Five embeddings of one simple group
Transactions of the American Mathematical Society. 2014. Vol. 366. No. 3. P. 1289-1331.
Ivan Cheltsov, Constantin Shramov.
We propose a new method to study birational maps between Fano varieties based on multiplier ideal sheaves. Using this method, we prove equivariant birational rigidity of four Fano threefolds acted on by the group A6. As an application, we obtain that the Cremona group of rank 3 has at least five non-conjugate subgroups isomorphic to A6.
Research target: Mathematics
Priority areas: mathematics
Keywords: Cremona groupBirational rigidity
Tori in the Cremona groups
Popov V. Izvestiya. Mathematics. 2013. Vol. 77. No. 4. P. 742-771.
We classify up to conjugacy the subgroups of certain types in the full, affine, and special affine Cremona groups. We prove that the normalizers of these subgroups are algebraic. As an application, we obtain new results in the linearization problem by generalizing Bia{\l}ynicki-Birula's results of 1966--67 to disconnected groups. We prove fusion theorems for n-dimensional tori in the affine and in special affine Cremona groups of rank n and introduce and discuss the notions of Jordan decomposition and torsion prime numbers for the Cremona groups.
Rationality of the quotient of ℙ2 by finite group of automorphisms over arbitrary field of characteristic zero
Andrey S. Trepalin. Central European Journal of Mathematics. 2014. Vol. 12. No. 2. P. 229-239.
Let $\bbk$ be a field of characteristic zero and $G$ be a finite group of automorphisms of projective plane over $\bbk$. Castelnuovo's criterion implies that the quotient of projective plane by $G$ is rational if the field $\bbk$ is algebraically closed. In this paper we prove that $\mathbb{P}^2_{\bbk} / G$ is rational for an arbitrary field $\bbk$ of characteristic zero.
Added: Dec 3, 2013
On stable conjugacy of finite subgroups of the plane Cremona group, II
Yuri Prokhorov. arxiv.org. math. Cornell University, 2013
We prove that, except for a few cases, stable linearizability of finite subgroups of the plane Cremona group implies linearizability.
Jordan groups and automorphism groups of algebraic varieties
Vladimir L. Popov. arxiv.org. math. Cornell University, 2013. No. 1307.5522.
This is an expanded version of my talk at the workshop ``Groups of Automorphisms in Birational and Affine Geometry'', October 29–November 3, 2012, Levico Terme, Italy. The first section is focused on Jordan groups in abstract setting, the second on that in the settings of automorphisms groups and groups of birational self-maps of algebraic varieties. The appendix is an expanded version of my notes on open problems posted on the site of this workshop. It contains formulations of some open problems and the relevant comments.
Added: Jul 21, 2013
Popov V. arxiv.org. math. Cornell University, 2012. No. arXiv:1207.5205v3.
We classify up to conjugacy the subgroups of certain types in the full, in the affine, and in the special affine Cremona groups. We prove that the normalizers of these subgroups are algebraic. As an application, we obtain new results in the Linearization Problem generalizing to disconnected groups Bialynicki-Birula's results of 1966-67. We prove ``fusion theorems'' for n-dimensional tori in the affine and in the special affine Cremona groups of rank n. In the final section we introduce and discuss the notions of Jordan decomposition and torsion prime numbers for the Cremona groups.
Added: Jan 9, 2013
"Del Pezzo fibrations", Appendix to "Five embeddings of one simple group" by I. Cheltsov and C. Shramov
Prokhorov Y. Transactions of the American Mathematical Society. 2014. Vol. 366. No. 3. P. 1289-1331.
Added: Apr 9, 2014
Cremona Groups and the Icosahedron
Cheltsov I., Shramov K. CRC Press, 2015.
Cremona Groups and the Icosahedron focuses on the Cremona groups of ranks 2 and 3 and describes the beautiful appearances of the icosahedral group A5 in them. The book surveys known facts about surfaces with an action of A5, explores A5-equivariant geometry of the quintic del Pezzo threefold V5, and gives a proof of its A5-birational rigidity.
The authors explicitly describe many interesting A5-invariant subvarieties of V5, including A5-orbits, low-degree curves, invariant anticanonical K3 surfaces, and a mildly singular surface of general type that is a degree five cover of the diagonal Clebsch cubic surface. They also present two birational selfmaps of V5 that commute with A5-action and use them to determine the whole group of A5-birational automorphisms. As a result of this study, they produce three non-conjugate icosahedral subgroups in the Cremona group of rank 3, one of them arising from the threefold V5.
This book presents up-to-date tools for studying birational geometry of higher-dimensional varieties. In particular, it provides readers with a deep understanding of the biregular and birational geometry of V5.
