﻿ Skoltech Center for Advanced Studies | The purpose of this meeting is mainly to develop existing and to start new collaborations among people from Moscow and Pisa in different branches of mathematics, including representation theory, algebraic geometry, probability and analysis, low dimensional topology, dynamical systems.
The Colloquium will take place from October 1st to October 5th in HSE and Skoltech (Moscow)

Also, the main emphasis is supposed to be given to the communication among the participants and ideas exchange more than to a formal presentation of the obtained results, so if you would like to come but do not want to give a talk you are also welcome.

If our plan will work smoothly and successful, we are planning to run the second leg of the Colloquium in spring 2019 at the De Giorgi Mathematics Research Center in Pisa.

The Colloquium will be partially financed by SkolTech and Higher School of Economics.

Organizers:
– Higher School of Economics
– Skoltech

Organizing Committee:

– Sasha Skripchenko (HSE & Skoltech)
– Stefano Marmi (Scuola Normale Superiore, Pisa)

### Program

[ last update: October 2, 2018 ]

Schedule for Monday, October 1, 2018 / Representation theory /
Skoltech, 3 Nobel Str., hall 426 (cohort space)

10:45 – 11:45 / Evgeny Feigin / Semi-infinite flag varieties
We will describe basic properties of the semi-infinite flag varieties. We will show that they share many nice features of the classical flag varieties for simple Lie groups. Geometric, algebraic and combinatorial parts of the story will show up in our talk

12:15 – 13:15 / Jacopo Gandini / The Bruhat order on Hermitian symmetric varieties and on abelian ideals
Given a Hermitian symmetric variety for a simple algebraic group G, I will describe its decomposition into B-orbits (where B is a Borel subgroup of G), and the corresponding partial order defined by inclusions of orbit closures. A similar related problem is to describe the orbits of B in the abelian ideals of the Lie algebra of B, together with the corresponding partial order. I will explain how the two problems are connected, and how to solve them in terms of the combinatorics of the Weyl group of G. Based on joint works with A. Maffei, P. Moseneder-Frajria and P. Papi

13-15 – 14-30 / Lunch

14-30 – 15-30 / Michael Finkelberg / Microlocal cells in affine Weyl groups
The classical construction due to Steinberg, Spaltenstein and Springer, gives a partition of the Weyl group W of a simple Lie algebra into the cells numbered by the nilpotent orbits. Lusztig proposed two similar constructions giving a partition of the affine Weyl group into cells numbered by the conjugacy classes of W. We prove the equivalence of the two constructions. This is a joint work with D.Kazhdan and Y.Varshavsky
(joint session with the seminar of Skoltech Center for Advanced Study)

Schedule for Tuesday, October 2, 2018 / Dynamical systems /
Skoltech, 3 Nobel Str., hall 303

11.30 – 12.30 / Alexander Bufetov / The Hoelder property for the spectral measures of translation flows
The main result of the talk is a Hoelder estimate on the spectral measure for a generic translation flow in genus two. A new cocycle is introduced that governs the spectral asymptotics. Joint work with Boris Solomyak

13.00 – 14.00 / Paolo Giulietti / Parabolic dynamics and anisotropic Banach spaces
In recent years, the study of parabolic flows has been sharpened by a functional analytic approach grounded on anisotropic spaces. I will show a general relation between the distributions appearing in the study of ergodic averages of parabolic flows and the eigendistributions arising from transfer operators of hyperbolic nature

14.00 – 15.30 / Lunch

15.30 – 16.30 / Stefano Marmi / Diofantine condition for interval exchange transformations /

17.00 – 18.00 / Vladlen Timorin / Slices of the parameter space of cubic polynomials
We study slices of the parameter space of cubic polynomials, up to affine conjugacy, given by a fixed value of the multiplier at a non-repelling fixed point. In particular, we study the location of the MAIN CUBIOID in this parameter space. The MAIN CUBIOID consists of affine conjugacy classes of complex cubic polynomials that have certain dynamical properties generalizing those of quadratic polynomials, whose parameters are in the (filled) main cardioid. Based on a joint work with A. Blokh, L. Oversteegen and R. Ptacek

Schedule for Wednesday, October 3, 2018 / Low-dimensional topology /
HSE, Faculty of Mathematics, 6 Usacheva str.

