Difference between revisions of "Colloquia 2012-2013"
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All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.
== Spring 2012 ==
!align="left" | host(s)
|'''Jan 23, 4pm'''
|'''Tues, Jan 15, B139'''
|[http://maeresearch.ucsd.edu/spagnolie/ Saverio Spagnolie] (Brown)
|[http://people.maths.ox.ac.uk/piercel/ Lillian Pierce] (Oxford)
|''Hydrodynamics of Self-Propulsion Near a Boundary: Construction of a Numerical and Asymptotic Toolbox''
|[[#Pierce|A new twist on the Carleson operator]]
|Denissov
|'''Thurs, Jan 17, 2pm, 901VV'''
|[http://www-personal.umich.edu/~jblasiak/ Jonah Blasiak] (Michigan)
|[[#Blasiak|Positivity, complexity, and the Kronecker problem]]
|Terwilliger
|Jan 27
|[http://ccom.ucsd.edu/~astern/ Ari Stern] (UCSD)
|[http://www.maths.usyd.edu.au/u/afish/ Alexander Fish] (Sydney)
|''Numerical analysis beyond Flatland: semilinear PDEs and problems on manifolds''
|[[#Fish|Product sets in amenable groups through a dynamical approach]]
|Jean-Luc / Julie
|Gurevich
|Feb 3
|[http://www.math.ubc.ca/~magyar/ Akos Magyar] (UBC)
|[http://www.math.wisc.edu/~dymarz/ Tullia Dymarz] (Madison)
|''On prime solutions to linear and quadratic equations''
|[[#Dymarz|Quasisymmetric vs Bi-Lipschitz maps]]
|Street
|[http://www.math.columbia.edu/~lhhuang/index/Welcome.html Lan-Hsuan Huang] (Columbia U)
|[http://www.math.wisc.edu/~jeanluc/ Jean-Luc Thiffeault] (Madison)
|''Positive mass theorems and scalar curvature problems''
|[[#Thiffeault|pseudo-Anosovs with small or large dilatation]]
|Sean
|Roch
|Feb 15
|[http://maeresearch.ucsd.edu/lauga/ Eric Lauga] (UCSD)
|[[#Lauga|Optimization in fluid-based locomotion]]
|Spagnolie
|[http://math.umn.edu/~svitlana/ Svitlana Mayboroda] (University of Minnesota)
|[[#Mayboroda|Elliptic PDEs, analysis, and potential theory in irregular media]]
|Stovall
|Mar 1
|[http://www.math.harvard.edu/~kwickelg/ Kirsten Wickelgren] (Harvard)
|[[#Wickelgren|Grothendieck's anabelian conjectures]]
|March 8
|[http://homepages.cae.wisc.edu/~negrut/ Dan Negrut] (UW - Mechanical Engineering)
|[[#Negrut|Using Advanced Computing in Applied Dynamics: From the Dynamics of Granular Material to the Motion of the Mars Rover]]
|Caldararu
|March 15
|[http://www.icmat.es/kurusch/ Kurusch Ebrahimi Fard] (Madrid)
|[[#Fard|Spitzer-type identities in non-commutative Rota-Baxter algebras]]
|Gurvich
|<strike> March 22 </strike>
|[http://www2.warwick.ac.uk/fac/sci/maths/people/staff/neil_oconnell/ Neil O'Connell] (Warwick)
|CANCELLED
|Timo Seppalainen
|'''Spring Break'''
|No Colloquium
|April 5
|[http://hobbes.la.asu.edu/ John Jones] (ASU)
|[[#Jones| Number fields with prescribed ramification]]
|Boston
|April 12
|[http://math.mit.edu/~asnowden/ Andrew Snowden] (MIT)
|[[#Snowden|Large algebraic structures with large amounts of symmetry]]
|Moe Hirsch (Honorary Fellow -- UW Madison)
|[[#Hirsch|The Divine Madness: Mathematics, Myths and Metaphors]]
|'''Tuesday, April 30, 4PM'''
|[http://www.ens.fr/spip.php?article1296&lang=en Laure Saint-Raymond] (ENS)]
|[[#Saint-Raymond|Wasow Lecture: The Irreversibility in Gas Dynamics, a Matter of Probability]]
|Wasow Lecture
|May 3
|Davesh Maulik (Columbia)
|[[#Maulik|Plane curve singularities and knot invariants]]
|May 10
|[http://www.wisdom.weizmann.ac.il/~gelbar/ Steve Gelbart] (Weizmann Institute)
|[[#Gelbart| "Zeta"]]
|[http://www.math.wisc.edu/~mmwood/ Melanie Wood] (UW Madison)
|[http://www.math.wisc.edu/~ellenber/ Jordan Ellenberg] (Madison)
|''Counting polynomials and motivic stabilization''
|[[#Ellenberg|''FI-modules: an introduction'']]
|local
|<strike>'''Sept 20, 4pm'''</strike>
|[http://www.math.uconn.edu/~hering/ Milena Hering] (University of Connecticut)
|[http://www-stat.stanford.edu/~cgates/PERSI/ Persi Diaconis] (Stanford)
|''The moduli space of points on the projective line and quadratic Groebner bases''
|Andrei
|[http://www.math.ubc.ca/~malabika/ Malabika Pramanik] (University of British Columbia)
|[http://eaton.math.rpi.edu/faculty/J.McLaughlin/mclauj.html Joyce McLaughlin] (RPI)
|''Analysis on Sparse Sets''
|[[#McLaughlin|''Mathematics for Imaging Biomechanical Parameters in Dynamic Elastography'']]
|Benguria
|WIMAW
|[http://www.comsec.uwaterloo.ca/~ggong/ Guang Gong] (Electrical Engineering, University of Waterloo)
|[http://math.mit.edu/~emarberg/ Eric Marberg] (MIT)
|''Polyphase Sequence Families with Low Correlation and DFT for Spreading OFDM''
|[[#Marberg|''Supercharacters for algebra groups: applications and extensions'']]
|Shamgar
|Isaacs
|[http://pages.cs.wisc.edu/~sifakis/ Eftychios Sifakis] (UW-Madison, CS Dept.)
|[http://math.uchicago.edu/~masur/ Howard Masur] (Chicago)
|''TBA''
|[[#Masur|''Winning games for badly approximable real numbers and billiards in polygons'']]
|Nigel
|Dymarz
|'''Wed, Oct 10, 4pm'''
|[http://www.charlesdoran.net/ Charles Doran] (University of Alberta)
|[http://www.kent.ac.uk/smsas/maths/our-people/profiles/lemmens_bas.html Bas Lemmens] (Univ. of Kent)
|[[#Lemmens|From hyperbolic geometry to nonlinear Perron-Frobenius theory]]
|Matt Ballard
|LAA lecture
|'''Thur, Oct 11, 4pm'''
|[http://web.williams.edu/Mathematics/cadams/ Colin Adams] and [http://web.williams.edu/Mathematics/tgarrity/ Thomas Garrity] (Williams College)
|[http://www.ma.utexas.edu/users/gamba/ Irene Gamba] (UT-Austin)
|''Which is better, the derivative or the integral?''
