## Freezing transition for the Riemann zeta function on a short interval

In this talk, we will present a proof of the freezing transition for the Riemann zeta function as conjectured by Fyodorov, Hiary & Keating. The connection with log-correlated fields will be emphasized. The problem is related to understanding moments of zeta on a typical short interval. The proof relies on techniques developed to understand the leading order of the maximum of zeta. If time permits, we will discuss the “one-step replica symmetry breaking behaviour” (1-RSB) which can be proved for a simplified model of zeta.

## Mini-course: Riemann zeta function and log-correlated fields, Part 2

For info, the Part 1 will take place in Lipschitz Saal, Uni Bonn (Endenicher Allee 60),

Tuesday 12th June, 9h00-10h45.

I will review some aspects of the theory of the Riemann zeta-function, especially those aspects relating to the value distribution of the zeta function on the critical line and the connections with Random Matrix Theory. I will then review conjectural connections between the extreme value statistics of the zeta function and those of log-correlated Gaussian fields, recent progress towards developing a rigorous understanding of these, and open problems.

## Short talk: The Fyodorov-Bouchaud formula and Liouville conformal field theory

Starting from the restriction of a 2d Gaussian free field (GFF) to the unit circle one can define a Gaussian multiplicative chaos (GMC) measure whose density is formally given by the exponential of the GFF. In 2008 Fyodorov and Bouchaud conjectured an exact formula for the density of the total mass of this GMC. In this talk we will explain how to prove rigorously this formula by using the techniques of conformal field theory. The key observation is that the moments of the total mass of GMC on the circle are equal to one-point correlation functions of Liouville theory in the unit disk. The same techniques also allow to derive a similar result on the unit interval [0,1] (in collaboration with Tunan Zhu). Finally we will briefly discuss applications to random matrix theory and to the asymptotics of the maximum of the GFF.

## Mini-course: Random matrices and logarithmically correlated fields, Part 2

For info: the Part 1 will take place in Lipschitz Saal at Uni Bonn (Endenicher Allee 60), Monday 11th June, 14h00-15h45.

A connection between branching structures and characteristic polynomials of random matrices emerged in the past few years. We will illustrate this for two models of random matrices, corresponding to dimension 1 and 2 spectra: the Circular Unitary Ensemble and Ginibre random matrices. The discussed topics will include the second moment method, extrema, the Gaussian free field and Gaussian multiplicative chaos.

## Short talk: First order asymptotic of Liouville four-point correlation function and the conformal bootstrap

Based on the rigorous path integral formulation of Liouville conformal field theory introduced by David-Kupiainen-Rhodes-Vargas, we compute the first order asymptotic of the four-point correlation function on the sphere as two insertions get close together, which is expected to describe the density of vertices around the root of large random planar maps. Moreover, our results are consistent with predictions from the framework of theoretical physics known as the conformal bootstrap, featuring the DOZZ formula and logarithmic corrections in the distance between the insertions.

## Basics of the BV formalism - Part 2

If the Batalin-Vilkovisky formalism is a cohomological handle for field theories in the Lagrangian formalism, a similar construction can be set up for the associated Hamiltonian picture and goes under the name of Batalin, Fradkin and Vilkovisky (BFV).

The link between the two has been made explicit recently by Cattaneo, Mnev and Reshetikhin (CMR) as a tool to treat field theories on manifolds with boundary.

In this talk I will review the basics of the BFV and CMR constructions and show how they relate to what we have done so far.

## The Hopf-invariant one problem and applications

## tba

## Introduction to elliptic cohomology

## p-adic dimentions

## tba

## Classical torsion and L^2-torsion I

## Construction of harmonic maps between Riemannian manifolds

By definition, harmonic maps between Riemannian manifolds are critical points of the energy functional.

In local coordinates the condition for a map to be harmonic constitutes a system of non-linear,

elliptic partial differential equations of second order.

Clearly, solutions to such differential equations are typically such maps are hard to construct.

In this talk we present a few construction methods of harmonic maps between Riemannian manifolds.

Furthermore, we will briefly explain applications of harmonic maps to other research areas (in Mathematics).

## On a non-abelian analogue of the Polya-Vinogradov inequality

We consider an analogue of the classical Polya-Vinogradov inequality for character sums in

the context of the group GL(2) over finite fields. This is joint work with Satadal Ganguly.

