## Archive AG&Phy - SoSe 22

*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.

**Isolated date: 15 Mar. 2022**

*A new proof of positivity of DT invariants of symmetric quivers*

**Isolated date: 22 Mar. 2022**

*The Hecke correspondence via Yang-Mills-Higgs flow lines*

**Sommersemester 2022**

**19. Apr. 2022**

Jun-Yong Park (MPIM Bonn)*Arithmetic topology of the moduli stack of Weierstraß fibrations over global function fields*We will first consider the formulation of the moduli of fibered algebraic surfaces as the Hom space of algebraic curves on moduli stacks of curves. Cohomology with weights on these moduli naturally allows us to enumerate elliptic & hyperelliptic curves over global function fields ordered by bounded discriminant height. In the end, we formulate analogous heuristics for parallel countings over number fields through the global fields analogy. This is a joint work with Oishee Banerjee (Bonn) and Johannes Schmitt (Zürich).

**26. Apr. 2022**

*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.

**3. May 2022**

*The mysterious loop invariants of a knot*

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.

**10. May 2022**

**NEW ROOM 007 (IRIS Building)**

**17. May 2022**

free cumulants, which are fundamental quantities in non-commutative versions of probability theory.

**24. May 2022**

*The Barr-Beck Theorem in Symplectic Geometry*

_{1}) and Fuk(M

_{2}) associated to a Lagrangian correspondence L

_{12}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.

**31. May 2022**

*Bimodules over categories enriched over closed monoidal categories and A∞-categories*

_{∞}-category has to be extended in order to have wider applications.

We propose to extend it to the context of bimodules over V-categories. The ground category V is assumed to be additive, closed symmetric monoidal, complete and cocomplete. We propose a framework for enriched A

_{∞}-categories: we choose a V-category D, then we define an A

_{∞}-category as a tensor D-bimodule equipped with the deconcatenation comultiplication and coderivation whose square in a sense is 0.

**7. June 2022**

*Flat F-manifolds, F-CohFT and integrable systems*

_{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.

**14. June 2022**

*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.

**21. June 2022**

*Veronese subalgebras and Veronese morphisms for a class of Yang-Baxter algebras*

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.

**28. June 2022**

**Special room just for this day: 1.011 (Rudower Chaussee 25, Johann von Neumann-Haus)**

*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.

**5. July 2022**

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.