|[[#Gamba|Analytical and numerical issues associated with the dynamics of the non-linear Boltzmann equation]]
|Maxim
|[http://www.math.temple.edu/~lorenz/ Martin Lorenz] (Temple University)
|[http://user.math.uzh.ch/rosenthal/ Joachim Rosenthal] (Univ. of Zurich)
|''Prime ideals and group actions in noncommutative algebra''
|[[#Rosenthal|Linear Random Network Codes, a Grassmannian Approach]]
|Don Passman
|[http://www.math.uchicago.edu/~schlag/ Wilhelm Schlag] (University of Chicago)
|[http://www.math.wisc.edu/~spagnolie/uw/index.html Saverio E Spagnolie] (Madison)
|[[#Spagnolie|Elastic slender bodies in fluids and slender bodies in elastic fluids]]
|Spring recess
|[http://math.berkeley.edu/~oeding/Home.html Luke Oeding] (UC Berkeley)
|[[#Oeding|The Trifocal Variety]]
|'''Tues, Oct 30'''
|[http://www.math.tulane.edu/~cortez/ Ricardo Cortez] (Tulane)
|[http://www.math.nyu.edu/faculty/majda/ Andrew Majda] (Courant)
|[[#Majda|Data Driven Methods for Complex Turbulent Systems]]
|Mitchell
|Smith, Stechmann
|'''Thurs, Nov 1'''
|[http://www.math.harvard.edu/~gross/ Benedict H. Gross] (Harvard)
|[http://math.uchicago.edu/~ryzhik/ Lenya Ryzhik] (Chicago)
|[[#Ryzhik|The role of a drift in elliptic and parabolic equations]]
|'''distinguished lecturer'''
|Kiselev
|[http://www.math.umn.edu/~sverak/ Vladimir Sverak] (Minnesota)
|[[#Sverak|On scale-invariant solutions of the Navier-Stokes equations]]
|[http://www-bcf.usc.edu/~guralnic/ Robert Guralnick] (University of Southern California)
|[http://www.math.wisc.edu/~andrews/ Uri Andrews] (Madison)
|''Maps from the Generic Riemann surface to the Riemann sphere''
|[[#Andrews|Computable Stability Theory]]
|Lempp
|'''Mon, Nov 12'''
|[http://www.maths.ox.ac.uk/people/profiles/gui-qiang.chen Gui-Qiang Chen] (Oxford)
|[http://math.mit.edu/~smart/ Charles Smart] (MIT)
|[[#Smart|Regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity]]
|Feldman
|[http://www.math.sunysb.edu/~mde/ Mark Andrea de Cataldo] (Stony Brook)
|[http://www-personal.umich.edu/~erman/ Daniel Erman] (U.Michigan)
|'''Wed, Nov 28'''
|''Tentatively Scheduled''
|[http://www.nd.edu/~chill10/ Cameron Hill] (Notre Dame)
|[[#Hill|Genericity in Discrete Mathematics]]
|[http://www.math.columbia.edu/~namle/ Nam Le] (Columbia)
|[[#Le|Boundary regularity for solutions to the linearized Monge-Ampere equations and applications]]
|[http://users.math.yale.edu/~alf8/ Amanda Folsom] (Yale)
|[[#Folsom|q-series and quantum modular forms]]
|Ellenberg/WIMAW
== Abstracts ==
===Mon, Jan 23: Saverio Spagnolie (Brown)===
===<span id="Ellenberg"></span>Sept 14: Jordan Ellenberg (UW-Madison) ===
"Hydrodynamics of Self-Propulsion Near a Boundary: Construction of a Numerical and Asymptotic Toolbox"
''FI-modules: an introduction''
(joint work with T Church, B Farb, R Nagpal)
In topology and algebraic geometry one often encounters phenomena of _stability_. A famous example is the cohomology of the moduli space of curves M_g; Harer proved in the 1980s that the sequence of vector spaces H_i(M_g,Q), with g growing and i fixed, has dimension which is eventually constant as g grows with i fixed.
In many similar situations one is presented with a sequence {V_n}, where the V_n are not merely vector spaces, but come with an action of S_n. In many such situations the dimension of V_n does not become constant as n grows -- but there is still a sense in which it is eventually "always the same representation of S_n" as n grows. The preprint
shows how to interpret this kind of "representation stability" as a statement of finite generation in an appropriate category; we'll discuss this set-up and some applications to the topology of configuration spaces, the representation theory of the symmetric group, and diagonal coinvariant algebras. Finally, we'll discuss recent developments in the theory of FI-modules over general rings, which is joint work with (UW grad student) Rohit Nagpal.
===<span id="Diaconis"></span> '''Thu, Sept 20''': Persi Diaconis (Stanford) ===
''Spatial mixing: problems and progress''
One standard way of mixing (cards, dominos, Mahjong tiles) is to 'smoosh' them around on the table with two hands. I will introduce some models for this, present data (it's surprisingly effective) and some first theorems. The math involved is related to fluid flow and Baxendale-Harris random homeomorphisims.
===<span id="McLaughlin"></span>Sept 21: Joyce R. McLaughlin (Rensselaer Polytechnic Institute) ===
''Mathematics for Imaging Biomechanical Parameters in Dynamic Elastography''
Elastography, the imaging of biomechanical parameters in
tissue, is motivated by the doctor's palpation exam where the doctor
presses against the skin to detect stiff and abnormal tissue changes. In
dynamic elastography experiments, the tissue is in motion with
displacement amplitudes on the order of tens of microns. The
displacement(s) are determined with sequences of MR data sets or
sequences of ultrasound RF/IQ data sets within the tissue; and the data
shows a dispersive effect indicating that tissue is viscoelastic. A
choice of viscoelastic model must be made. For each model the
biomechanical parameters satisfy a first order, linear or nonlinear,
partial differential equation (system) with real or complex coefficients.
We discuss the mathematical properties of these equations and how those
properties lead to successful interpretation of the data, and to
successful algorithms and images. We show biomechanical images of breast
cancer and prostate cancer and compare those images to ultrasound images
and histology slides with marked cancerous inclusions.
===<span id="Marberg"></span>Sept 28: Eric Marberg (MIT) ===
''Supercharacters for algebra groups: applications and extensions''
The group U_n(F_q) of unipotent upper triangular matrices over a
finite field belongs to the same list of fundamental examples as the
symmetric or general linear groups. It comes as some surprise, therefore,
that the group's irreducible characters are unknown, and considered in some
sense unknowable. In order to tackle problems normally requiring knowledge
of a group's irreducible characters, Diaconis and Isaacs developed the
notion of the supercharacters of an algebra group, generalizing work of
Andre and Yan. Algebra groups form a well-behaved class of p-groups
including U_n(F_q) as a prototypical example, and supercharacters are
certain reducible characters which form a useful approximation to the set
of irreducible characters. In this talk I will survey several equivalent
definitions of the supercharacters of an algebra group, and discuss some
applications and extensions of these approaches. On one end of things, I
intend briefly to introduce the recent discovery of how certain
representation theoretic operations on the supercharacters of U_n(F_q)
naturally define a Hopf algebra structure, which has been studied under a
different name by combinatorialists. In another direction, I will explain
how one can view the supercharacters of an algebra group as the first step
in a more general reduction process, which can be used to shed light on
some mysterious properties of U_n(F_q).
===<span id="Masur"></span>Oct 04 Howard Masur (Chicago) ===
''Winning games for badly approximable real numbers and billiards in polygons''
Wolfgang Schmidt invented the notion of a winning subset of Euclidean space in a game between two players. Winning sets have nice
properties such as full Hausdorff dimension. The basic example of a winning set considered by Schmidt are those reals badly
approximated by rationals. An equivalent formulation by Artin is that badly approximable reals correspond to geodesics that stay in a bounded set in the modular curve. There is also an equivalent formulation in terms of sets of directions for the linear flow on a square torus or equivalently, billiard trajectories in a square. In joint work with Yitwah Cheung and Jon Chaika we extend this notion of winning to flows on flat surfaces of higher genus with applications to billiards in rational angled polygons. My intention in this talk is to give the background on the Schmidt game, describe the classical results before introducing the more recent work.
===<span id="Lemmens"></span>'''Wed, Oct 10''': Bas Lemmens (University of Kent) ===
''From hyperbolic geometry to nonlinear Perron-Frobenius theory''
In a letter to Klein Hilbert remarked that the logarithm of the cross-ratio is a metric on
any open, bounded, convex set in Euclidean space. These metric spaces are nowadays
called Hilbert geometries. They are a natural non-Riemannian generalization of Klein's
model of the hyperbolic plane, and play a role in the solution of Hilbert's fourth problem.