11.00 – 12.00 / Sergey Lando / Algebraic combinatorial structures underlying finite order
room 306
Finite order invariants of knots introduced by V. Vassiliev around 1990 can be expressed in terms of weight systems. These are functions on chord diagrams satisfying so-called 4-term relations. The vector space formed by chord diagrams modulo the 4-term relations can be endowed with a structure of a commutative cocommutative Hopf algebra graded by the number of chords. This Hopf algebra has a graph counterpart, which is the Hopf algebra of graphs modulo 4-term relations.
Vassiliev’s 4-term relations have a natural extension to finite order invariants of links. Similarly to the case of knots, a finite order knot invariant of links defines a function on embedded graphs with arbitrarily many vertices (as opposed to chord diagrams, which are embedded graphs with a single vertex). The talk is devoted to the description of the Hopf algebra of binary delta-matroids associated to embedded graphs in a way similar to the one associating intersection graph to a chord diagram. The combinatorial notion of delta-matroid was introduced by A. Bouch´et around 1980. Binary delta-matroids span a graded commutative cocommutative Hopf algebra, which, in turn, admits factorization modulo 4-term relations

12.15 – 13. 15 / Carlo Petronio / On the Hurwitz existence problem / room 213
To a branched cover between closed surfaces one can associate a combinatorial datum, given by the genera and orientability of the source and target surfaces, the total degree, the number of branching points, and the partitions of the total degree given by the local degrees at the preimages of the branching points. This datum must satisfy some necessary conditions, notably the Riemann-Hurwitz formula, and some other ones in the non-orientable
case. An old and still unsettled problem asks for which combinatorial data these necessary conditions are actually sufficient for the existence of a corresponding branched cover.
Starting from a 3-dimensional motivation for picking up this problem, I will report on recent and less recent progress on it, involving several different techniques

13.15 – 14.30 / Lunch

14.30 – 15.30 / Ivan Dynnikov / Exchange classes of rectangular diagrams, Legendrian knots, and the knot symmetry group / room 306
Rectangular diagrams are a particularly nice way to represent knots and links in the three-space. The crucial property of this presentation is the existence of a monotonic simplification algorithm for recognizing the unknot [I.D., 2006]. The present research (joint with M.Prasolov) is motivated by an attempt to extend the monotonic simplification procedure to arbitrary knot types.
It turnes out that rectangular diagrams have a very strong connection with Legendrian knots. Namely, every diagram carries two Legendrian knots of the same topological type, one with respect to the standard contact structure and the other with respect to its mirror image. Exchange moves preserve both Legendrian types, whereas stabilizations and
destabilizations alters one of them and preserve the other, depending on the type of the (de)stabilization.
We reduce the classification of rectangular diagrams up to exchange moves to the classification of Legendrian knots up to Legendrian isotopy (and of their symmetries). In particular, the classification of rectangular diagrams that do not admit any simplification by elementary moves is reduced to the classification of Legendrian knots that do not admit a destabilization

15.45 – 16.45 / Alexander Gaifullin / On infinitely generated homology of Torelli groups / room 213
The Torelli group of an oriented closed surface of genus g is the kernel of the action of the mapping class group of the surface on the first integral homology group of it.
We prove that the k-th integral homology group of the Torelli group of genus g contains a free Abelian subgroup of infinite rank, provided that g>2 and 2g-4<k<3g-5. Earlier the same property was known only for k=3g-5 (Bestvina, Bux, Margalit, 2007) and in the special case g=k=3 (Johnson, Millson, 1992).
As a consequence, we obtain that a classifying space for the Torelli group cannot have a finite (2g-3)-skeleton. The proofs are based on the study of the Cartan-Leray spectral sequence for the action of the Torelli group on the complex of cycles constructed by Bestvina, Bux, and Margalit

Schedule for Thursday, October 4, 2018 / Analysis and Probability /
HSE, Faculty of Mathematics, 6 Usacheva str.

