Humboldt-Universität zu Berlin - Mathematisch-Naturwissenschaftliche Fakultät - Institut für Mathematik

Special seminar: in honor of Yuri I. Manin 85th birthday
8 Mar. 2022
Matilde Marcolli (Caltech)

Categorical dynamics: from Hopfield to Pareto
In recent work with Yuri Manin, we proposed a model of neural information networks based on functorial
assignments of resources to networks, originating in Segal's notion of summing functors and Gamma spaces, and a corresponding categorical form of Hopfield equations on networks. I will review some properties and examples of such categorical framework and dynamics, and I will show how it can applied to a form of Pareto optimization.

 

Vladimir Dotsenko (Universite de Strasbourg)

I shall talk about a new interpretation of refined Donaldson-Thomas invariants of symmetric quivers, in particular re-proving their positivity (conjectured by Kontsevich and Soibelman, and proved by Efimov). This interpretation has two key ingredients. The first is a certain Lie (super-)algebra, for which we have two interpretations, in the context of Koszul duality theory and in the context of vertex Lie algebras. The second is an action of the Weyl algebra of polynomial differential operators on that Lie algebra, for which the characters of components of the space of generators give precisely the refined DT invariants. This is a joint project with Evgeny Feigin and Markus Reineke, partially relying on my recent work with Sergey Mozgovoy.

 

Graeme Wilkin (University of York)

The Hecke correspondence plays a central role in the Geometric Langlands program. In this talk I will show that the Hecke correspondence for Higgs bundles has a natural analytic interpretation in terms of spaces of flow lines for the Yang-Mills-Higgs functional. This requires first constructing ancient solutions to the Yang-Mills-Higgs flow, and then classifying the isomorphism classes that appear in this construction. Finally I will also describe a geometric criterion for distinguishing between broken and unbroken flow lines using secant varieties associated to the underlying Riemann surface.

 

Pavel Etingof (MIT)
Hecke operators over local fields and an analytic approach to the geometric Langlands correspondence

I will review an analytic approach to the geometric Langlands correspondence, following my work with E. Frenkel and D. Kazhdan, arXiv:1908.09677, arXiv:2103.01509, arXiv:2106.05243. This approach was developed by us in the last couple of years and involves ideas from previous and ongoing works of a number of mathematicians and mathematical physicists, Kontsevich, Langlands, Teschner, and Gaiotto-Witten. One of the goals of this approach is to understand single-valued real analytic eigenfunctions of the quantum Hitchin integrable system. The main method of studying these functions is realizing them as the eigenbasis for certain compact normal commuting integral operators the Hilbert space of L2 half-densities on the (complex points of) the moduli space Bun_G of principal G-bundles on a smooth projective curve X, possibly with parabolic points. These operators actually make sense over any local field, and over non-archimedian fields are a replacement for the quantum Hitchin system. We conjecture them to be compact and prove this conjecture in the genus zero case (with parabolic points) for G=PGL(2). I will first discuss the simplest non-trivial example of Hecke operators over local fields, namely G=PGL(2) and genus 0 curve with 4 parabolic points. In this case the moduli space of semistable bundles Bun_G^{ss} is P^1, and the situation is relatively well understood; over C it is the theory of single-valued eigenfunctions of the Lame operator with coupling parameter -1/2 (previously studied by Beukers and later in a more functional-analytic sense in our work with Frenkel and Kazhdan). I will consider the corresponding spectral theory and then explain its generalization to N>4 points and conjecturally to higher genus curves.

 

Stavros Garoufalidis (MPIM Bonn/SUSTech Shenzhen)

A main problem in quantum topology is the Volume Conjecture which asserts that an evaluation of the colored Jones polynomial (known as the Kashaev invariant) is a sequence of complex numbers that grows exponentially at the rate of the hyperbolic volume of a knot complement. This conjecture connects the Jones polynomial with hyperbolic geometry. The loop invariants are the refinement of the above conjecture to all orders in perturbation theory, and take values in the trace field of a knot. Hence, the loop invariants have topological, but also mysteriously geometric origin. A geometric definition of them is

currently unknown. In the talk we will discuss how these invariants behave under finite cyclic covers, and give clues about their possible geometric definition. Joint work with Seokbeom Yoon.

 

Tom Bridgeland (Sheffield University)

Whenever we have a divergent power series we can attempt to produce an analytic function by using Borel resummation. This process depends on a choice of ray in the complex plane, and in general we obtain a variety of different Borel sums related to each other by discrete jumps across critical rays. There are a number of examples now known where these Stokes jumps are precisely described by the DT invariants of some associated category. In the first half of the talk I will describe a particular example of this phenomenon relating to the topological string partition function of the resolved conifold. In the second half I will explain how viewing DT invariants as non-linear Stokes data suggests a way to encode them in a geometric structure which is a kind of non-linear Frobenius manifold.

 

Philippe Biane (Université Marne-la-Vallée)

The Quantum Symmetric Simple Exclusion Process (QSSEP) is a model of fermionic quantum particles hopping on a finite interval. D. Bernard and T. Jin have shown that the fluctuations of the invariant measure for this process, when the number of sites goes to infinity, are encoded into polynomials, with a strong combinatorial flavour. In this talk I give an explicit combinatorial formula for these polynomials in terms of associahedra which, quite surprisingly, shows that they can be interpreted as free cumulants of a family of commuting random variables. I will explain the physical model in the talk as well as what are
free cumulants, which are fundamental quantities in non-commutative versions of probability theory.