In the nineteen fifties Garrett Birkhoff and Hans Samelson independently discovered that
one can use Hilbert's metric and the contraction mapping principle to prove the existence
and uniqueness of a positive eigenvector for a variety of linear operators that leave a closed
cone in a Banach space invariant. Their results are a direct extension of the classical
Perron-Frobenius theorem concerning the eigenvectors and eigenvalues of nonnegative
matrices. In the past decades this idea has been further developed and resulted in strikingly detailed nonlinear extensions of the Perron-Frobenius theorem. In this talk I will discuss
the synergy between metric geometry and (nonlinear) operator theory and some of the
recent results and open problems in this area.
===<span id="Gamba"></span>'''Thur, Oct 11''': Irene Gamba (University of Texas-Austin) ===
''Analytical and numerical issues associated with the dynamics of the non-linear Boltzmann equation''
The non-linear Boltzmann equation models the evolution of a statistical flow associated with particle systems in a rarefied or mesoscopic regimes. Its analytical local as well as long time behavior depends strongly on the growth conditions of the collisional kernels, as functions of the intermolecular potentials and scattering mechanisms. We will present recent analytical results, such as classical convolutional inequalities and sharp moments estimates, which imply propagation of $W^{k,p}$ norms and exponential decay of high energy tails, and their consequences on the existence, regularity, and stability of solutions for initial value problems as well as control of decay rates to equilibrium. In particular we present a numerical approximation to the non-linear Boltzmann problem by a conservative spectral scheme, and show spectral accuracy as well as error estimates.
===<span id="Rosenthal"></span>Oct 12: Joachim Rosenthal (Univ of Zurich) ===
''Linear Random Network Codes, a Grassmannian Approach''
A novel framework for random network coding has been introduced by
Koetter and Kschischang. In this framework information is encoded in
subspaces of a given ambient space over a finite field. A natural
metric is introduced where two subspaces are `close to each other' as
soon as their dimension of intersection is large. This framework poses
the challenge to come up with new codes with optimal or near optimal
distance and to develop efficient decoding algorithms.
In a first part of the talk we will provide a survey. In a second part
of the talk we report on progress constructing spread codes
and orbit codes. The decoding problem of orbit codes can be
interpreted as a problem in Schubert calculus over a finite field.
===<span id="Spagnolie"></span>'''Fri, Oct 19''': Saverio E. Spagnolie (Madison) ===
''Elastic slender bodies in fluids and slender bodies in elastic fluids''
The scientific study of elastic materials dates back to Galileo, and fluid mechanics to Archimedes, but the interaction of elastic bodies and viscous fluids remains a topic at the frontier of modern research. We will discuss two problems on this topic of recent interest. First, when a flexible filament is confined to a fluid interface, the balance between capillary attraction, bending resistance, and tension from an external source can lead to a self-buckling instability. We will walk through an analysis of this elastocapillary instability, and analytical formulae will be shown that compare favorably with the results of detailed numerical computations. Second, we will discuss the motility of a swimming helical body in a viscoelastic fluid, wherein the fluid itself exhibits an elastic response to deformation. The helical geometry is exploited to generate a highly accurate numerical method, and we will show that the introduction of viscoelasticity can either enhance or retard the swimming speed depending on the body geometry and the properties of the fluid (through a dimensionless Deborah number). Our findings bridge the gap between studies showing situationally dependent enhancement or retardation of swimming speed, and may help to clarify phenomena observed in systems from spermatozoan swimming to mechanical drilling.
===<span id="Oeding"></span>'''Fri, Oct 26''': Luke Oeding (UC Berkeley) ===
''The Trifocal Variety''
In Computer Vision one considers many cameras looking at the same scene. From this setup many interesting geometric and algebraic questions arise. In this talk we will focus on the case of 3 cameras and study the so called trifocal tensors. Trifocal tensors are constructed from a bilinear map defined using the trifocal setup. A natural question is, given a particular tensor, how can one determine if it is a trifocal tensor? This question can be answered by finding implicit defining equations for the trifocal variety. From an algebraic standpoint, it is also interesting to know the minimal generators of the defining ideal of the trifocal variety.
In this talk I will explain our use of symbolic and numerical computations aided by Representation Theory and Numerical Algebraic Geometry to find the minimal generators of the ideal of the trifocal variety. This is joint work with Chris Aholt (Washington). Our work builds on the work of others (such as Hartley-Zisserman, Alzati-Tortora and Papadopoulo-Faugeras) who have already considered this problem set-theoretically.
My goal is to make most of the talk accessible to anyone with a modest background in Linear Algebra.
===<span id="Majda"></span>'''Tues, Oct 30''': Andrew Majda (Courant) ===
''Data Driven Methods for Complex Turbulent Systems''
An important contemporary research topic is the development of physics
constrained data driven methods for complex, large-dimensional turbulent
systems such as the equations for climate change science. Three new
approaches to various aspects of this topic are emphasized here. 1) The
systematic development of physics constrained quadratic regression models
with memory for low-frequency components of complex systems; 2) Novel
dynamic stochastic superresolution algorithms for real time filtering of
turbulent systems; 3) New nonlinear Laplacian spectral analysis (NLSA) for
large-dimensional time series which capture both intermittency and
low-frequency variability unlike conventional EOF or principal component
analysis. This is joint work with John Harlim (1,2), Michal Branicki (2),
and Dimitri Giannakis (3).
=== <span id="Ryzhik"></span>'''Thurs, Nov 1''': Lenya Ryzhik (Chicago) ===
''The role of a drift in elliptic and parabolic equations''
The first order partial differential equations are closely connected to the underlying characteristic ODEs. The second order elliptic and parabolic equations are as closely connected to the Brownian motion and more general diffusions with a drift. From a variety of points of view, the drift does not really matter - all diffusions look more or less similar. As a caveat to this reasonable line of thought, I will describe a menagerie of problems: linear and nonlinear, steady and time-dependent, compressible and incompressible, where the drift and diffusion confirm the Mayakovsky thesis "Woe to one alone!" Together, they lead to enhanced mixing and improved regularity that are impossible
for each one of them to attain.
===<span id="Sverak"></span>Nov 2: Vladimir Sverak (Minnesota) ===
''On scale-invariant solutions of the Navier-Stokes equations''
The solutions of the Navier-Stokes equations which are invariant under the scaling symmetry of the equations provide an interesting window into non-linear regimes which are not accessible by perturbation theory. They appear to give valuable hints concerning the old question about the uniqueness of weak solutions. In the lecture we outline a recent proof of the result that for every scale-invariant initial data there is a global scale-invariant solution (smooth for positive times), and we explain connections to the uniqueness problem. This is joint work with Hao Jia.
===<span id="Andrews"></span>Nov 9: Uri Andrews (Madison) ===
''Computable Stability Theory''
Stability theory attempts to classify the underlying structure of
mathematical objects. The goal of computable mathematics is to understand
when mathematical objects or constructions can be demonstrated computably.
I'll talk about the relationship between underlying structure and
computation of mathematical objects.
===<span id="Smart"></span>Monday, Nov 12: Charles Smart (MIT) ===
''Regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity''
I will discuss regularity of fully nonlinear elliptic
equations when the usual uniform upper bound on the ellipticity is
replaced by bound on its $L^d$ norm, where $d$ is the dimension of the
ambient space. Our
estimates refine the classical theory and require several new ideas
that we believe are of independent interest. As an application, we
prove homogenization for a class of stationary ergodic strictly
elliptic equations.
===<span id="Hill"></span>Wednesday, Nov 28: Cameron Hill (Notre Dame) ===
''Genericity in Discrete Mathematics''
I will discuss the intriguing role that ``generic'' objects can play in finitary discrete mathematics. Using the framework of model theory, we will see that there are at least two reasonable but very different notions of genericity relative to a class of finite structures (like a class of finite graphs or a class of finite groups). Through discussions of zero-one laws and structural Ramsey theory, I will try to illustrate the importance of generically-categorical classes -- those for which the various notions of genericity coincide. To finish, I will state some results that follow from the assumption of generic-categoricity.