Введение в математическую статистику
Ивченко Г. И., Медведев Ю. И. М.: ЛКИ, 2010.
p-elementary subgroups of the Cremona group of rank 3
Prokhorov Y. In bk.: Classification of Algebraic Varieties. Zürich: European Mathematical Society Publishing house, 2010. P. 327-338.
For the subgroups of the Cremona group $\mathrm{Cr}_3(\mathbb C)$ having the form $(\boldsymbol{\mu}_p)^s$, where $p$ is prime, we obtain an upper bound for $s$. Our bound is sharp if $p\ge 17$.
Model for organizing cargo transportation with an initial station of departure and a final station of cargo distribution
Khachatryan N., Akopov A. S. Business Informatics. 2017. No. 1(39). P. 25-35.
A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traffic is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the final node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a finite-dimensional system of differential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the "correct" extension of solutions of a system of differential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.
Nullstellensatz over quasi-fields
Trushin D. Russian Mathematical Surveys. 2010. Vol. 65. No. 1. P. 186-187.
Энатская Н. Ю., Хакимуллин Е. Р. М.: МИЭМ, 2011.
Деловой климат в оптовой торговле во II квартале 2014 года и ожидания на III квартал
Лола И. С., Остапкович Г. В. Современная торговля. 2014. № 10.
Деловой климат в оптовой торговле в I квартале 2012 года
Лола И. С., Лукашина Ж. Е., Пахомова О. П. М.: НИУ ВШЭ, 2012.
Laminations from the Main Cubioid
Timorin V., Blokh A., Oversteegen L. et al. arxiv.org. math. Cornell University, 2013. No. 1305.5788.
According to a recent paper \cite{bopt13}, polynomials from the closure $\ol{\phd}_3$ of the {\em Principal Hyperbolic Domain} ${\rm PHD}_3$ of the cubic connectedness locus have a few specific properties. The family $\cu$ of all polynomials with these properties is called the \emph{Main Cubioid}. In this paper we describe the set $\cu^c$ of laminations which can be associated to polynomials from $\cu$.
Entropy and the Shannon-McMillan-Breiman theorem for beta random matrix ensembles
Bufetov A. I., Mkrtchyan S., Scherbina M. et al. arxiv.org. math. Cornell University, 2013. No. 1301.0342.
Новые информационные технологии. Тезисы докладов XVIII международной студенческой конференции-школы-семинара
М.: МИЭМ, 2010.
Метод параметрикса для диффузий и цепей Маркова
Конаков В. Д. STI. WP BRP. Издательство попечительского совета механико-математического факультета МГУ, 2012. № 2012.
Is the function field of a reductive Lie algebra purely transcendental over the field of invariants for the adjoint action?
Colliot-Thélène J., Kunyavskiĭ B., Vladimir L. Popov et al. Compositio Mathematica. 2011. Vol. 147. No. 2. P. 428-466.
Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.
Absolutely convergent Fourier series. An improvement of the Beurling-Helson theorem
Vladimir Lebedev. arxiv.org. math. Cornell University, 2011. No. 1112.4892v1.
We obtain a partial solution of the problem on the growth of the norms of exponential functions with a continuous phase in the Wiener algebra. The problem was posed by J.-P. Kahane at the International Congress of Mathematicians in Stockholm in 1962. He conjectured that (for a nonlinear phase) one can not achieve the growth slower than the logarithm of the frequency. Though the conjecture is still not confirmed, the author obtained first nontrivial results.
Обоснование адиабатического предела для гиперболических уравнений Гинзбурга-Ландау
Пальвелев Р., Сергеев А. Г. Труды Математического института им. В.А. Стеклова РАН. 2012. Т. 277. С. 199-214.
Hypercommutative operad as a homotopy quotient of BV
Khoroshkin A., Markaryan N. S., Shadrin S. arxiv.org. math. Cornell University, 2012. No. 1206.3749.
We give an explicit formula for a quasi-isomorphism between the operads Hycomm (the homology of the moduli space of stable genus 0 curves) and BV/Δ (the homotopy quotient of Batalin-Vilkovisky operad by the BV-operator). In other words we derive an equivalence of Hycomm-algebras and BV-algebras enhanced with a homotopy that trivializes the BV-operator. These formulas are given in terms of the Givental graphs, and are proved in two different ways. One proof uses the Givental group action, and the other proof goes through a chain of explicit formulas on resolutions of Hycomm and BV. The second approach gives, in particular, a homological explanation of the Givental group action on Hycomm-algebras.
Cross-sections, quotients, and representation rings of semisimple algebraic groups
V. L. Popov. Transformation Groups. 2011. Vol. 16. No. 3. P. 827-856.
Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.
Новые информационные технологии. Тезисы докладов XIX международной студенческой конференции-школы-семинара