Valery Alexeev (University of Georgia). Compact moduli of K3 surfaces.

Let F be a moduli space of lattice-polarized K3 surfaces. Suppose that one has chosen a canonical effective ample divisor R on a general K3 in F. We call this divisor "recognizable" if its flat limit on Kulikov surfaces is well defined. We prove that the normalization of the stable pair compactification F_R for a recognizable divisor is a Looijenga semitoroidal compactification. For polarized K3 surfaces (X,L) of degree 2d, we show that the sum of rational curves in the linear system |L| is a recognizable divisor, giving a modular semitoroidal compactification of F_2d for all d. This is a joint work with Philip Engel.

Alexey Bondal (Steklov Mathematical Institute RAS, NRU HSE). Singularities and Chimaeras.

I will discuss non commutative resolutions of singularities via exact categories and various new mathematical objects that we got when discussing this approach with Agnieszka Bodzenta. Among those are chimaera categories, which generalise the singularity categories to the nonGorenstein case, and categorically symmetric singularities.

Chris Brav (NRU HSE).  Non-commutative calculus, connections, and loop spaces.  

We rework the non-commutative Cartan calculus of Hochschild cochains acting on Hochschild chains, and show it is compatible with the commutative Cartan calculus of vector fields acting on differential forms. Along the way, we generalise a theorem of Ben-Zvi and Nadler, relating equivariant sheaves on a free loop space to complete filtered D-modules. This is joint work with Nick Rozenblyum.

Sergey Galkin (NRU HSE).  An introduction to graph potentials.  

A functional equation  f(x,y,s) + f(z,w,s) = f(x,z,t) + f(y,w,t)  is an analogue of WDVV equation in birational geometry. Its solutions give new constructions of topological quantum field theories. In particular, f(x,y,z) = xy/z + xz/y + yz/x + q/xyz, i.e. a generating function for quantum Clebsch-Gordan conditions (spherical triangle inequalities + parity) produces a hierarchy of Laurent polynomials, enumerated by trivalent graphs, that are mirror dual to monotone Lagrangian tori, the fibers of integrable systems on character varieties of flat SU(2) connections on Riemann surfaces, and (their categories of matrix factorizations) shall be new algebro-geometric building blocks for (hypothetical) Donaldson-Floer theories. I will define graph potentials and discuss their elementary properties, and time permits will sketch a construction of the respective integrable systems and toric degenerations based on the sheaves of SU(2) WZW conformal blocks over Deligne-Mumford moduli spaces, that were build in a classical work of Tsuchiya-Ueno-Yamada. The talk is based on "Graph potentials and moduli spaces of rank two bundles on a curve” (arXiv:2009.05568), my joint work with Pieter Belmans and Swarnava Mukhopadhyay.

Dmitry Kaledin (Steklov Mathematical Institute RAS, NRU HSE).  Bokstedt, Bokstein and Bott.   

The well-known and fundamental Bostedt periodicity theorem asserts that the Topological Hochschild Homology of a finite field is a free commutative algebra in one generator of degree 2. An essential part of the proof is constructing this generator, and proving that it is not nilpotent. None of the existing constructions is too difficult, but none of them is elementary either, and some are quite amuzing. I am going to present a couple of them, and relate the construction to other well-known objects such as the Bott periodicity generator and the Bokstein homomorphism. I will not assume any prior knowledge of THH. Partially based on joint work with A. Fonarev.

Igor Krichever (NRU HSE, CAS Skoltech).  On algebraic integrability of the elliptic two-dimensional O(N) sigma model.  

Harmonic maps of two-dimensional Riemann surface Sigma to a Riemann manifold M are of interest both in physics and mathematics. They are critical points of the Dirichlet functional, the sigma model action. In the talk a new approach to the study of these models will be presented. In particular we show that the Novikov-Veselov hierarchy can be seen as a family of commuting symmetries of the O(N) sigma model.   As a corollary we prove that   the spectral curves associated with harmonic maps of two-torus to spheres are algebraic. The talk is based on a joint work with Nikita Nekrasov.

Sergey Lando (NRU HSE).  On stratification of Hurwitz spaces of stable maps.  

Hurwitz spaces are universal families of stable curve-to-curve maps. They are stratified according to singularities the maps are allowed to have. The cohomology class Poincar'e dual to the locus of any given singularity admits a universal expression in terms of certain basic characteristic classes, and these expressions can be explicitly computed. Our expressions hold for any family of curve-to-curve maps satisfying certain transversality conditions. They extend universal expressions for isolated singularities, which exist due to R. Thom's principle, to the nonisolated case. We also obtain universal expressions for residual polynomials describing classes Poincar'e dual to multisingularities. The results lead to new formulas for series of Hurwitz numbers. The aim of the project is to develop a tool for computing cohomology classes in spaces of mappings of curves required for computing Gromov-Witten invariants of curves. This is a joint work in progress with M. Kazarian and D.Zvonkine, the last two papers being   “Double Hurwitz Numbers and Multisingularity Loci in Genus 0”, IMRN 2021, and “Universal Cohomological Expressions for Singularities in Families of Genus 0 Stable Maps”, IMRN 2018.

Yuri Prokhorov (Steklov Mathematical Institute RAS, NRU HSE).  Birational transformations of conic bundles.

A conic bundle is a at morphism f : X  S of smooth varieties whose fibers are plane conics. In this talk, I will discuss application of Sarkisov program to the rationality problem of algebraic varieties having conic bundle structures. I concentrate on conic bundles, which lie on the “boundary” between birationally rigid and non-rigid ones and are especially interesting for this reason. The talk is based on the joint work in progress with V. Shokurov.

Viсtor Przyjalkowski  (Steklov Mathematical Institute RAS, NRU HSE). Singular Landau--Ginzburg models.

We discuss a way how to construct Landau--Ginzburg models using toric geometry. We discuss benefits given by the discussed procedure. We also discuss an example when the compactified Landau--Ginzburg models have unavoidable singularity over infinity.

Ilia Zharkov (Kansas State University). Topological SYZ fibrations with discriminant in codimension two.

To date only for K3 surfaces (trivial) and the quintic threefold ('01 M. Gross) the discriminant can be made to be in codimension two. I will outline the source of the problem and how to resolve it in much more general situations using phase and over-tropical pairs-of-pants. Joint project with Helge Ruddat. If time permits, I'll explain an application to lifting tropical cycles from the SYZ base to "holomorphic" and "Lagrangian" type objects in the torus fibrations.

Schedule