# Laboratory Colloquium

March 23 2018, room 306

17:00 Valery Lunts (Indiana University, LMS HSE). On three notions of a dimension of triangulated categories.

I will report on a project with Alexey Elagin.

In the project we study different notions of a dimension of triangulated categories. THat is, we consider Rouquier dimension rdim, Serre dimension sdim, and the diagonal dimension ddim. It is clear that these numbers do not coincide in general. We consider some examples and study their "basic" properties, so this is just a start of the project. However it is clear something at the moment, which enables one to make some predictions.

December 4 2017, room 427

**15:30 R. Schoen** (University of California, Irvine). Harmonic mappings and applications

In this talk we will survey the theory of harmonic mappings, including the theory of mappings to CAT(0) spaces and the special structure of such mappings in case the target is a Euclidean or hyperbolic building.

17:00 S.Lando (NRU HSE). Combinatorial solutions to integrable hierarchies

Abstract: Since Witten’s work around 1990, it is well known that properly collected Gromov-Witten invariants (of all genera) of certain varieties constitute solutions to integrable hierarchies of partial differential equations. This is true, in particular, for the Kontsevich-Witten potential of a point, which is a solution to the Korteweg – de Vries hierarchy, and, as proven by Okounkov in 2000, for simple Hurwitz numbers, which form a solution to the Kadomtsev-Petviashvili hierarchy. Hurwitz numbers, which enumerate ramified coverings of the 2-sphere, also can be expressed in terms of properly equipped graphs. In the talk, we will discuss a natural question about which classes of graph invariants possess a similar property. The talk is based on a joint work with S.Chmutov (Ohio State University) and M.Kazarian. No specific preliminary knowledge is required.

November 17 2017, room 306

**17:00 **Artan Sheshmani (Harvard). Nested Hilbert schemes, local Donaldson-Thomas theory, Vafa-Witten / Seiberg-Witten correspondence

Abstract: We report on the recent rigorous and general construction of the deformation-obstruction theories and virtual fundamental classes of nested (flag) Hilbert scheme of one dimensional subschemes of a smooth projective algebraic surface. This construction will provide one with a general framework to compute a large class of already known invariants, such as Poincare invariants of Okonek et al, or the reduced local invariants of Kool and Thomas in the context of their local surface theory. We show how to compute the generating series of deformation invariants associated to the nested Hilbert schemes, and via exploiting the properties of vertex operators, prove that in some cases they are given by modular forms. We finally establish a connection between the Vafa-Witten invariants of local-surface threefolds (recently analyzed Tanaka and Thomas) and such nested Hilbert schemes. This construction (via applying Mochizuki's wall- crossing techniques) enables one to obtain a relations between the generating series of Seiberg-Witten invariants of the surface, the Vafa-Witten invariants and some modular forms. This is joint work with Amin Gholampour and Shing-Tung Yau following arXiv:1701.08902 and arXiv:1701.08899.

**18:30 **S.Lando (NRU HSE). Combinatorial solutions to integrable hierarchies

Abstract: Since Witten’s work around 1990, it is well known that properly collected Gromov-Witten invariants (of all genera) of certain varieties constitute solutions to integrable hierarchies of partial differential equations. This is true, in particular, for the Kontsevich-Witten potential of a point, which is a solution to the Korteweg – de Vries hierarchy, and, as proven by Okounkov in 2000, for simple Hurwitz numbers, which form a solution to the Kadomtsev-Petviashvili hierarchy. Hurwitz numbers, which enumerate ramified coverings of the 2-sphere, also can be expressed in terms of properly equipped graphs. In the talk, we will discuss a natural question about which classes of graph invariants possess a similar property. The talk is based on a joint work with S.Chmutov (Ohio State University) and M.Kazarian. No specific preliminary knowledge is required.

November 10 2017, room 306

17:00 Sergey Gorchinskiy (ILMS, Steklov Institute). Categorical measures for varieties with finite group actions

The talk is based on a common work with D. Bergh, M. Larsen, and V. Lunts. Given a variety with a finite group action, we compare categorical measures of the corresponding quotient stack and the extended quotient. Under some conditions the measures are the same and there are examples showing that they might be non-equal in general. We will dicsuss related technique and auxiliary results including the Grothendieck group of Deligne-Mumford stacks.

November 3 2017, room 306

**15:30 **John Alexander Cruz Morales (Universidad Nacional de Colombia) On Stokes matrices for Frobenius manifolds

In this talk we will discuss how to compute the Stokes matrices for some semisimple Frobenius manifolds by using the so-called monodromy identity. In addition, we want to discuss the case when we get integral matrices and their relations with mirror symmetry. This is part of an ongoing project with Maxim Smirnov which extends previous work with Marius van der Put for the case of quantum cohomology of projective and weighted projective spaces to other Frobenius manifolds not necessarily of quantum cohomology type.

**17:00 **Grigory Mikhalkin (Geneva). Examples of tropical-Lagrangian correspondences

Согласно методологии "Эс-Игрек-Зет" (Штромингер-Яу-Заслов), тропические объекты могут быть воплощены в классическом мире двумя способами: как объекты в комплексной, и как объекты в симплектической геометриях. Каждое из таких воплощений должно быть математически описано своей теоремой соответствия. В то время как тропически-комплексные соответствия изучались и изучаются достаточно интенсивно (в частности, для кривых, точек, и полных пересечений), тропически-симплектические соответствия относительно малоизучены. В докладе мы рассмотрим некоторые простейшие примеры таких соответствий. В качестве применения мы передокажем теорему Гивенталя (доказанную около 30 лет назад) о лагранжевых вложениях связных сумм бутылок Клейна в C^{2}.

