preserving isomorphism on the entire F^n. This has led to a variety of generalizations, namely to submodules over certain rings and/or different weight functions. In this talk, I will discuss the extension theorem for Frobenius rings and poset weights. Both notions, Frobenius rings and poset weights, will be introduced.
Oct 9, 2013 Akihiro Higashitani (Osaka University)
Integer decomposition property of dilated polytopes
Abstract: We say that P has the integer decomposition property (IDP, for short) if any integer point in mP can be written as the sum of m integer points in P, where m is an arbitrary positive integer. In this talk, we discuss the problem when an integral convex polytope without IDP has IDP by integral dilation.
Oct 2, 2013 Augustine O'Keefe (University of Kentucky)
An algebraic study of Cameron-Walker graphs
Abstract: Given a finite simple graph G, two commonly studied invariants in graph theory are the matching number, m(G), and the induced matching number of a graph, i(G). These combinatorial invariants provide upper and lower bounds, respectively, for the (Castelnuovo-Mumford) regularity of the quotient of the edge ideal associated to the graph, R/I(G). Cameron and Walker characterize all graphs where the matching number is the same as the induced matching number and therefore the regularity can be explicitly calculated. In this talk we will examine other algebraic and combinatorial properties of R/I(G) where G satisfies m(G)=i(G), such as Cohen-Macaulayness, shellability, and vertex decomposability.
Sep 25, 2013 Furuzan Ozbek (University of Kentucky)
A sufficient condition for covering ideals
Abstract: The concepts of envelope and cover were introduced independently by Enochs and Auslander-Smalo for classes of modules. Since then the definition has been applied to different classes of categories. One of the recent application was introduced by Asensio, Herzog, Fu and Torrecillas where the theory of covers and envelopes is extended to ideals. In this talk, we will show how identifying an ideal I with a certain class of objects in the quiver A_2 can help us to obtain sufficient conditions for I to be a covering ideal.
Sep 25, 2013 Bill Robinson (University of Kentucky)
On a Class of Determinantal Ideals
Abstract: We will discuss a class of ideals determined by taking minors in a subregion of a matrix of indeterminates, called a skew tableau, and also in a reflected version of this considered as a subregion of a symmetric matrix. We will use liaison-theoretic tools to investigate properties of these ideals, and study their liaison classification.
Sep 18, 2013 Sema Gunturkun (University of Kentucky)
A Construction of Homogeneous Gorenstein Ideals
Abstract: A way of constructing Gorenstein ideals from small Gorenstein ideals in local Gorenstein rings is shown by A. Kustin and M. Miller in 1983. In this talk, we show a variant of their construction for graded case and avoid the ring extension. We see that how the liaison theory helps us immensely to modify their construction.
Sep 18, 2013 Carolyn Troha (University of Kentucky)
Irreducible Cyclic Orbit Codes
Abstract: After subspace codes were introduced in 2008 by Koetter and Kschischang most constructions involved the lifting of matrix codes. However, Rosenthal et al. introduced in 2011 a new method of constructing constant dimension subspace codes by using a group action of GL_n(F_q) on the projective geometry PG(q,n), called orbit codes. A specific subset of these codes, which have been studied more in depth, are irreducible cyclic orbit codes. In this talk, I will introduce the construction of an irreducible cyclic orbit code as well as explore a method to find the cardinality and distance of such a code.
Sep 4, 2013 Stephen Sturgeon (University of Kentucky)
Cellular Resolutions of the Cyclic Polytopes
Abstract: Cellular resolutions have been an area of active interest in the study of monomial ideals in the past few years. A cellular resolution is a way of encoding the information of the free resolution of an ideal in a cell complex. We will study the Stanley-Reisner rings arising from the simplicial polytopes known as the cyclic polytopes. These polytopes have the interesting property that they maximize all entries in the f-vector of the set of polytopes with fixed dimension and vertices. I will explain a little of the background theory and then cover the main idea of the construction.
Apr 10, 2013 Jonathan Constable (University of Kentucky)
The Class Number and Binary Quadratic Forms
Abstract: Let F be a positive definite binary quadratic form. One may classify such forms with fixed discriminant Δ, up to equivalence in GL2(Z) or refine this further to proper equivalence in SL2(Z). These are classical results developed by Lagrange and Gauss and lead to well-known statements about the class number h(Δ). In his paper "Über Bilineare Formen Mit Vier Variabeln", Kronecker introduces the finer notion of complete equivalence, which is used to study the class number of positive definite forms with integer coefficients of the type ax^2+2bxy+cy^2. In this talk we will discuss KroneckerÂ's development of the class number via complete equivalence and compare it with the classical results of Lagrange and Gauss.
Mar 27, 2013 Jing Xi (University of Kentucky)
Sequential importance sampling
Abstract: Sequential importance sampling (SIS) is a procedure which can be used to sample contingency tables with constraints. It proceeds by simply sampling cell entries of the contingency table sequentially and terminate at the last cell such that the final distribution is approximately uniform. I will first introduce this procedure in both statistical and polyhedral geometry view, and explain its advantages and problems. Then I will introduce the SIS procedure via conditional Poisson (CP) distribution which is used to sample zero-one contingency tables with fixed marginal sums. In this case, the procedure proceeds by sampling one column after another sequentially and terminate at the last column. I will explain both two-way and multi-way cases, and also why it performs better than the general SIS procedure when we have zero-one constraints.
