## Enumerative geometry, matrix models and universality classes via topological recursion

## First discussions

## The ideal theorem, and Dedekind zeta functions on the critical line

We give a new second moment estimate for the Dedekind zeta function on the critical line. As an application, we improve the error term in the counting theorem for integral ideals of norm $\leq x$ in a fixed number field $K$. This is joint work with Roger Baker.

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## First meeting/Working group on Grothendieck-Teichmüller groups

Concerning the working group on Grothendieck-Teichmuller, I suggest that

we meet this/Friday 21/09/2018 at 14h00,/ in the /seminarraum./

We shall discuss how we organize our group.

The idea of this working group is to have an exchange of points of views

concerning a tool which connects both geometric and arithmetic areas.

People who do not necessarily want to give a talk, are also welcome.

Noémie Combe

## Introductory Short Talks, Part II

## Introductory Short Talks, Part 1

## Quantum Matrix Algebras via compatible R-matrices

The notion of compatible R-matrices was introduced by Isaev-Ogievetsky-Pyatov.

It enables one to introduced a large family of new quantum algebras and the

corresponding Bethe subalgebras. Also, this notion is useful for defining

the so-called half-quantum algebras (Isaev-Ogievetsky) which generalize the

notions of the Manin and q-Manin matrices. In my talk I plan to introduce all

these objects and to exhibit the role of the mentioned algebras in

studying the Yangians and their different generalizations.

## The size of the primes obstructing the existence of rational points and Brownian motion

## State integrals, the quantum dilogarithm and knots

State integrals (and their building block, the quantum dilogarithm) express the partition function of complex Chern-Simons theory of triangulated 3-manifolds with boundary. I will give an introduction to the subject, focusing on examples, as well as recent results on expressing state integrals in terms of Neumann-Zagier data and in terms of q-series of Nahm type. This is work joint in parts with R. Kashaev and D. Zagier.

## New guests at the MPIM

## New guests at the MPIM

## Statistics of Diophantine problems and the Cohen-Lenstra heuristics

## On binary additive problems with prime variables in short intervals

In this talk, we consider some asymptotic behaviour of binary additive

problems with prime variables in short intervals. We start with a

brief review on exceptional set estimates in short intervals and

continue to the recent work of Languasco and Zaccagnini on the short

interval asymptotic formulas. Although the preceding results were

mainly obtained via the circle method, we see that a more direct

method gives better results for some cases.

## A survey on the evaluation of the values of Dirichlet $L$-functions and of their logarithmic derivatives on the line of $1$

Let $q$ be a positive integer $q>1$, and let $\chi$ be a Dirichlet character modulo $q$. Let $L(s, \chi)$ be the attached Dirichlet $L$-functions,

and let $L^\prime(s, \chi)$ denote its derivative with respect to the complex variable $s$. In this talk, we survey certain known results on the evaluation of values of Dirichlet $L$-functions and of their logarithmic derivatives at $1+it_0$ for fixed real number $t_0$.

## Asymptotics and inequalities for partitions into squares

We show that the number of partitions into squares with an even number of parts is asymptotically equal to that of partitions into squares with an odd number of parts. We further prove that, for $ n $ large enough, the two quantities are different and which of the two is bigger depends on the parity of $ n. $ This solves a recent conjecture formulated by Bringmann and Mahlburg (2012)

## Minicourse 2

## Coffee break

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