===<span id="Le"></span>Nov 30: Nam Le (Columbia) ===
''Boundary regularity for solutions to the linearized Monge-Ampere equations and applications''
In this talk, I will discuss boundary regularity of solutions to the linearized Monge-Ampere equations, and their applications to nonlinear, fourth order, geometric Partial Differential Equations (PDE).
First, I will present my regularity results in joint work with O. Savin and T. Nguyen including:
boundary Holder gradient estimates and global $C^{1,\alpha}$ estimates, global Holder estimates and global $W^{2,p}$ estimates.
Then, I will describe applications of the above regularity results to several nonlinear, fourth order, geometric PDE such as: global second derivative estimates for the second
boundary value problem of the prescribed affine mean curvature and Abreu's equations; and global regularity for minimizers having prescribed determinant of certain convex functionals motivated by
the Mabuchi functional in complex geometry.
===<span id="Folsom"></span>Dec 14: Amanda Folsom (Yale) ===
''q-series and quantum modular forms''
While the theory of mock modular forms has seen great advances in the last decade, questions remain. We revisit Ramanujan's last letter to Hardy, and prove one of his remaining conjectures as a special case of a more general result. Surprisingly, Dyson's combinatorial rank function, the Andrews-Garvan crank functions, mock theta functions, and quantum modular forms, all play key roles. Along these lines, we also show that the Rogers-Fine false theta functions, functions that have not been well understood within the theory of modular forms, specialize to quantum modular forms. This is joint work with K. Ono (Emory U.) and R.C. Rhoades (Stanford U.).
===<span id="Pierce"></span>Tues, Jan 15, B139: Lillian Pierce (Oxford) ===
''A new twist on the Carleson operator''
Must the Fourier series of an L^2 function converge pointwise almost everywhere? In the 1960's, Carleson answered this question in the affirmative, by studying a particular type of maximal singular integral operator, which has since become known as a Carleson operator. In the past 40 years, a number of important results have been proved for generalizations of the original Carleson operator. In this talk we will introduce the Carleson operator and survey several of its generalizations, and then describe new joint work with Po Lam Yung that introduces curved structure to the setting of Carleson operators.
===<span id="Blasiak"></span>Thurs, Jan 17, 2pm, 901VV: Jonah Blasiak (Michigan) ===
''Positivity, complexity, and the Kronecker problem''
Positivity problems in algebraic combinatorics ask to find positive combinatorial formulae
for nonnegative quantities arising in geometry and representation theory like cohomological dimensions and dimensions of algebras and their irreducible representations.
A famous open positivity problem in representation theory is the Kronecker problem, which asks for a positive combinatorial formula for decomposing tensor products of irreducible representations of the symmetric group.
We will begin with a general discussion of positivity problems and an intriguing new motivation for these problems from complexity theory.
We will then present our solution to a special case of the Kronecker problem that substantially improves on previous results.
===<span id="Fish"></span>Fri, Jan 25: Alexander Fish (Sydney) ===
''Product sets in amenable groups through a dynamical approach''
We will describe a new correspondence between product sets in a countable amenable group and product sets in compact groups. This approach enables us to obtain quantitative results concerning product sets in amenable groups. The talk is for general mathematical audience.
(based on a joint work with M. Bjorklund)
===<span id="Dymarz"></span>Fri, Feb 1: Tullia Dymarz (Madison) ===
''Quasisymmetric vs Bi-Lipschitz maps''
On a metric space, there are various classes of functions which respect aspects of the metric space structure. One of the most basic classes is the bi-Lipschitz maps (Lipschitz maps whose inverses are also Lipschitz). Another possibly much larger class consists of the so-called quasisymmetric maps (these are closely related quasiconformal maps). On both Euclidean space and the p-adics, there are many quasisymmetric maps which are not bi-Lipschitz. However, on the product of Euclidean space with the p-adics, we show that all quasisymmetric maps are bi-Lipschitz. Furthermore, our proof does not use any direct analysis
but instead uses coarse topology and results from negative curvature.
===<span id="Thiffeault"></span>Feb 8: Jean-Luc Thiffeault (Madison) ===
''pseudo-Anosovs with small or large dilatation''
Homeomorphisms of a surface to itself can be classified using the
well-known Thurston-Nielsen theorem. The most interesting topological
class contains pseudo-Anosov mappings: they stabilize a pair of
transverse singular foliations with a finite number of singularities.
These foliations are called unstable and stable, and are respectively
expanded and compressed by an algebraic constant called the
dilatation. Characterizing the possible values of these dilatations
for a given suface is an open problem. Here I discuss a method to
find the minimum value of the dilatation on closed surfaces of a given
genus, for the special case where the foliations are orientable. I
will then address the opposite problem: how to find pseudo-Anosovs
with large dilatations. Unlike the minimizer problem, this is not
well-defined -- the answer is infinity -- unless we add a constraint.
Constraints can arise from practical optimization problems in
engineering, and I will show some optimal solutions that can be
incorporated in devices called taffy pullers. (This is joint work with
Erwan Lanneau and Matt Finn.)
===<span id="Lauga"></span>Feb 15: Eric Lauga (UCSD) ===
''Optimization in fluid-based locomotion''
The world of self-propelled low-Reynolds number swimmers is inhabited by a myriad of microorganisms such as bacteria, spermatozoa, ciliates, and plankton. In this talk, we focus on the locomotion of ciliated cells. Cilia are short slender whiplike appendages (a few microns long, one tenth of a micron wide) internally actuated by molecular motors (dyneins) which generate a distribution of bending moments along the cilium length and produce time-varying shape deformations. In most cases cilia are not found individually but instead in densely packed arrays on surfaces. In this work we will ask the question: can the individual and collective dynamics of cilia on the surface of an individual microorganism be rationalized as the solution to an optimization problem? We first address the deformation of individual cilia anchored on surfaces before characterizing the locomotion and feeding by surface distortions of swimmers covered by cilia array. We demonstrate, as solution to the optimization procedure, the appearance of the well-known two-stroke kinematics of an individual cilium, as well as waves in cilia array reminiscent of experimentally-observed metachronal waves.
===<span id="Mayboroda"></span>Feb 22: Svitlana Mayboroda (University of Minnesota) ===
''Elliptic PDEs, analysis, and potential theory in irregular media''
Elliptic boundary value problems are well-understood in the
case when the boundary, the data, and the coefficients exhibit
smoothness. However, perfectly uniform smooth systems do not exist in
nature, and every real object inadvertently possesses irregularities
(a sharp edge of the boundary, an abrupt change of the medium, a
defect of the construction).
The analysis of general non-smooth elliptic PDEs gives rise to
decisively new challenges: possible failure of maximal principle and
positivity,
breakdown of boundary regularity, lack of the classical L^2 estimates,
to mention just a few. Further progress builds on an involved blend of
harmonic analysis, potential theory and geometric measure theory
techniques. In this talk we are going to discuss some highlights of
the history, conjectures, paradoxes, and recent discoveries such as
the higher-order Wiener criterion and maximum principle for higher
order PDEs, solvability of rough elliptic boundary problems, as well
as an intriguing phenomenon of localization of eigenfunctions --
within and beyond the limits of the famous Anderson localization.
Parts of the talk are based on joint work with S. Hofmann, M. Filoche,
C. Kenig, V. Maz'ya, and J. Pipher.
The swimming kinematics and trajectories of many microorganisms are altered by the presence of nearby boundaries, be they solid or deformable, and often in perplexing fashion. When an organism's swimming dynamics vary near such boundaries a question arises naturally: is the change in behavior fluid mechanical, biological, or perhaps due to other physical laws? We isolate the first possibility by exploring a far-field description of swimming organisms, providing a general framework for studying the fluid-mediated modifications to swimming trajectories. Using the simplified model we consider trapped/escape trajectories and equilibria for model organisms of varying shape and propulsive activity. This framework may help to explain surprising behaviors observed in the swimming of many microorganisms and synthetic micro-swimmers. Along the way, we will discuss the numerical tools constructed to analyze the problem of current interest, but which have considerable potential for more general applicability.