Mar 22, 2013 Anton Dochtermann (University of Miami)
Laplacian ideals, arrangements, and resolutions
Abstract: The classical Laplacian matrix of a graph G describes the combinatorial dynamics of the Abelian Sandpile Model and the more general Riemann-Roch theory of G. The lattice ideal associated to the Laplacian provides a more recently considered algebraic perspective on this (re)emerging field. The Laplacian ideal has a distinguished monomial initial ideal that has also been studied in connection with G-parking functions. We study homological properties and show that a minimal free resolution of the initial ideal is supported on the bounded subcomplex of a section of the graphical arrangement of G. As a corollary we obtain a combinatorial characterization of the Betti numbers in terms of acyclic orientations. This generalizes constructions from Postnikov and Shaprio (for the case of the complete graph) and connects to work of Manjunath and Sturmfels, and Perkinson on the commutative algebra of Sandpiles. This is joint work with Raman Sanyal.
Mar 20, 2013 Jaimal Thind (University of Toronto)
Quantum McKay Correspondence and Equivariant Sheaves on the Quantum Projective Line
Abstract: The McKay correspondence gives a bijection between finite subgroups of SU(2) and affine A,D,E Dynkin diagrams. There is a quantum version of this statement (due to Kirillov Jr and Ostrik) which relates "finite subgroups" of quantum sl(2) and finite A,D,E Dynkin diagrams. We use this correspondence to construct the category of "equivariant" coherent sheaves on the quantum projective line. This is done by defining analogues of the symmetric algebra and the structure sheaf, and using them to define a triangulated category which is a natural analogue of the derived category of equivariant sheaves on the projective line. We then produce natural objects in this triangulated category, and relate our category to the derived category of representations of the corresponding A,D,E quiver. This can be thought of as a quantum analogue of the projective McKay correspondence of Kirillov Jr. We will first review the classical constructions, then discuss the "quantum" analogues.
Feb 27 & Mar 6, 2013 Uwe Nagel (University of Kentucky)
The Macaulay-Matlis duality
Abstract: The Macaulay-Matlis duality is the basis for many duality results in algebra and geometry. It takes a very specific form for ideals in a polynomial ring by interpreting polynomials as differential operators. It will be discussed how it can be used to produce irreducible decompositions and to derive certain symmetry properties of irreducible ideals aka Gorenstein ideals. All concepts will be explained in the talk.
Feb 20, 2013 Stephen Sturgeon (University of Kentucky)
Cellular Resolutions of the n-gon
Abstract: A cellular resolution is a way of representing the free resolution of a monomial ideal by associating it with a cell complex. In this talk we will discuss a cellular resolution of the Stanley-Reisner ring of the n-gon. This is an interesting example because this ring is Gorenstein (symmetric betti table) and the cellular resolution is a self-dual polytope (symmetric f-vector). The talk will focus on the primary ideas used in the construction of the polytope.
Nov 28, 2012 Bill Robinson (University of Kentucky)
Liaison Theory and Determinantal Ideals
Abstract: The central objects of study in classical algebraic geometry are varieties. Liaison theory is a classification theory of varieties and their defining ideals. This talk will introduce some ideas and results from liaison theory, with a focus on the algebraic side of the story. We will also present the proof of a theorem of Elisa Gorla about the liaison class of ladder determinantal varieties.
Nov 7/14, 2012 Heide Gluesing-Luerssen (University of Kentucky)
About Various MacWilliams Identities for Codes over Finite Commutative Rings
Abstract: MacWilliams identities play a prominent role in algebraic coding theory because they tell how certain information about a code, encoded in enumerators, can be used to deduce information about the dual code. We will provide a unified approach to MacWilliams identities for various weight enumerators of linear block codes over Frobenius rings. Such enumerators count the number of codewords having a pre-specified property, and MacWilliams identities yield a transformation between such an enumerator and the corresponding enumerator of the dual code. All identities can be derived from a MacWilliams identity for the full weight enumerator using the concept of an F-partition, as introduced by Zinoviev and Ericson (1996).
Oct 31, 2012 Nicholas Armenoff (University of Kentucky)
Linear Codes and Commutative Algebra
Abstract: One goal of algebraic coding theory is to find linear error-correcting codes with maximum error-correcting capacity. Correlated to the error-correcting capacity of a code C are the generalized Hamming weights of C. Thus, using Hochster's formula and techniques of commutative algebra, we will derive the generalized Hamming weights for one class of BCH codes and partially extend these results to special cases of another class of BCH codes.