October 20 2017, room 306

**17:00 **Sergey Arkhipov (Aarhus University). Braid relations in the affine Hecke category and differential forms with logarithmic singularities

Abstract: We recall the even and odd algebro-geometric realizations of the affine Hecke category - one via equivariant coherent sheaves on the Steinberg variety and the other in terms of some equivariant DG-modules over the DG-algebra of differential forms on a reductive group G. The latter one has a toy analog called the coherent Hecke category. It contains certain canonical objects satisfying braid relations via convolution. The proof uses simple facts from the geometry of Bott-Samelson varieties. Our goal is to provide a similar proof of braid relations in the affine Hecke category. It turns out that canonical braid group generators are given by certain DG-modules of logarithmic differential forms and braid relations follow immediately from a general statement which seems to be new: direct image of the DG-module of logarithmic differential forms does not depend on a resolution of singularities.

October 06 2017

A.Mironov. Обыкновенные коммутирующие дифференциальные операторы с полиномиальными коэффициентами и автоморфизмы первой алгебры Вейля.

В докладе будет рассказано об обыкновенных коммутирующих дифференциальных операторах, и в частности, о методе построения коммутативных подалгебр в первой алгебре Вейля. В докладе также будет обсуждаться задача об описании орбит действия автоморфизмов первой алгебры Вейля на множестве коммутирующих операторов с полиномиальными коэффициентами при фиксированной спектральной кривой.

Доклад основан на совместной работе с А.Б.Жегловым.

A.Zheglov. Algebro-geometric spectral data for planar Calogero-Moser systems.

My talk (based on a joint work with Igor Burban) is devoted to the algebraic analysis of planar rational Calogero-Moser systems. This class of quantum integrable systems is known to be superintegrable. This means that the underlying Schrödinger operator with Calogero-Moser potential can be included into a large family of pairwise commuting partial differential operators such that the space of joint power series eigenfunctions is generically one-dimensional.

More algebraically, any such system is essentially determined by a certain algebro-geometric datum: the projective spectral surface (defined by the algebra of planar quasi-invariants with natural filtration) and the spectral sheaf (defined by a module known to be Cohen-Macaulay of rank one). This geometric datum has very special algebro-geometric properties, the most important of which is a very special form of the Hilbert polynomial of the module (sheaf). Moreover, the spectral variety appears to be rational but very singular (only Cohen-Macaulay, even not normal). It turns out that all rank one Cohen-Macaulay modules over the algebra of planar quasi-invariants can be explicitly described in terms of very natural moduli parameters, and this description looks in some sence very similar to to the description of the generalised Jacobian for singular rational curves. The spectral module of a planar Calogero-Moser system is actually projective, and its underlying moduli parameters are explicitely determined.

Unlike the case of curves, not every Cohen-Macaulay module is spectral. The moduli space of spectral sheaves appears to be much more subtle, but its structure indicates the existence of integrable deformations of Calogero-Moser systems. I am going to explain how the classification of CM modules, combined with tools of the algebraic inverse scattering method, leads to certain new integrable deformations of Calogero-Moser systems in the algebra of differential-difference operators.

September 29 2017

**17:00 **Don Zagier (MPI for Mathematics, Bonn). Poor Man's Adeles and Multiple Zeta Values.

The "poor man's adeles" of the title is the informal name of the ring whose elements are "numbers" having a well-defined value modulo almost every prime number.

It turns out that examples of elements of this ring show up in many places in mathematics.

In the lecture I will describeseveral examples of this, most notably a finite-field version of the well-known multiple zeta values invented by Euler and much studied in recent years (this part is joint work with Masanobu Kaneko), but also examples coming from areas as different as quantum invariants of homology 3-spheres and transition matrices between different bases of the space of solutions of a linear differential equation with regular singularities.

**18:30 **Alessio Corti (Imperial College, London) Fano varieties and mirror symmetry

After discussing the words in the title, I will sketch a project to classify Fano varieties inspired by mirror symmetry.

June 19 2017

**18:15 **E. Lupercio (Cinvestav). Quantum Toric Varieties.

In this talk I present in more detail the theory of quantum toric varieties developed by Katzarkov, Meersseman, Verjosvsky and myself. I will explain what a quantum fan is and how it correspond to a quantum toric variety. Additionally, I will present the moduli of quantum P^{1}.

**19:30 **D.Kaledin (MI RAS, HSE). Brown representability for groupoids

Brown representability theorem in topology was very important when it appeared 50 years ago, but now is reduced to a bit of a historical curiosity and not used much. Part of the reason is that it only applies to functors from pointed connected CW complexes to abelian groups. We will show how to upgrade the theorem to functors from arbitrary complexes to groupoids. This gives a very simple, concise and purely categorical description of homotopy types and enhanced categories, without any need to choose any specific model and impose any notion of a weak equivalence or a model structure. In particular, in this description, it is easy to put some additional algebraic or geometric structure on the target groupoids

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