===<span id="Wickelgren"></span>March 1: Kirsten Wickelgren (Harvard) ===
''Grothendieck's anabelian conjectures''
===Fri, Feb 3: Akos Magyar (UBC)===
Grothendieck's anabelian conjectures predict that the solutions to certain polynomial equations over Q are determined by the loops on the corresponding space of all solutions, or more precisely that the etale fundamental group is a fully faithful functor from certain anabelian schemes to profinite groups with Galois action. This is analogous to an equivalence between fixed points and homotopy fixed points for Galois actions. We will introduce the anabelian conjectures and their topological analogues, and relate certain nilpotent obstructions to the existence of rational points introduced by Jordan Ellenberg to higher cohomology operations. These cohomology operations encode when the existence of particular Galois extensions implies the existence of others, and are connected with the formality of the etale cochains. As a corollary one has that the order n Massey product <x,x,..x,1-x,x,...x> vanishes, where x denotes the image of x in k* under the Kummer map k* -> H^1(Gal(kbar/k), Z_l(1)).
''On prime solutions to linear and quadratic equations''
The classical results of Vinogradov and Hua establishes prime solutions of linear and diagonal quadratic equations in sufficiently many variables. In the linear case there has been a remarkable progress over the past
===<span id="Negrut"></span>March 8: Dan Negrut (UW-Mechanical Engineering)===
few years by introducing ideas from additive combinatorics. We will discuss
''Using Advanced Computing in Applied Dynamics: From the Dynamics of Granular Material to the Motion of the Mars Rover''
some of the key ideas, as well as their use to obtain multidimensional extensions
of the theorem of Green and Tao on arithmetic progressions in the primes. We
will also discuss some new results on prime solutions to non-diagonal quadratic
equations of sufficiently large rank. Most of this is joint work with B. Cook.
===Wed, Feb 8: Lan-Hsuan Huang (Columbia U)===
This talk outlines numerical solution methods and their implementation in a high performance computing-enabled software infrastructure aimed at supporting physics-based simulation for virtual design in Engineering. The applications of interest include granular dynamics, rigid/flexible many-body dynamics, and fluid-solid interaction problems. CHRONO, the software infrastructure developed as part of this ongoing effort, partitions the problem of interest into a number of sub-problems that are solved in parallel using Graphics Processing Unit (GPU) cards, or multi-core CPUs. The five components at the cornerstone of the vision that eventually led to CHRONO are: (a) modeling support for multi-physics phenomena; (b) scalable numerical methods for multi-GPU and multi-core hardware architectures; (c) methods for proximity computation and collision detection; (d) support for domain decomposition and load balance; and (e) tools for carrying out visualization and post-processing in a distributed manner. Several engineering applications will be used to demonstrate how these five components are implemented to leverage a heterogeneous CPU/GPU supercomputer available at the Wisconsin Applied Computing Center. The talk will conclude with a brief discussion of current trends in high performance computing and how they are poised to change the field of Computational Science in the near future.
''Positive mass theorems and scalar curvature problems''
More than 30 years ago, Schoen-Yau and later Witten made major breakthroughs in
===<span id="Fard"></span>March 15: Kurusch Ebrahimi Fard (Madrid) ===
proving the positive mass theorem. It has become one of the most important theorems in
''Spitzer-type identities in non-commutative Rota-Baxter algebras''
general relativity and differential geometry. In the first part of the talk, I will
introduce the positive mass theorem and present our recent work that extends the
classical three-dimensional results to higher dimensions. In the second part, I will
discuss how the observation from general relativity enables us to solve classical
geometric problems related to the scalar curvature.
===Fri, Feb 10: Melanie Wood (local)===
Gian-Carlo Rota suggested in one of his last articles the problem of developing the notion of integration algebra, complementary to the already existing theory of differential algebras. This idea was mainly motivated by Rota's deep appreciation for Chen's fundamental work on iterated integrals. As a starting point for such a theory Rota proposed to consider a particular operator identity first introduced in 1960 by the mathematician Glen Baxter. It was later coined Rota-Baxter identity. Examples range from algebras with a direct decomposition into subalgebras to algebras of functions equipped with the ordinary Riemann integral or its discrete analogs.
''Counting polynomials and motivic stabilization''
Rota-Baxter algebras feature a genuine factorization property. It is intimately related to linear fixpoint equations, such as those, for instance, appearing in the renormalization problem in perturbative quantum field theory. For arbitrary commutative Rota-Baxter algebras, proper exponential solutions of such fixpoint equations are described by what is known as the classical Spitzer identity. The similar classical Bohnenblust-Spitzer identity involves the symmetric group, and generalizes the simple observation that the n-fold iterated integral of a function is proportional to the n-fold product of the primitive of this function. Recently, the seminal Cartier-Rota theory of classical Spitzer-type identities has been generalized to noncommutative Rota-Baxter algebras. Pre-Lie algebras (also known as Vinberg or Gerstenhaber algebras) play a crucial role in this approach.
In this talk we will provide a short introduction to Rota-Baxter algebras, and review recent work on Spitzer-type identities.
This talk is based on joint work with Frederic Patras (CNRS, Nice, France) and Dominique Manchon (CNRS, Clermont-Ferrand, France).
We will begin with the problem of counting polynomials modulo a prime
===<span id="Jones"></span>April 5: John Jones (ASU) ===
p with a given pattern of root multiplicity. Here we will discover
''Number fields with prescribed ramification''
phenomena that point to vastly more general patterns in configuration
spaces of points. To see these patterns, one has to work in the ring
of motives--so we will describe this place where a space is equivalent
to the sum of its pieces. We will then be able to describe how these
patterns in the ring of motives are related to theorems in topology on
the homological stability of configuration spaces. This talk is based
on joint work with Ravi Vakil.
===Fri, Feb. 17: Milena Hering (UConn)===
Finite extensions of the rational numbers are basic objects of algebra and
''The moduli space of points on the projective line and quadratic Groebner bases''
number theory. We discuss the interaction of two basic invariants
associated to such a field, its set of ramified primes and its Galois
group. After describing the basic problem, we present computational and
theoretical results.
The ring of invariants for the action of the automorphism group of the
===<span id="Snowden"></span>April 12: Andrew Snowden (MIT) ===
projective line on the n-fold product of the projective line is a
classical object of study. The generators of this ring were determined
by Kempe in the 19th century. However, the ideal of relations has been
only understood recently in work of Howard, Millson, Snowden and
Vakil. They prove that the ideal of relations is generated by
quadratic equations using a degeneration to a toric variety.
I will report on joint work with Benjamin Howard where we further study
the toric varieties arising in this degeneration. As an application we show
that the second Veronese subring of the ring of invariants admits a presentation
whose ideal admits a quadratic Groebner basis.
===Fri, Feb. 24: Malabika Pramanik (UBC)===
In recent years, it has been realized that various large algebraic
''Analysis on Sparse Sets''
structures admitting large amounts of symmetry behave as if they were
small. For example, when symmetries are taken into account, polynomial
rings in infinitely many variables retain some of the favorable
properties of polynomial rings in finitely many variables. This
observation has been applied to obtain uniformity results in diverse
fields. I will describe some of the recent work in this area.
[http://www.math.wisc.edu/~street/pramanik-abstract.pdf Abstract]
===<span id="Hirsch"></span>April 19: Moe Hirsch (Honorary Fellow -- UW Madison) ===
===Fri, March 2: Guang Gong (Department of EE University of Waterloo)===
I will discuss various ideas about mathematics which many of its
''Polyphase Sequence Families with Low Correlation and DFT for Spreading OFDM''
practitioners seem to believe, including the Myth of Truth, the Myth of
Proof, the Myth of Certainty, and so forth.
Spreading sequences are employed for providing frequency diversity for an orthogonal
This talk is available as a pdf file:
frequency division multiplexing (OFDM) system and relieving its high peak power. In this talk, I will present the current status of the constructions of polyphase sequences and sequence families with low correlation and discrete Fourier transform (DFT) spectrum or large zero correlation zone and their applications in the spreading OFDM system.