Oct 10/24, 2012 Alberto Corso (University of Kentucky)
Integrality of quasi socle ideals
Abstract: Given an ideal I of a local ring (R,m) we are interested in determining when the socle I:m, and in more generality the quasi socle ideals I:m^s where s is a positive integer, is integral over the ideal I. In the first part of the talk we will review what integrality of ideals means and what is known about the problem in the special cases when I is a complete intersection, height two perfect, Gorenstein ideal. The focus will be in looking for an explicit way to determine the generators of these quasi-socle ideals in terms of a presentation matrix of the ideal and in a characteristic free fashion. Time permitting, I will describe some of the new result obtained in an ongoing work joint with Shiro Goto, Craig Huneke, Claudia Polini and Bernd Ulrich.
Sep 26 and Oct 3, 2012 Augustine O'Keefe (University of Kentucky)
Toric models in Algebraic Statistics
Abstract: Algebraic Statistics is a relatively new field in mathematics that aims to use algebraic geometry, commutative algebra and combinatorics to solve statistical problems. This talk will focus on the use of toric ideals - which can be thought of as being generated by homogeneous binomials - in the study of statistical models. No expertise in statistics is required as we will define what we need as we go. There will be many examples, mostly involving conditional independence, graphical and ranking models.
Apr 25, 2012 Augustine O'Keefe (Tulane University)
Characterizing graphs admitting Cohen-Macaulay toric rings
Abstract: Given a finite discrete graph G=(V,E) one can construct a toric ring, denoted K[G], via a monomial map from a polynomial ring over the edge set to a polynomial ring over the vertex set. Hibi and Ohsugi (1999) showed that generators of the defining ideal of this ring, I_G, arise from binomials associated to even closed walks in the graph G. Because of this association we are hopeful that we can also characterize invariants of K[G] via the combinatorial structure of the graph. In particular, we are interested when a graph G admits a Cohen-Macaulay toric ring K[G].
Apr 13, 2012 Elizabeth Weaver (University of Kentucky); PhD defense
Minimality and Duality of Tail-biting Trellises for Linear Codes
Abstract: Codes can be represented by edge-labeled directed graphs called trellises, which are used in decoding with the Viterbi algorithm. We will first examine the well-known product construction for trellises and present an algorithm for recovering the factors of a given trellis. To maximize efficiency, trellises that are minimal in a certain sense are desired. It was shown by Koetter and Vardy that one can produce all minimal tail-biting trellises for a code by looking at a special set of generators for a code. These generators along with a set of spans comprise what is called a characteristic pair, and we will discuss how to determine the number of these pairs for a given code. Finally, we will look at trellis dualization, in which a trellis for a code is used to produce a trellis representing the dual code. The first method we discuss comes naturally with the known BCJR construction. The second, introduced by Forney, is a very general procedure that works for many different types of graphs and is based on dualizing the edge set in a natural way. We call this construction the local dual, and we show the necessary conditions needed for these two different procedures to result in the same dual trellis.
Apr 12, 2012 Dennis Moore (University of Kentucky); PhD defense
Hilbert polynomials and strongly stable ideals
Abstract: Strongly stable ideals are important in algebraic geometry, commutative algebra, and combinatorics. Prompted, for example, by combinatorial approaches for studying Hilbert schemes and the existence of maximal total Betti numbers among saturated ideals with a given Hilbert polynomial, three algorithms are presented. Each of these algorithms produces all strongly stable ideals with some prescribed property: the saturated strongly stable ideals with a given Hilbert polynomial, the almost lexsegment ideals with a given Hilbert polynomial, and the saturated strongly stable ideals with a given Hilbert function. Bounds for the complexity of our algorithms are included. Also included are some applications for these algorithms and some estimates for counting strongly stable ideals with a fixed Hilbert polynomial.
Apr 10, 2012 Aleams Barra (University of Kentucky); PhD defense
Equivalence Theorems and the Local-Global Property
Abstract: In this thesis we revisit some classical results about MacWilliams equivalence theorems for codes over fields and rings. These theorems deal with the question whether, for a given weight function, weight preserving isomorphisms between codes can be described explicitly. We will show that a condition, which was already known to be sufficient for the MacWilliams equivalence theorem, is also necessary. Furthermore we will study local-global extensions that naturally generalize the MacWilliams equivalence theorems. Making use of the Fourier-transform and F-partitions we will prove that for various subgroups of the group of invertible matrices the local-global extension principle is valid.
Apr 5, 2012 David Cook II (University of Kentucky); PhD defense
The Lefschetz properties in positive characteristic
Abstract: The Lefschetz properties are algebraic properties that artinian standard graded algebras may enjoy. They are present if certain multiplication maps induce linear maps of maximal rank between the appropriate degree components of the algebra. While the majority of research on the Lefschetz properties has focused on characteristic zero, I consider the presence of the properties in positive characteristic. In particular, I study the Lefschetz properties by considering the prime divisors of determinants of critical maps. In both cases presented, the determinants are seen to be enumerations of combinatorial significance.
