## Algebraic geometry of the Lagrangian correspondence of Gaiotto

## A simplicial and categorical background for Hopf algebras in number theory and physics

We will discuss how simplicial and categorical structures lead to Hopf algebras.

## tba

## Rationality of Fano 3-folds over nonclosed fields

The rationality problem for smooth Fano threefolds over algebraically closed fields is basically solved.

In this talk I will discuss rationality of forms of these Fanos over nonclosed fields of characteristic 0.

I will concentrate on the case where the Picard number equals 1.

The talk is based on joint work with Alexander Kuznetsov (in progress).

## Harmonic analysis on GLn over finite fields

There are many formulas that express interesting properties of a finite group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ** character ratio**:

## Deformations of Abelian categories in algebra, geometry & physics

W. Lowen and M. Van den Bergh developed a deformation theory for Abelian categories, such that deformations of the category of (quasi)coherent sheaves on an affine variety Spec A correspond to deformations of the algebra A, which in turn are controlled by the Hochschild cochain complex equipped with the Gerstenhaber bracket.

## Some topics in Kobayashi–Hitchin correspondences

Recently, we studied the Kobayashi–Hitchin correspondences for monopoles and harmonic bundles. In this talk, we would like to discuss some related topics. We plan to explain equivalences between twisted monopoles and twisted difference modules, generalizing the equivalences in the untwisted case. We also plan to discuss some examples of Higgs bundles to which we may apply our equivalences.

## Special values of Asai L-functions

## Unoriented topological field theory in low dimensions, II

This talk is an introduction to unoriented topological field theory. I will give an overview of known examples and highlight some open problems. I will also describe a new class of examples, namely unoriented extensions of twisted Dijkgraaf-Witten theory, and the resulting topological interpretation of the character theory of categorical representations.

## Quantum theta functions and signal analysis

Representations of the celebrated Heisenberg commutation relations and their exponentiated versions form the starting point for a number of basic constructions, both in mathematics and mathematical physics (geometric quantization, quantum tori, classical and quantum theta functions) and signal analysis (Gabor analysis). In this talk I explain how Heisenberg relations bridge the noncommutative geometry and signal analysis. After providing a brief comparative dictionary of the two languages, I will show e.g.

## Knots and q-series, II

## Knots and q-series, I

## Relative shifted symplectic structures

I will discuss the notion of relative shifted symplectic structures on sheaves of derived stacks over locally compact Hausdorff topological spaces. I will describe a general pushforward construction of relative symplectic forms and in the constructible case will

explain explicit techniques for computing such forms. As applications I will discuss a relative lift of recent results of Shende--Takeda on topological Fukaya categories, and a universal construction of symplectic structures on derived irregular character varieties. This

is a joint work with Dima Arinkin and Bertrand Toen.

## Asymptotics and lower bounds for complex and real counts of singular surfaces in $\mathbb{P}^3$

We use tropical geometry and floor decomposed tropical surfaces to study the asymptotics of counts of singular surfaces in $\mathbb{P}^3$ satisfying point conditions. In the real case, we give a lower bound for the number of real singular surfaces satisfying point conditions. This is joint work with Thomas Markwig, Kristin Shaw, Eugenii Shustin.

## Aspects of non compact Gromov–Witten–Floer theory

Gromov–Witten invariants and Lagrangian Floer theory for compact symplectic manifolds are now very much established as far as general theory concerns. In symplectic geometry, non-compact symplectic manifolds (such as the cotangent bundle) are as important as compact ones. There are many versions and variants for non-compact Gromov–Witten–Floer theory and many of them are still not yet completely established. In this talk, I want to explain some of such versions and issues to study them.

## Motivic Donaldson–Thomas invariants of the moduli stacks of parabolic Higgs bundles and bundles with connections

Using the ideas of motivic integration I will explain how to compute the "number" of semistable Higgs bundles (maybe with parabolic structure) of fixed rank and degree on a smooth complex curve. Based on that result I am going to discuss a similar problem in the case of bundles with connections. If time permits, I will explain open questions, including a conjectural relation to Satake correspondence for affine Grassmannians over 2-dimensional local rings. This is a joint project with Roman Fedorov and Alexander Soibelman.

## Very stable Higgs bundles, the nilpotent cone and mirror symmetry

I will discuss the existence of very stable Higgs bundles how it implies a precise formula for the multiplicity of the components of the nilpotent cone and its relationship to mirror symmetry. Joint project with Nigel Hitchin.

## From the Hitchin component to opers

Let $C$ be a compact Riemann surface. A holomorphic quadratic differential on $C$ determines a spectral curve $\Sigma$ over $C$. Given a holomorphic quadratic differential, we can associate two types of differential operators. One is an oper on $C$, also known as a holomorphic Schrödinger operator.

## Weil–Petersson geometry and the BGW KdV tau function

It was conjectured by Witten and proven by Kontsevich that a generating function for intersection numbers on the moduli space of curves is a tau function of the KdV hierarchy, now known as

the Kontsevich–Witten tau function. Mirzakhani reproved this theorem via the study of

## Lagrangians within Higgs bundles and hyperpolygons

Through different group actions, we will construct Lagrangian submanifolds of the moduli space of principal Higgs bundles contained over the singular locus of the Hitchin fibration, as well as of the space of hyperpolygons. As an application, we will consider correspondences between these spaces coming from mirror symmetry as well from other dualities and correspondences. The talk is based on different pieces of work in progress with Steve Rayan, Sebastian Heller, Indranil Biswas, Steve Bradlow and Lucas Branco.

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