I will first introduce different constructions for those polyphase sequence families. Then I present some of our recent work on power residue sequences, Sidelnikov sequences, the Weil representation sequences, and new sequences with low correlation and DFT which can be obtained by directly using the Weil bounds, together with the shift-and-add method for increasing the sizes of the sequence families with a graceful degrading correlation and DFT. Following this, I show our new findings on the Golay sequences with large zero autocorrelation zones. Finally, I will conclude my talk by exhibiting some unsolved problems along this line and some experimental results in the applications of spreading OFDM systems with these aforementioned sequences.
===Fri, March 23: Martin Lorenz (Temple University)===
<http://sprott.physics.wisc.edu/chaos-complexity/hirsch12.pdf>.
''Prime ideals and group actions in noncommutative algebra''
Having originated from number theory, the notion of a prime ideal has become central in many different branches of algebra. This talk will focus on the role of prime ideals in the representation theory of noncommutative algebras and the use of group actions as an efficient tool in organizing the spectrum of all prime ideals of a given algebra. In many cases, the group in question is an affine algebraic group and geometric methods are essential.
===<span id="Maulik"></span>May 3: Davesh Maulik (Columbia) ===
'''Plane curve singularities and knot invariants'''
Given a polynomial function $f(x,y)$ in two complex variables, with a critical point at the origin, by studying the nearby points, one can associate a knot (or link) inside $\mathbb{R}^3$. In nice situations, we can try to understand analytic properties of the singularity in terms of topological properties of this knot. For example, Milnor showed that the codimension of the ideal generated by the partial derivatives is related to the degree of the Alexander polynomial of the link. I will explain a broad generalization of this example, first conjectured by Oblomkov and Shende, which relates the geometry of the Hilbert scheme of the singularity to the HOMFLY polynomial of the link. As time permits, I will discuss further extensions (some known, some conjectural).
===Fri, April 20: Bob Guralnick (USC)===
===<span id="Gelbart"></span>May 10: Steve Gelbart (Weizmann Institute, Israel) ===
''Maps from the Generic Riemann surface to the Riemann sphere''
''"Zeta"''
Zariski, in his thesis, proved that any map from the generic Riemann surface
This will be an introductory (and incomplete) lecture on the story of "zeta". Starting with
of genus g > 6 could not be solvable. Using more sophisticated permutation
the fundamental work of Euler, Kummer and Riemann, and the telling work of Artin, Hecke and Hasse-Weil, we end up with the ever growing mysteries of Iwasawa, Langlands, and beyond.
group theory, we can prove a much stronger result: if f is an indecomposable
map of degree n from the generic Riemann surface of genus g > 3 to the
Riemann sphere, then the monodromy group of f is either the symmetric group
of degree n with n > (g+2)/2 or the alternating group of degree n with n > 2g.
We will discuss the main ideas used in solving this problem and some related
problems. We will also discuss the analog of this problem in positive
characteristic.
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, unless otherwise indicated.
Tues, Jan 15, B139 Lillian Pierce (Oxford) A new twist on the Carleson operator Denissov
Thurs, Jan 17, 2pm, 901VV Jonah Blasiak (Michigan) Positivity, complexity, and the Kronecker problem Terwilliger
Jan 25 Alexander Fish (Sydney) Product sets in amenable groups through a dynamical approach Gurevich
Feb 1 Tullia Dymarz (Madison) Quasisymmetric vs Bi-Lipschitz maps Street
Feb 8 Jean-Luc Thiffeault (Madison) pseudo-Anosovs with small or large dilatation Roch
Feb 15 Eric Lauga (UCSD) Optimization in fluid-based locomotion Spagnolie
Feb 22 Svitlana Mayboroda (University of Minnesota) Elliptic PDEs, analysis, and potential theory in irregular media Stovall
Mar 1 Kirsten Wickelgren (Harvard) Grothendieck's anabelian conjectures Street
March 8 Dan Negrut (UW - Mechanical Engineering) Using Advanced Computing in Applied Dynamics: From the Dynamics of Granular Material to the Motion of the Mars Rover Caldararu
March 15 Kurusch Ebrahimi Fard (Madrid) Spitzer-type identities in non-commutative Rota-Baxter algebras Gurvich
March 22 Neil O'Connell (Warwick) CANCELLED Timo Seppalainen
March 29 Spring Break No Colloquium
April 5 John Jones (ASU) Number fields with prescribed ramification Boston
April 12 Andrew Snowden (MIT) Large algebraic structures with large amounts of symmetry Street
April 19 Moe Hirsch (Honorary Fellow -- UW Madison) The Divine Madness: Mathematics, Myths and Metaphors Street
Tuesday, April 30, 4PM Laure Saint-Raymond (ENS)] Wasow Lecture: The Irreversibility in Gas Dynamics, a Matter of Probability Wasow Lecture
May 3 Davesh Maulik (Columbia) Plane curve singularities and knot invariants Street
May 10 Steve Gelbart (Weizmann Institute) "Zeta" Gurevich
Sept 14 Jordan Ellenberg (Madison) FI-modules: an introduction local
Sept 20, 4pm Persi Diaconis (Stanford) CANCELLED Jean-Luc
Sept 21 Joyce McLaughlin (RPI) Mathematics for Imaging Biomechanical Parameters in Dynamic Elastography WIMAW
Sept 28 Eric Marberg (MIT) Supercharacters for algebra groups: applications and extensions Isaacs
Oct 5 Howard Masur (Chicago) Winning games for badly approximable real numbers and billiards in polygons Dymarz
Wed, Oct 10, 4pm Bas Lemmens (Univ. of Kent) From hyperbolic geometry to nonlinear Perron-Frobenius theory LAA lecture
Thur, Oct 11, 4pm Irene Gamba (UT-Austin) Analytical and numerical issues associated with the dynamics of the non-linear Boltzmann equation WIMAW
Oct 12 Joachim Rosenthal (Univ. of Zurich) Linear Random Network Codes, a Grassmannian Approach Boston
Oct 19 Saverio E Spagnolie (Madison) Elastic slender bodies in fluids and slender bodies in elastic fluids Roch
Oct 26 Luke Oeding (UC Berkeley) The Trifocal Variety Gurevich
Tues, Oct 30 Andrew Majda (Courant) Data Driven Methods for Complex Turbulent Systems Smith, Stechmann
Thurs, Nov 1 Lenya Ryzhik (Chicago) The role of a drift in elliptic and parabolic equations Kiselev
Nov 2 Vladimir Sverak (Minnesota) On scale-invariant solutions of the Navier-Stokes equations Kiselev
Nov 9 Uri Andrews (Madison) Computable Stability Theory Lempp
Mon, Nov 12 Charles Smart (MIT) Regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity Feldman
Mon, Nov 26 Daniel Erman (U.Michigan) TBA Boston
Wed, Nov 28 Cameron Hill (Notre Dame) Genericity in Discrete Mathematics Lempp
Nov 30 Nam Le (Columbia) Boundary regularity for solutions to the linearized Monge-Ampere equations and applications Feldman
Dec 14 Amanda Folsom (Yale) q-series and quantum modular forms Ellenberg/WIMAW
Sept 14: Jordan Ellenberg (UW-Madison)
FI-modules: an introduction (joint work with T Church, B Farb, R Nagpal)
Sept 21: Joyce R. McLaughlin (Rensselaer Polytechnic Institute)
Mathematics for Imaging Biomechanical Parameters in Dynamic Elastography
Elastography, the imaging of biomechanical parameters in tissue, is motivated by the doctor's palpation exam where the doctor presses against the skin to detect stiff and abnormal tissue changes. In dynamic elastography experiments, the tissue is in motion with displacement amplitudes on the order of tens of microns. The displacement(s) are determined with sequences of MR data sets or sequences of ultrasound RF/IQ data sets within the tissue; and the data shows a dispersive effect indicating that tissue is viscoelastic. A choice of viscoelastic model must be made. For each model the biomechanical parameters satisfy a first order, linear or nonlinear, partial differential equation (system) with real or complex coefficients. We discuss the mathematical properties of these equations and how those properties lead to successful interpretation of the data, and to successful algorithms and images. We show biomechanical images of breast cancer and prostate cancer and compare those images to ultrasound images and histology slides with marked cancerous inclusions.