Oct 13, 2011 David Conti (University College Dublin)
Matrix representations of trellises and enumerating trellis pseudocodewords
Abstract: The performance of powerful decoding algorithms derived from special graph representations of codes is linked to so-called pseudocodewords and pseudoweights. Trellises are amongst the most notable of such graph representations of codes, for both theoretical and decoding purposes. In this talk, after giving a basic introduction, we will discuss some facets of the problem of enumerating trellis pseudocodewords and their pseudoweights. We will approach this problem by considering a natural matrix representation of trellises.
Sep 20, 2011 Uwe Nagel (University of Kentucky)
Criteria for componentwise linearity
Abstract: Stable monomial ideals are defined by a combinatorial property. They arise surprisingly often in various constructions in algebra and combinatorics. In 1999 Herzog and Hibi proposed the concept of componentwise linear ideals. Since then it turned out that componentwise linear ideals may be viewed as the polynomial ideals that are analogous to the stable ideals among the monomial ideals. This is partially due to a criterion for componentwise linear ideals by Aramonva, Herzog, and Hibi that applies over fields of characteristic zero. In the talk we introduce these concepts and results, and we discuss an extension of the mentioned criterion that is independent of the characteristic of the base field. The latter is joint work with Tim Roemer.
Apr 18, 2011 Laura Steil (University of Kentucky); PhD defense
Isometry Classes of Quadratic Forms defined over p-adic Rings
Abstract: Let f be a quadratic form defined over Z_p, the ring of p-adic integers with p a prime number. Define N_i(f) to be the number of solutions of the congruence f\equiv0\mod p^i. When p is not 2 we derive a formula for N_i(f) for i greater than 1 based on the known formula for N_1(f). This formula then gives a sequence (N_i(f))_{i=1}^{\infty} for the quadratic form f. We then find when the dimension of f and the sequence (N_i(f))_{i=1}^{\infty} determine the isometry class of f. When p is not 2, we show that the Jordan splitting gives information to determine exactly in which cases the isometry class is thus determined. We also demonstrate a case in which two non-isometric forms give the same sequence (N_i(f))_{i=1}^{\infty}. When p=2 and if the form f is regular when reduced mod 2 we can extend most of these results to forms defined over Z_2.
Apr 11, 2011 Jinjia Li (University of Louisville)
Asymptotic behavior of socle under Frobenius iterations
Abstract: Let (R,m) be a standard-graded local algebra over a field of positive characteristic p. Suppose I is an m-primary ideal of R. We study how the top socle degree and the socle length of R/J behave asymptotically, where J is a (varying) Frobenius power of I. We will also discuss their relations with Hilbert-Kunz multiplicity/function and asymptotic behaviors (with respect to Frobenius iteration) of some other homological invariants.
Mar 21, 2011 Elizabeth Weaver (University of Kentucky)
Counting and Dualizing Characteristic Pairs
Abstract: Trellis representations for linear block codes are widely used in decoding algorithms. To maximize efficiency, trellises that are minimal in a certain sense are desired. It was shown by Koetter and Vardy that one can produce all minimal tail-biting trellises for a code by looking at a special set of generators contained in characteristic pairs for a code. We will discuss how to determine the number of these pairs for a given code and examine a process for dualizing these pairs to produce pairs for the dual code.
Mar 10, 2011 David Conti (University College Dublin, Ireland)
Codes on graphs, pseudocodewords, and pseudoweights
Abstract: Algebraic codes are mathematical objects designed to reliably transmit data under presence of noise. A prominent approach in modern coding theory focuses on efficient iterative decoding algorithms based on graphical representations of codes. Therein it has been shown that decoding failure is determined by so-called pseudocodewords (particularly those with small pseudoweight) arising from the graph representation. In this talk we take a mathematical stroll on this topic, introducing the main questions along with a motivating conjecture on the Golay code. We will focus on the two paramount cases of Tanner Graph and Tail-Biting Trellis representations, and we will show in particular how an algebraic formalization can be developed for the latter. We will hint as well at how invariant theory and semi-algebraic geometry can play a role.
Mar 7, 2011 Heide Gluesing-Luerssen (University of Kentucky)
Tail-biting trellises for linear block codes
Abstract: Graphical models for linear block codes have proven to be a powerful tool for decoding. The most common of these models are trellises. These are graphs where the vertices are layered along a (potentially circular) time axis, and edges may only connect vertices at consecutive times. We will show how to construct minimal trellises for a given code. Thereafter, we will address the question of how to dualize a trellis in order to obtain a trellis for the dual code.
Feb 28, 2011 Ines B. Henriques (University of California at Riverside)
Ascent and descent modulo exact zero divisor
Abstract: An element a in a commutative ring R is said to be an exact zero divisor if it satisfies that R/a R is non-trivial, proper and isomorphic to (0 :_R a). It will be proved that homological and structural properties pass both ways between R and R/aR. This is a joint work with Luchezar Avramov and Liana Sega.