## On a universal Torelli theorem for elliptic surfaces

## On a universal Torelli theorem for elliptic surfaces

Given two semistable elliptic surfaces over a smooth, projective curve $C$ defined over a field of characteristic zero or finitely generated over its prime field, we show that any compatible family of effective isometries of the N{\'e}ron-Severi lattices of the base changed elliptic surfaces for all finite separable maps $B\to C$ arises from

an isomorphism of the elliptic surfaces. Without the effectivity hypothesis, we show that the two elliptic surfaces are isomorphic.

We also determine the group of universal automorphisms of a semistable elliptic surface. In particular, this includes showing that the Picard-Lefschetz transformations corresponding to an irreducible component of a singular fibre, can be extended as universal isometries. In the process, we get a family of homomorphisms of the affine Weyl group associated to $\tilde{A}_{n-1}$ to that of $\tilde{A}_{dn-1}$, indexed by natural numbers $d$, which are closed under composition.

This is joint work with S. Subramanian.

## A universal Torelli-Tate theorem for elliptic surfaces

Given two semistable elliptic surfaces over a smooth, projective curve $C$ defined over a field of characteristic zero or finitely generated over its prime field, we show that any compatible family of effective isometries of the N{\'e}ron-Severi lattices of the base changed elliptic surfaces for all finite separable maps $B\to C$ arises from

an isomorphism of the elliptic surfaces. Without the effectivity hypothesis, we show that the two elliptic surfaces are isomorphic.

We also determine the group of universal automorphisms of a semistable elliptic surface. In particular, this includes showing that the Picard-Lefschetz transformations corresponding to an irreducible component of a singular fibre, can be extended as universal isometries. In the process, we get a family of homomorphisms of the affine Weyl group associated to $\tilde{A}_{n-1}$ to that of $\tilde{A}_{dn-1}$, indexed by natural numbers $d$, which are closed under composition.

This is joint work with S. Subramanian.

## Computation without Representation -- change of time --

A major part of "quantum topology" (you don't have to know what's that) is the definition and computation of various knot invariants by carrying out computations in quantum groups (you don't have to know what are these). Traditionally these computations are carried out "in a representation", but this is very slow: one has to use tensor powers of these representations, and the dimensions of powers grow exponentially fast. I will describe a direct-participation method for carrying out these computations without having to choose a representation and explain why in many ways the results are better and faster. The two key points we use are a technique for composing infinite-order "perturbed Gaussian" differential operators, and the little-known fact that every semi-simple Lie algebra can be approximated by solvable Lie algebras, where computations are easier.

This is joint work with Roland van der Veen and continues work by Rozansky and Overbay. For further details, see http://drorbn.net/b18.

## Braids and the Grothendieck-Teichmüller Group -- change of time --

I will explain what are associators (and why are they useful and natural) and what is the Grothendieck-Teichmüller group, and why it is completely obvious that the Grothendieck-Teichmüller group acts simply transitively on the set of all associators. Not enough will be said about how this can be used to show that "every bounded-degree associator extends", that "rational associators exist", and that "the pentagon implies the hexagon".

In a nutshell: the filtered tower of braid groups (with bells and whistles attached) is isomorphic to its associated graded, but the isomorphism is neither canonical nor unique - such an isomorphism is precisely the thing called "an associator". But the set of isomorphisms between two isomorphic objects *always* has two groups acting simply transitively on it - the group of automorphisms of the first object acting on the right, and the group of automorphisms of the second object acting on the left. In the case of associators, that first group is what Drinfel'd calls the Grothendieck-Teichmüller group GT, and the second group, isomorphic (though not canonically) to the first, is the "graded version" GRT of GT.

Almost everything I will talk about is in my old paper "On Associators and the Grothendieck-Teichmüller Group I" For further details, see http://drorbn.net/b18.

## An example of BV quantization in action

The purpose of this talk is to survey the machinery developed

by Costello, with an emphasis on the procedure by which one starts with

an action functional and ends up with higher algebraic structure at the

end. The running example will take as input the curved beta-gamma

system, a 2d sigma model of maps from a Riemann surface to a complex

manifold. I will sketch how to construct its BV quantization and how to

analyze its factorization algebra, which determines a sheaf of vertex

algebras known as "chiral differential operators." The talk aims to be

conversational and not very technical, as I want to communicate the big

picture.

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