 

Nate Bottman (MPIM Bonn)

The Barr–Beck theorem gives conditions under which an adjunction F -| G is monadic. Monadicity, in turn, means that the category on the right can be computed in terms of the data of F and its endomorphism GF. I will present joint work-in-progress with Abouzaid, in which we consider this theorem in the case of the functors between Fuk(M₁) and Fuk(M₂) associated to a Lagrangian correspondence L₁₂ and its transpose. These functors are often adjoint, and under the hypothesis that a certain map to symplectic cohomology hits the unit, the hypotheses of Barr-Beck are satisfied. This can be interpreted as an extension of Abouzaid’s generation criterion, and we hope that it will be a useful tool in the computation of Fukaya categories.

 

Volodymyr Lyubashenko (Zürich Universität)

 

Paolo Rossi (Universita degli Studi di Padova)

Cohomological field theories are families of cohomology classes on the moduli space of stable curves of genus g, with n distinct marked points. The parameters in these families live on the n-fold tensor product of a vector space V. They must satisfy boundary conditions prescribing their behaviour at the boundary divisors of the moduli space, parametrizing stable curves with separating and nonseparating nodes. F-CohFTs differ from cohomological field theories in that one marked point plays a special role (S_n equivariance is broken to S_n-1-equivariance) and a boundary condition is imposed at separating nodes only (the node is formed by attaching the special point on one component and a normal one on the other). If CohFTs reduce to Dubrovin-Frobenius manifolds when restricted to genus 0, F-CohFTs reduce to flat F-manifolds. In this talk I will explain how the well known-relation between CohFTs and integrable systems can be extended to F-CohFTs, at the cost of losing the Hamiltonian structure. I will also mention that the well known Givental-Teleman reconstruction result of a semisimple CohFT from its Frobenius manifold can be generalized to semisimple F-CohFTs. All this is the fruit of a joint work with A. Arsie, A. Buryak and P. Lorenzoni.

 

Tyler Kelly (Birmingham University)
Open enumerative geometry for Landau-Ginzburg models and Mirror Symmetry
A Landau-Ginzburg (LG) model is a triplet of data (X,W,G) consisting of a regular complex-valued function W from a quasi-projective variety X with a group G acting on X so that W is invariant. An enumerative theory developed by Fan, Jarvis and Ruan gives FJRW invariants, an analogue of Gromov-Witten invariants, for LG models. We define an open enumerative theory for certain LG models, building on the FJRW point of view. Roughly speaking, our theory involves computing specific integrals on certain moduli of discs with boundary and interior marked points. One can then construct a mirror LG model to the original one using these invariants. This allows us to prove a mirror symmetry result analogous to that established by Cho-Oh, Fukaya-Oh-Ohta-Ono and Gross in the context of mirror symmetry for toric Fano manifolds. If time permits, I will explain some key features that this enumerative geometry enjoys (e.g., open topological recursion relations and wall-crossing). This is joint work with Mark Gross and Ran Tessler.

 

Tatiana Gateva-Ivanova (MPIM Bonn)

We study d-Veronese subalgebras A^(d) of quadratic algebras A_{X=A(K, X, r)} related to finite nondegenerate involutive set-theoretic solutions (X, r) of the Yang-Baxter equation, where K is a field and d > 1 is an integer. We find an explicit presentation of the d-Veronese A^(d) in terms of one-generators and quadratic relations.

We introduce the notion of a d-Veronese solution (Y, r_Y), canonically associated to (X, r) and use its Yang-Baxter algebra A_Y= A(K, Y, r_Y) to define a Veronese morphism v_n,d:A_Y→A_X. We prove that the image of v_n,d is the d-Veronese subalgebra A^(d), and find explicitly a minimal set of generators for its kernel. Finally, we show that the Yang-Baxter algebra A(K, X, r) is a PBW algebra if and only if (X, r) is a square-free solution. In this case the d-Veronese A^(d) is also a PBW algebra.

 

Yvain Bruned (University of Edinburgh)

Renormalisation of stochastic partial differential equations
In this talk, we will present the main ideas of the renormalisation of stochastic partial differential equations (SPDEs), as it appears in the theory of regularity structures. It is crucially based on the notion of a model that is a collection of stochastic integrals recentered around a base point and renormalised. They are used for Taylor-expanding solutions of singular SPDEs. We will discuss the transformation of the canonical model to the renormalised one and the underlying algebraic structure which are Hopf algebras on decorated trees.

 

Hussein Mourtada (Institut Mathématique de Jussieu)
Rogers-Ramanujan type identities and arc spaces

In this talk, I will show a link between an invariant of singularities (involving arc spaces) and Rogers-Ramanujan identities. These latter are identities concerning the theory of integer partitions which have a long history in classical number theory; an integer partition of an integer n being simply a decreasing sequence of positive integers whose sum is equal to n. I will then explain how this link allows to find and prove new families of partition identities. The talk is directed at a broad public. It concerns various works with C. Brucheck and J. Schepers, with P. Afsharijoo and with P. Afsharijoo, J. Dousse and Frédéric Jouhet.