Feb 21, 2011 David Leep (University of Kentucky)
Sums of squares in quadratic number rings
Abstract: Studying number theory in quadratic extensions of the rational numbers has a long history. Many of the famous questions and theorems concerning sums of squares in the integers can be posed and often solved in the integers of a quadratic number field. This talk will deal with the problem of identifying which integers in a quadratic number field can be written as a sum of squares of integers. A necessary condition for such a representation is that the integer be totally positive. The condition is not sufficient however and several criteria are developed to guarantee a representation as a sum of squares. The results are based on theorems of I. Niven, C. Siegel from the 1940's, and R. Scharlau from 1980.
Feb 14, 2011 David Cook II (University of Kentucky)
Stability of the syzygy module of a monomial ideal
Abstract: Stability and semi-stability are desirable properties which a vector bundle may enjoy. In this talk we will explore the stability and semi-stability of the first syzygy module of a monomial ideal. In particular, we will see Brenner's combinatorial criterion and related results.
Feb 7, 2011 Stephen Sturgeon (University of Kentucky)
Cellular resolutions
Abstract: The free resolution of an ideal is a way of deriving information about an ideal. For some ideals we can find certain geometric objects that store the information for the free resolution of an ideal (i.e. Cellular Resolutions). I will cover these basic definitions and explain a cellular resolution of the class of Ferrers ideals. With this basis I will explain a method for obtaining a cellular resolution for the edge ideal of the complement of the n-gon which is a recent result by Bierman.
Hyperplane sections and the Socle Lemma
Abstract: It is a natural and classical idea to study a geometric object by slicing it with a (general) hyperplane and then trying to infer properties of the original object from properties of the slice. In 1993 Huneke and Ulrich found a new technique for using this approach. It is based on a linear algebra result. We will discuss Huneke's and Ulrich's ideas.
Dec 13, 2010 Laura Steil and Elizabeth Weaver (University of Kentucky)
Points in Uniform Position and Maximum Distance Separable Codes
Abstract: We will be looking at evaluation codes in projective space as defined in a paper by Johan P. Hansen. We will examine the relationship between the parameters of the code and the Hilbert function. Further, we will consider how the geometric position of the scheme affects the distance of the resulting code.
Nov 15, 2010 Enrico Carlini (Politecnico di Torino, Italy)
Non-negative rank of matrices
Abstract: Given a matrix P, its rank has many interesting interpretations. Among these, we know that P can be written as the sum of rk(P) matrices of rank one and no fewer. If P is a non-negative matrix, we can consider the non-negative rank of P, namely rk+(P). The non-negative rank measures the number of non-negative rank one matrices we need to write P as a non-negative linear combination of them. The non-negative rank is of particular interest to statisticians and to people working in optimization theory. But there is no good way to compute rk+(P). Clearly rk+(P)>=rk(P), but how big can the non-negative rank be? How is the non-negative rank affected by perturbation of P? In this talk, we will try to address these questions and many other issues about the non-negative rank. In particular, we will introduce a graphical approach to try and get some insight on rk+(P). The graphical approach can be found in a paper by Chu and Moody. The original results of this talk are obtained in collaboration with Bocci and Rapallo.
Nov 8, 2010 David W. Cook II (University of Kentucky)
Punctured hexagons and almost complete intersections
Abstract: We establish a connection between Artinian monomial almost complete intersections in three variables and hexagons with triangular punctures. It turns out that the prime factors of the number of certain tilings of the punctured hexagon are exactly the field characteristics in which the algebra fails to have the weak Lefschetz property. We use this to prove several new cases, with combinatorial explanations, of a conjecture by Migliore, Miró-Roig, and Nagel pertaining to characteristic zero.
Oct 18, 2010 Ed Enochs (University of Kentucky)
Variations of the Euclidean Algorithm
Abstract: Given an Euclidean domain, we have the associated division algorithm and Euclidean algorithm. The integer matrix version of the Euclidean algorithm is called Smith's theorem. Smith's theorem gives a quick proof of the fundamental theorem of abelian groups. The polynomial matrix version of Smith's theorem gives the Jordan canonical forms for matrices. We will prove a version of Smith's theorem for matrices with Laurent polynomials as entries that implies Grothendieck's theorem about locally free coherent sheaves on the projective line. Then we will indicate a relation with sheaves on Grassmannian varieties of the form G(n,2n).
Oct 11, 2010 Manoj Kumini (Purdue University)
Dependence of Betti tables on characteristic
Abstract: We will look at some examples of monomial ideals whose Betti tables depend on the characteristic of the base field. We will prove that if an ideal has componentwise linear resolution in all characteristics, then its Betti table is independent of the characteristic. This is joint work with K. Dalili.
Oct 4, 2010 Ben Nill (University of Georgia)
Cellular resolutions of ideals defined by simplicial homomorphisms
Abstract: In this talk I will give a survey on Gorenstein polytopes. These are lattice polytopes that exhibit a natural duality. They can also be characterized purely algebraically or via Ehrhart theory. I will start by introducing a big class of important examples: reflexive polytopes - and possibly some new generalizations. The arguably most-studied Gorenstein polytope is the Birkhoff polytope: the polytope of doubly stochastic matrices. I will give a brief view on how a long-standing conjecture by Stanley on the unimodality of its so-called h*-polynomial has been proved by Athanasiadis and generalized to Gorenstein polytopes by Bruns and Roemer. Finally, I would like to talk about recent results and open questions on Gorenstein polytopes as combinatorial models of topologically mirror-symmetric Calabi-Yau manifolds.
