Abstracts of upcoming talks at the MPIM. For an overview see also the calendar.

## Recent developments in Quantum Topology -- Cancelled --

We will review the basics of quantum topology such as the colored Jones polynomial of a knot, its standard conjectures relating to asymptotics, arithmeticity and modularity, as well as the recent quantum hyperbolic invariants of Kashaev et al, their state-integrals and their structural properties. The course is aimed to be accessible by graduate students and young researchers.

## Some adjoint L-values and Hilbert modular Eisenstein congruences

I will start from the situation of a cuspidal Hecke eigenform f of real quadratic character, congruent to its complex conjugate modulo a prime P ramified in the coefficient field.

## The m-step solvable anabelian geometry of finitely generated fields

A celebrated theorem of Neukirch and Uchida states that two number fields are isomorphic if their absolute Galois groups are isomorphic. The Grothendieck birational conjecture predicts a similar result for all finitely generated fields. This has been proved by Pop, using in an essential way the Neukirch-Uchida theorem for global fields.

## -- Cancelled -- Powers of the Dedekind eta function and the Bessenrodt-Ono inequality

In this talk I present recent results obtained with Markus Neuhauser towards the non-vanishing of the coefficients of the Dedekind eta function in the spirit of G.-C. Rota. This includes Serre’s table, pentagonal numbers, results of Kostant in the context of simple affine Lie algebras and the Lehmer conjecture. In the second part I will talk about partition numbers and the Bessenrodt-Ono inequality.

## IMPRS seminar on various topics: Period domains

## Local behaviour of the set of the values of Euler's totient function

Let $V$ be the set of the values of the Euler's $\varphi$ function. It is known that the natural density of $V$, namely the limit of $V(x)/x$, as $x$ tends to infinity, is $0$.

In a paper to appear, M. K. Das, P. Eyyunni and B. R. Patil claim that this also applies to a local density of $V$, known as the uniform upper density, or the Banach density, defined by

$$

\delta^{*(}V) = \lim_{H \rightarrow \infty} \limsup_{x \rightarrow \infty} \frac{1}{H}\left(V(x+H) -V(x)\right).

$$

## Higgs bundles on Riemann surfaces, II

On a Riemann Surface $\Sigma$, the moduli space of polystable $\mathrm{SL}_n(\mathbb{C}$)-Higgs bundles can be identified with the space of reductive representations $\pi _1 (\Sigma) \to \mathrm{SL}_n(\mathbb{C})$. In this talk, we discuss a proof of this so called non-abelian Hodge correspondence. Our goal is to understand how

to construct a Higgs bundle from a given representation and how this construction relates to the theory of harmonic maps.

## Lower Bounds for Discrete Negative Moments of the Riemann zeta Function

I will talk about lower bounds for the discrete negative 2k-th moment of the derivative of the Riemann zeta function for all

fractional k > 0. The bounds are in line with a conjecture of Gonek and Hejhal. This is a joint work with Winston Heap and

Junxian Li.

## Coxeter Groups

## 1. Singular Hodge theory of matroids; 2. Logarithmic concavity of weight multiplicities for irreducible sln(C)-representations

Talk 1: Title: Singular Hodge theory of matroids

If you take a collection of planes in R

3, then the number of lines you get by intersecting the planes is at least the number of planes. This is an example of a more general statement, called the “Top-Heavy Conjecture”, that Dowling and Wilson conjectured in 1974.On the other hand, given a hyperplane arrangement, I will explain how to uniquely associate a certain polynomial (called its Kazhdan–Lusztig polynomial) to it.

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