09.45 – 10.45 / Franco Flandoli / Probability and PDEs: some examples / room 212
Three general themes where Probability meets PDEs will be touched:
i) existence and/or uniqueness results for almost every initial condition
ii) regularization by noise
iii) selection by zero-noise limit.
Examples wil be given, taken mainly from fluid mechanics. The many open problems in all these directions and also on intrinsic stochasticity will be outlined

11.00 – 12.00 / Alexander Kolesnikov / Logarithmic Minkowski problem and optimal transportation / room 212
The logarithmic Minkowski problem is a variant of the classical Minkowski problem. Given a finite measure m on the unit sphere it asks for existence and uniqieness of a convex body K such that the associated cone measure coincides with m. The existence was recently proved in the class of even
measures by Boroczky, Lutwak, Yang and Zhang, whereas the uniqueness is still an open problem,
related to another open conjecture: log Brunn-Minkowski inequality. We show that the associated PDE for the log-Minkowski problem is an equation of the Kaehler-Einstein type and propose a variational functional which minimal points are solutions to the problem. This functional contains a mass transportational functional associated to a special cost function on the sphere. This fact allows to attack this problem using the tools of the optimal transportation theory. In particular, we prove a new transportation Talagrand-type inequality

12.00 – 13.20 / Lunch

13.20 – 14.20 / Matteo Novaga / Almost minimal configurations for the Heitmann-Radin energy / room 212
We describe low energy configurations for the Heitmann-Radin sticky discs functional, in the limit of diverging number of discs. We show a compactness result for configurations whose energy scales like the perimeter, after suitable renormalization. In this case, the empirical measures converge to a set of finite perimeter, while a microscopic variable, representing the orientation of the underlying lattice, converges to a locally constant function of bounded variation

14.30 – 15.30 / Mauro Mariani / Gamma-expansion of finite-dimensional Dirichlet energies / room 212
I will discuss the expansion by Gamma-convergence of a classical object: Dirichlet forms in a so-called low temperature limit. The talk will focus on a case wherea full expansion can be characterized explicitly. The result can be regarded as a variational counterpart of classical and recent estimates on the low-lying spectrum (and the associated eigenfunctions) of Schroedinger’s operator’s. Many of the tools used had their origin in the Pisa’s Pisa mathematical community

Schedule for Friday, October 5, 2018 / Algebraic geometry /
HSE, Faculty of Mathematics, 6 Usacheva str.

13.00 – 14.00 / Alexander Kuznetsov / Semiorthogonal decompositions for singular surfaces / room 211
I will describe semiorthogonal decompositions for a wide class of singular rational surfaces with components equivalent to derived categories of local finite-dimensional algebras

14.00 – 15.30 / Lunch

15.30 – 16.30 / Vadim Vologodsky / Finite subgroups of algebraic groups and the group of birational automorphisms of a surface / room 108
This is a joint work with Constantin Shramov. We prove that, for a perfect field K that contains all roots of unity and an anisotropic algebraic group G over K, finite subgroups of G(K) have bounded order. As an application we prove that the group of birational automorphisms of a pointless surface over such field (of characteristic not equal to 2) has bounded finite subgroups

17.00 – 18.00 / Angelo Vistoli / Essential dimension of finite groups / room 306
Essential dimension is a fundamental way of measuring the complexity of a finite group G. It measures how many independent parameters are needed to describe all Galois extensions E/K with group G, where K is an extension of a fixed base field k. For example, if k contains a primitive n-th root of 1 and G is a cyclic group of order n, all Galois extensions E/K with Galois group G is obtained by taking an n-th root of an element of K; this means that the essential dimension of a cyclic group of order n is 1.
I will review some of what is known about essential dimension of finite groups and the main techniques that are used in the subject, including some very recent applications of birational geometry

(joint session with the seminar of HSE Laboratory of Algebraic Geometry)
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