Sep 27, 2010 Ben Braun (University of Kentucky)
Abstract: In a recent preprint, Dochtermann and Engstrom applied the homomorphism complex construction to construct cellular resolutions of hyper-edge ideals of a class of hyper-graphs they called cointerval. In this talk I will introduce cellular resolutions and homomorphism complexes, and discuss a generalization of their results. If time permits, I will also discuss some topological and combinatorial properties of cointerval simplicial complexes, a generalization of cointerval hypergraphs. This work is joint with Jonathan Browder of University of Washington and Steve Klee of University of California, Davis.
Identifying certain Rees algebras
Abstract: Last week we have seen that the Max-Flow Min-Cut property of a clutter is related to the Rees algebra of the clutter. Rees algebras are rather abstractly defined. In the talk we will discuss some ideals whose Rees algebras admit a nice description. These ideals include Ferrers ideals and strongly stable ideals generated in degree two.
Sep 13, 2010 Bonnie Smith (University of Kentucky)
The Packing Problem for Monomial Ideals
Abstract: In 1993, Conforti and Cornuejols conjectured that a clutter has the Max-Flow Min-Cut property (a linear programming condition) if and only if all of its minors pack. Cornuejols went on to offer $5,000 to whomever could prove or disprove the conjecture before 2020... a prize which is still up for grabs! The problem was later cast in algebraic terms by Gitler, Valencia and Villarreal. I will give an introduction to this problem and an overview of progress which has be made so far.
May 3, 2010 Bonnie Smith (University of Notre Dame)
The core of a strongly stable ideal
Abstract: A reduction of an ideal I can be thought of as a simpler ideal, and is an important tool for studying I and its powers. The core of I is the intersection of all of its reductions. We will look at ways in which the core arises naturally, and observe that in certain cases the core encodes geometric information about the ideal. However, the core is usually very difficult to describe explicitly, since by definition it is an infinite intersection. We will consider strongly stable ideals of degree two, a class of ideals coming from graph theory. After examining the properties of these ideals, we will show that there is a surprisingly simple formula for their cores.
Apr 26, 2010 Vassily Gorbounov (University of Aberdeen, UK)
Toward the universal deformation of Schubert calculus
Abstract: Schubert calculus has been in the intersection of several fast developing areas of mathematics for a long time. Originally invented as the description of the cohomology of homogeneous spaces it needs to be redesigned when applied to other generalized cohomology theories such as the equivariant, the quantum cohomology, K-theory, and cobordism. All this cohomology theories are different deformations of the ordinary cohomology. In this talk we show that there is the universal deformation of Schubert calculus which produces the above mentioned by specialization of the appropriate parameters. We build on the work of D.Gepner and E.Witten on the relation between the fusion algebra of the group $SU(n)$ at the level $k$ and the quantum cohomology of the Grassmann manifold $Gr(n,k)$. The main observation there was that the classical cohomology of the Grassmann manifold is a Jacobi ring of an appropriate potential and the quantum cohomology is the Jacobi ring of a particular deformation of the this potential. We extend this result to hermitian symmetric spaces. Namely we show that the cohomology of the hermitian symmetric space is a Jacobi ring of a certain potential and the equivariant and the quantum cohomology and K-theory is a Jacobi ring of a particular deformation of this potential. This suggests to study the most general deformations of the Frobenius algebra of cohomology of these manifolds by considering the versal deformation of the appropriate potential. The structure of the Jacobi ring of such potential is a subject of well developed singularity theory. This gives a potentially new way to look at the classical, the equivariant, the quantum and other flavors of Schubert calculus.
Apr 19, 2010 Detlev Hoffmann (University of Nottingham, UK)
Level and sublevels of rings
Abstract: The level (resp. sublevel) of a ring is the smallest number n such that -1 (resp. 0) can be written as a sum of n (resp. n+1) nonzero squares in the ring if such an n exists, otherwise it is defined to be infinity. A famous result by Pfister from the 1960s states that the level of a field, if finite, is always a 2-power, and each 2-power can in fact be realized as level of a suitable field. This answered a question by van der Waerden posed in the 1930s. In the case of fields, level and sublevel coincide, but this need not be true for other types of rings. We will give a survey of various known results about levels and sublevels of rings and mention some open problems.
Apr 12, 2010 Theodoros Kyriopoulos (University of Kentucky)
The Jacobian conjecture, II
Abstract: We continue the discussion of the Jacobian conjecture.