Sept 28: Eric Marberg (MIT)
Supercharacters for algebra groups: applications and extensions
The group U_n(F_q) of unipotent upper triangular matrices over a finite field belongs to the same list of fundamental examples as the symmetric or general linear groups. It comes as some surprise, therefore, that the group's irreducible characters are unknown, and considered in some sense unknowable. In order to tackle problems normally requiring knowledge of a group's irreducible characters, Diaconis and Isaacs developed the notion of the supercharacters of an algebra group, generalizing work of Andre and Yan. Algebra groups form a well-behaved class of p-groups including U_n(F_q) as a prototypical example, and supercharacters are certain reducible characters which form a useful approximation to the set of irreducible characters. In this talk I will survey several equivalent definitions of the supercharacters of an algebra group, and discuss some applications and extensions of these approaches. On one end of things, I intend briefly to introduce the recent discovery of how certain representation theoretic operations on the supercharacters of U_n(F_q) naturally define a Hopf algebra structure, which has been studied under a different name by combinatorialists. In another direction, I will explain how one can view the supercharacters of an algebra group as the first step in a more general reduction process, which can be used to shed light on some mysterious properties of U_n(F_q).
Oct 04 Howard Masur (Chicago)
Winning games for badly approximable real numbers and billiards in polygons
Wolfgang Schmidt invented the notion of a winning subset of Euclidean space in a game between two players. Winning sets have nice properties such as full Hausdorff dimension. The basic example of a winning set considered by Schmidt are those reals badly approximated by rationals. An equivalent formulation by Artin is that badly approximable reals correspond to geodesics that stay in a bounded set in the modular curve. There is also an equivalent formulation in terms of sets of directions for the linear flow on a square torus or equivalently, billiard trajectories in a square. In joint work with Yitwah Cheung and Jon Chaika we extend this notion of winning to flows on flat surfaces of higher genus with applications to billiards in rational angled polygons. My intention in this talk is to give the background on the Schmidt game, describe the classical results before introducing the more recent work.
Wed, Oct 10: Bas Lemmens (University of Kent)
From hyperbolic geometry to nonlinear Perron-Frobenius theory
In a letter to Klein Hilbert remarked that the logarithm of the cross-ratio is a metric on any open, bounded, convex set in Euclidean space. These metric spaces are nowadays called Hilbert geometries. They are a natural non-Riemannian generalization of Klein's model of the hyperbolic plane, and play a role in the solution of Hilbert's fourth problem.
In the nineteen fifties Garrett Birkhoff and Hans Samelson independently discovered that one can use Hilbert's metric and the contraction mapping principle to prove the existence and uniqueness of a positive eigenvector for a variety of linear operators that leave a closed cone in a Banach space invariant. Their results are a direct extension of the classical Perron-Frobenius theorem concerning the eigenvectors and eigenvalues of nonnegative matrices. In the past decades this idea has been further developed and resulted in strikingly detailed nonlinear extensions of the Perron-Frobenius theorem. In this talk I will discuss the synergy between metric geometry and (nonlinear) operator theory and some of the recent results and open problems in this area.
Thur, Oct 11: Irene Gamba (University of Texas-Austin)
Analytical and numerical issues associated with the dynamics of the non-linear Boltzmann equation
Oct 12: Joachim Rosenthal (Univ of Zurich)
Linear Random Network Codes, a Grassmannian Approach
Elastography, the imaging of biomechanical parameters in A novel framework for random network coding has been introduced by Koetter and Kschischang. In this framework information is encoded in subspaces of a given ambient space over a finite field. A natural metric is introduced where two subspaces are `close to each other' as soon as their dimension of intersection is large. This framework poses the challenge to come up with new codes with optimal or near optimal distance and to develop efficient decoding algorithms.
In a first part of the talk we will provide a survey. In a second part of the talk we report on progress constructing spread codes and orbit codes. The decoding problem of orbit codes can be interpreted as a problem in Schubert calculus over a finite field.
Fri, Oct 19: Saverio E. Spagnolie (Madison)
Elastic slender bodies in fluids and slender bodies in elastic fluids
Fri, Oct 26: Luke Oeding (UC Berkeley)
The Trifocal Variety
Abstract: In Computer Vision one considers many cameras looking at the same scene. From this setup many interesting geometric and algebraic questions arise. In this talk we will focus on the case of 3 cameras and study the so called trifocal tensors. Trifocal tensors are constructed from a bilinear map defined using the trifocal setup. A natural question is, given a particular tensor, how can one determine if it is a trifocal tensor? This question can be answered by finding implicit defining equations for the trifocal variety. From an algebraic standpoint, it is also interesting to know the minimal generators of the defining ideal of the trifocal variety.
Tues, Oct 30: Andrew Majda (Courant)
Data Driven Methods for Complex Turbulent Systems
An important contemporary research topic is the development of physics constrained data driven methods for complex, large-dimensional turbulent systems such as the equations for climate change science. Three new approaches to various aspects of this topic are emphasized here. 1) The systematic development of physics constrained quadratic regression models with memory for low-frequency components of complex systems; 2) Novel dynamic stochastic superresolution algorithms for real time filtering of turbulent systems; 3) New nonlinear Laplacian spectral analysis (NLSA) for large-dimensional time series which capture both intermittency and low-frequency variability unlike conventional EOF or principal component analysis. This is joint work with John Harlim (1,2), Michal Branicki (2), and Dimitri Giannakis (3).
Thurs, Nov 1: Lenya Ryzhik (Chicago)
The role of a drift in elliptic and parabolic equations
The first order partial differential equations are closely connected to the underlying characteristic ODEs. The second order elliptic and parabolic equations are as closely connected to the Brownian motion and more general diffusions with a drift. From a variety of points of view, the drift does not really matter - all diffusions look more or less similar. As a caveat to this reasonable line of thought, I will describe a menagerie of problems: linear and nonlinear, steady and time-dependent, compressible and incompressible, where the drift and diffusion confirm the Mayakovsky thesis "Woe to one alone!" Together, they lead to enhanced mixing and improved regularity that are impossible for each one of them to attain.
Nov 2: Vladimir Sverak (Minnesota)
On scale-invariant solutions of the Navier-Stokes equations
Nov 9: Uri Andrews (Madison)
Computable Stability Theory
Stability theory attempts to classify the underlying structure of mathematical objects. The goal of computable mathematics is to understand when mathematical objects or constructions can be demonstrated computably. I'll talk about the relationship between underlying structure and computation of mathematical objects.
Monday, Nov 12: Charles Smart (MIT)
Regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity
I will discuss regularity of fully nonlinear elliptic equations when the usual uniform upper bound on the ellipticity is replaced by bound on its $L^d$ norm, where $d$ is the dimension of the ambient space. Our estimates refine the classical theory and require several new ideas that we believe are of independent interest. As an application, we prove homogenization for a class of stationary ergodic strictly elliptic equations.
Wednesday, Nov 28: Cameron Hill (Notre Dame)
Genericity in Discrete Mathematics
I will discuss the intriguing role that ``generic objects can play in finitary discrete mathematics. Using the framework of model theory, we will see that there are at least two reasonable but very different notions of genericity relative to a class of finite structures (like a class of finite graphs or a class of finite groups). Through discussions of zero-one laws and structural Ramsey theory, I will try to illustrate the importance of generically-categorical classes -- those for which the various notions of genericity coincide. To finish, I will state some results that follow from the assumption of generic-categoricity.