Apr 5, 2010 Avinash Sathaye (University of Kentucky)
Recognizing the coordinate planes in three space
Abstract: A surface F(X,Y,Z)=0 is said to be a coordinate plane if k[X,Y,Z]=k[F,G,H] for some polynomials G,H. Naturally the coordinate ring k[X,Y,Z]/(F) of a coordinate plane is isomorphic to k[u,v], a polynomial ring in two variables. Any surface with this property is called an abstract plane. An abstract plane F has a parametrization X=p(u,v), y=q(u,v), z=r(u,v) such that F(p,q,r)=0 and k[p,q,r]=k[u,v]. The epimorphism problem asks if the converse is true, i.e. is every abstract plane a coordinate plane. We will describe the structure of the ring k[p,q] of an abstract plane and use it to prove some cases of this problem. The structure analyzes the jacobians of polynomials in the ring k[u,v].
Mar 29, 2010 Enrico Carlini (Politecnico di Torino, Italy)
Star configuration points
Abstract: The pairwise intersections of a family of lines in the plane is called a star configuration. A star configuration has the same Hilbert function as a set of generic points of the same cardinality. Thus it is natural to ask the following question: how special is a star configuration? I will show how to give a possible answer to this question. This is joint work with Adam van Tuyl at Lakehead, Canada.
Mar 8, 2010 Dennis Moore (University of Kentucky)
Saturated Strongly Stable Ideals with a given Hilbert Polynomial
Abstract: Strongly stable ideals have a simple combinatorial structure. The focus of the talk will be to explain how to find all the saturated strongly stable ideals which have a fixed Hilbert polynomial. To begin, the Hilbert polynomial and strongly stable ideals will be defined. Next, some basic operations on stable ideals are introduced. Then a few useful results are covered, culminating in an algorithm for finding the desired set of ideals. Also, I will mention why strongly stable ideals arise and how they are useful.
Mar 1, 2010 Theodoros Kyriopoulos (University of Kentucky)
The Jacobian Conjecture
Abstract: Let F: C^n -> C^n be a polynomial map (i.e. an endomorphism of affine n-space). The Jacobian conjecture claims that "if the determinant of the Jacobian matrix is a non-zero constant, then the map is an isomorphism of affine space" (i.e., it is a bijection and its inverse is a polynomial map. This conjecture was first stated by Eduard Ott-Heinrich Keller in 1939. Despite its simplicity, it has resisted many attempts to be solved, and many faulty proofs have been published by reputable mathematicians. We present various degree reductions, prove special cases, explain why injectivity or birationality would suffice to prove it, and relate it to differential topology.
Feb 22, 2010 Uwe Nagel (University of Kentucky)
Boij-Soederberg Theory
Abstract: The purpose of this talk is to discuss a recent breakthrough in commutative algebra. In 1890 Hilbert introduced free resolutions over a polynomial ring. They have been the object of intense research ever since, yet many questions have not been answered. In 2006 Boij and Soederberg proposed several conjectures about the numerical information encoded in a free resolution, the graded Betti numbers. Their amazing conjectures sparked a lot of interest, and in 2009 Eisenbud and Schreyer finally proved these conjectures. Free resolutions and graded Betti numbers will be introduced in the talk.
Feb 8, 2010 Zach Teitler (Texas A&M)
Ranks of polynomials
Abstract: The Waring rank of a polynomial of degree d is the least number of terms in an expression for the polynomial as a sum of d-th powers. The problem of finding the rank of a given polynomial and studying rank in general has been a central problem of classical algebraic geometry, related to secant varieties; in addition, there are applications to signal processing and computational complexity. Other than a well-known lower bound for rank in terms of catalecticant matrices, there has been relatively little progress on the problem of determining or bounding rank for a given polynomial (although related questions have proved very fruitful). I will describe new upper and lower bounds, with especially nice results for some examples including monomials and cubic polynomials. This is joint work with J.M. Landsberg.
Feb 1, 2010 David Cook II (University of Kentucky)
Clique-whiskering and face vectors
Abstract: The concept of whiskering a graph at every vertex in order to produce a Cohen-Macaulay graph was introduced by Villarreal. This result was strengthened to show that a fully whiskered graph is vertex-decomposable and pure by Doctermann and Engstroem.
We introduce a generalisation of whiskering which we call clique-whiskering. We show that a fully clique-whiskered graph is vertex-decomposable and has a surprising relation between its h-vector and the face vector of the independence complex of the base graph.
Dec 8, 2009 Josh Roberts (University of Kentucky)
A conjecture of Quillen and an algorithm for low dimensional group homology
Abstract: For certain rings R of algebraic integers, a conjecture of Quillen, as reformulated by Anton, states that certain classes in the mod p homology of D_1 vanish in the mod p homology of SL_2, where SL_n and D_n are the special linear group and group of diagonal matrices in GL_n respectively. We show that in homological dimension two, this is equivalent to the surjectivity of a certain transgression map.
In studying this problem, a very general algorithm was developed that finds an upper bound for the second homology, with coefficients in a finite field k, for any finitely presented group, and, in certain cases, calculates this homology exactly. Given a finitely presented group G, HopfÂ's formula expresses the second integral homology of G in terms of generators and relators. The algorithm we give exploits HopfÂ's formula to estimate H_2(G,k).
We conclude with a few example calculations.
Nov 10, 2009 Aleams Barra (University of Kentucky)
MacWilliams Extension Theorems
Abstract: Let C be a linear code of length n over a finite field F, that is, a subspace of F^n. In her thesis, MacWilliams showed that every linear map from C to F^n that preserves the Hamming weight extends to a monomial map on F^n, i.e., to a map that only permutes and rescales the coordinates. We will show that this result also holds true when we replace the field F by any finite Frobenius ring. We will discuss various weights on rings that have been studied in the literature, especially the Lee weight, the Euclidean weight and the homogeneous weight, and show that on certain rings, any linear map preserving the (Lee, Euclidean, or homogeneous) weight extends to a monomial map.
Nov 3, 2009 Alberto Corso (University of Kentucky)
Determinantal Equations Defining the Special Fiber Ring of Certain Ideals
Abstract: In a previous work, Corso and Nagel studied the algebraic properties of a class of monomial ideals arising from special bipartite graphs. These ideals, involving two distinct sets of variables, were dubbed Ferrers ideals. In particular, the special fiber ring of these ideals turned out to be defined by the two by two minors of a ladder. Using these equations we determine a special reduction of these ideals in a fashion that generalizes an old formula of Dedekind and Mertens about the relation of the contents of two polynomials and their product. Further, in some cases a specialization process produces an interesting class of monomial ideals generated in degree two. This class includes (square-free) strongly stable ideals. We show that the equations defining the special fiber ring of these new ideals are given by the two by two minors of a symmetric ladder, possibly with holes. This result extends some previous work of Conca and, later, Villarreal. As an application we will discuss how these results are needed in a proof of Bonnie Smith (U. of Notre Dame) who was able to give a description of the core of some strongly stable ideals. This is joint work with Uwe Nagel, Sonja Petrovic and Cornelia Yuen.
Oct 20, 2009 Elizabeth Weaver (University of Kentucky)
Trellis Representations for Linear Block Codes
Abstract: Linear block codes can be represented by a type of graph called a trellis which is used in decoding with the Viterbi algorithm. Since the computational complexity of this algorithm depends on the size of our trellis, it is useful to look at optimal trellises. In the literature, there exist mainly two types of trellises, conventional and tail-biting ones. While conventional trellises act on the linear time interval [0,n-1], tail biting trellises act on the circular time axis Z mod n. In the case of conventional trellises, there is a unique minimal trellis with many nice properties, and I will illustrate a construction by Forney for this unique minimal trellis. When tail-biting trellises are considered, the situation is more complicated, and there is no unique minimal trellis. I will introduce a method for creating the set of all minimal linear tail-biting trellises for a given code.
Orthogonal Decompositions of Categories
Abstract: An inner product on a finite dimensional vector space gives a notion of orthogonality of two vectors and so often provides orthogonal direct sum decompositions of such a vector space. Using the extension functor there is way to define an orthogonality relation between objects of a category. This in turn often leads to orthogonal decompositions of the category and so helps us get a better understanding of the category. In this talk I will give a brief history of these ideas and then will explain how they relate to my interests.
Sep 22/29, 2009 Uwe Nagel (University of Kentucky)
Pure O-sequences and Lefschetz properties
Abstract: Given a finite set of monomials of the same degree, a pure O-sequence is determined by counting the number of monomials that divide any of the given monomials. Pure O-sequences occur in various disguises, so ideally one would like to classify the possible pure O-sequences. This is considered a very difficult problem. In the talk an algebraic approach will be discussed. The first step is to reinterpret an O-sequence as the Hilbert function of a certain ring. Then the key is to investigate the properties of these rings. As a first application, this provides a new proof of Hibi's results about the growth of pure O-sequences. There remain many open problems. All relevant concepts will be explained in the talk.
Sep 14, 2009 David Leep (University of Kentucky)
The u-invariant of p-adic function fields
Abstract: The u-invariant of a field k, u(k), is the smallest integer n such that any quadratic form with coefficients from k having more than n variables has a nontrivial zero defined over k.
Calculating the u-invariant of an arbitrary field is often very difficult, but the u-invariant of fields arising in number theory (finite fields, p-adic fields, number fields) have been known for a long time. An interesting question is the computation of u-invariants of function fields over these fields. The u-invariants of function fields over finite fields are completely known, while essentially nothing is known about u-invariants of a function fields over number fields.
During the last decade, the u-invariants of function fields of curves over p-adic fields have been computed. A new result of Heath-Brown on systems of quadratic forms over p-adic fields has led to the computation of u-invariants of arbitrary function fields over p-adic fields. I will report on the background to this problem and my proof of this new result.
The talk will not be overly technical.
Jun 23, 2009 Jose Ignacio Iglesias Curto (University of Salamanca, Spain)
Coding theory and algebraic geometry
Abstract: After a brief review of the basics of coding theory we will explain how the use of algebraic geometric tools leads to certain interesting code constructions. The knowledge of the geometric elements involved can then be used to study the properties of the codes so obtained and to derive decoding algorithms.
We will also present a similar construction carried out in the context of convolutional codes and comment on the particularities of these codes.