Nov 30: Nam Le (Columbia)
Boundary regularity for solutions to the linearized Monge-Ampere equations and applications
In this talk, I will discuss boundary regularity of solutions to the linearized Monge-Ampere equations, and their applications to nonlinear, fourth order, geometric Partial Differential Equations (PDE). First, I will present my regularity results in joint work with O. Savin and T. Nguyen including: boundary Holder gradient estimates and global $C^{1,\alpha}$ estimates, global Holder estimates and global $W^{2,p}$ estimates. Then, I will describe applications of the above regularity results to several nonlinear, fourth order, geometric PDE such as: global second derivative estimates for the second boundary value problem of the prescribed affine mean curvature and Abreu's equations; and global regularity for minimizers having prescribed determinant of certain convex functionals motivated by the Mabuchi functional in complex geometry.
Dec 14: Amanda Folsom (Yale)
q-series and quantum modular forms
Tues, Jan 15, B139: Lillian Pierce (Oxford)
A new twist on the Carleson operator
Thurs, Jan 17, 2pm, 901VV: Jonah Blasiak (Michigan)
Positivity, complexity, and the Kronecker problem
Positivity problems in algebraic combinatorics ask to find positive combinatorial formulae for nonnegative quantities arising in geometry and representation theory like cohomological dimensions and dimensions of algebras and their irreducible representations. A famous open positivity problem in representation theory is the Kronecker problem, which asks for a positive combinatorial formula for decomposing tensor products of irreducible representations of the symmetric group. We will begin with a general discussion of positivity problems and an intriguing new motivation for these problems from complexity theory. We will then present our solution to a special case of the Kronecker problem that substantially improves on previous results.
Fri, Jan 25: Alexander Fish (Sydney)
Product sets in amenable groups through a dynamical approach
We will describe a new correspondence between product sets in a countable amenable group and product sets in compact groups. This approach enables us to obtain quantitative results concerning product sets in amenable groups. The talk is for general mathematical audience. (based on a joint work with M. Bjorklund)
Fri, Feb 1: Tullia Dymarz (Madison)
Quasisymmetric vs Bi-Lipschitz maps
On a metric space, there are various classes of functions which respect aspects of the metric space structure. One of the most basic classes is the bi-Lipschitz maps (Lipschitz maps whose inverses are also Lipschitz). Another possibly much larger class consists of the so-called quasisymmetric maps (these are closely related quasiconformal maps). On both Euclidean space and the p-adics, there are many quasisymmetric maps which are not bi-Lipschitz. However, on the product of Euclidean space with the p-adics, we show that all quasisymmetric maps are bi-Lipschitz. Furthermore, our proof does not use any direct analysis but instead uses coarse topology and results from negative curvature.
Feb 8: Jean-Luc Thiffeault (Madison)
pseudo-Anosovs with small or large dilatation
Homeomorphisms of a surface to itself can be classified using the well-known Thurston-Nielsen theorem. The most interesting topological class contains pseudo-Anosov mappings: they stabilize a pair of transverse singular foliations with a finite number of singularities. These foliations are called unstable and stable, and are respectively expanded and compressed by an algebraic constant called the dilatation. Characterizing the possible values of these dilatations for a given suface is an open problem. Here I discuss a method to find the minimum value of the dilatation on closed surfaces of a given genus, for the special case where the foliations are orientable. I will then address the opposite problem: how to find pseudo-Anosovs with large dilatations. Unlike the minimizer problem, this is not well-defined -- the answer is infinity -- unless we add a constraint. Constraints can arise from practical optimization problems in engineering, and I will show some optimal solutions that can be incorporated in devices called taffy pullers. (This is joint work with Erwan Lanneau and Matt Finn.)
Feb 15: Eric Lauga (UCSD)
Optimization in fluid-based locomotion
Feb 22: Svitlana Mayboroda (University of Minnesota)
Elliptic PDEs, analysis, and potential theory in irregular media
Elliptic boundary value problems are well-understood in the case when the boundary, the data, and the coefficients exhibit smoothness. However, perfectly uniform smooth systems do not exist in nature, and every real object inadvertently possesses irregularities (a sharp edge of the boundary, an abrupt change of the medium, a defect of the construction).
The analysis of general non-smooth elliptic PDEs gives rise to decisively new challenges: possible failure of maximal principle and positivity, breakdown of boundary regularity, lack of the classical L^2 estimates, to mention just a few. Further progress builds on an involved blend of harmonic analysis, potential theory and geometric measure theory techniques. In this talk we are going to discuss some highlights of the history, conjectures, paradoxes, and recent discoveries such as the higher-order Wiener criterion and maximum principle for higher order PDEs, solvability of rough elliptic boundary problems, as well as an intriguing phenomenon of localization of eigenfunctions -- within and beyond the limits of the famous Anderson localization.
Parts of the talk are based on joint work with S. Hofmann, M. Filoche, C. Kenig, V. Maz'ya, and J. Pipher.
March 1: Kirsten Wickelgren (Harvard)
Grothendieck's anabelian conjectures
March 8: Dan Negrut (UW-Mechanical Engineering)
Using Advanced Computing in Applied Dynamics: From the Dynamics of Granular Material to the Motion of the Mars Rover
March 15: Kurusch Ebrahimi Fard (Madrid)
Spitzer-type identities in non-commutative Rota-Baxter algebras
Gian-Carlo Rota suggested in one of his last articles the problem of developing the notion of integration algebra, complementary to the already existing theory of differential algebras. This idea was mainly motivated by Rota's deep appreciation for Chen's fundamental work on iterated integrals. As a starting point for such a theory Rota proposed to consider a particular operator identity first introduced in 1960 by the mathematician Glen Baxter. It was later coined Rota-Baxter identity. Examples range from algebras with a direct decomposition into subalgebras to algebras of functions equipped with the ordinary Riemann integral or its discrete analogs. Rota-Baxter algebras feature a genuine factorization property. It is intimately related to linear fixpoint equations, such as those, for instance, appearing in the renormalization problem in perturbative quantum field theory. For arbitrary commutative Rota-Baxter algebras, proper exponential solutions of such fixpoint equations are described by what is known as the classical Spitzer identity. The similar classical Bohnenblust-Spitzer identity involves the symmetric group, and generalizes the simple observation that the n-fold iterated integral of a function is proportional to the n-fold product of the primitive of this function. Recently, the seminal Cartier-Rota theory of classical Spitzer-type identities has been generalized to noncommutative Rota-Baxter algebras. Pre-Lie algebras (also known as Vinberg or Gerstenhaber algebras) play a crucial role in this approach. In this talk we will provide a short introduction to Rota-Baxter algebras, and review recent work on Spitzer-type identities. This talk is based on joint work with Frederic Patras (CNRS, Nice, France) and Dominique Manchon (CNRS, Clermont-Ferrand, France).
April 5: John Jones (ASU)
Number fields with prescribed ramification
Finite extensions of the rational numbers are basic objects of algebra and number theory. We discuss the interaction of two basic invariants associated to such a field, its set of ramified primes and its Galois group. After describing the basic problem, we present computational and theoretical results.
April 12: Andrew Snowden (MIT)
In recent years, it has been realized that various large algebraic structures admitting large amounts of symmetry behave as if they were small. For example, when symmetries are taken into account, polynomial rings in infinitely many variables retain some of the favorable properties of polynomial rings in finitely many variables. This observation has been applied to obtain uniformity results in diverse fields. I will describe some of the recent work in this area.
April 19: Moe Hirsch (Honorary Fellow -- UW Madison)
I will discuss various ideas about mathematics which many of its practitioners seem to believe, including the Myth of Truth, the Myth of Proof, the Myth of Certainty, and so forth.
May 3: Davesh Maulik (Columbia)
Plane curve singularities and knot invariants
May 10: Steve Gelbart (Weizmann Institute, Israel)
"Zeta"
This will be an introductory (and incomplete) lecture on the story of "zeta". Starting with the fundamental work of Euler, Kummer and Riemann, and the telling work of Artin, Hecke and Hasse-Weil, we end up with the ever growing mysteries of Iwasawa, Langlands, and beyond.
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