## Talk coaching for Postdocs

## 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.

## 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.

## -- 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.

## 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.

## Estimates for binary quadratic forms and Apollonian circle packings

Given a positive definite integral binary quadratic form, it is a classical problem in number theory to count the integers that are represented by this form. A modern treatment was given in 2006 by Valentin Blomer and Andrew Granville.

## Some Feynman diagrams in pure algebra

I will explain how the computation of compositions of maps of a certain natural class, from one polynomial ring into another, naturally leads to a certain composition operation of quadratics and to Feynman diagrams. I will also explain, with very little detail, how this is used in the construction of some very well-behaved poly-time computable knot polynomials.

## Algebraic knot theory

This will be a very "light" talk: I will explain why about 13 years ago, in order to have a say on some problems in knot theory, I've set out to find tangle invariants with some nice compositional properties. In my second talk in Bonn I will explain how such invariants were found - though they are yet to be explored and utilized.

## 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.

## tba

## Regulators of number fields and abelian varieties

In the general study of regulators, we present three inequalities. We first bound from below the regulators of number fields, following previous works of Silverman and Friedman. We then bound from below the regulators of Mordell-Weil groups of abelian varieties defined over a number field, assuming a conjecture of Lang and Silverman. Finally we explain how to prove an unconditional statement for elliptic curves of rank at least 4. This third inequality is joint work with Pascal Autissier and Marc Hindry.

## tba

## tba

## Localization techniques for singular algebraic varietieties

The standard ways of computing, or indeed defining, invariants of singular varietis use the procedure of resolution of singularities. To explicitly know a resolution, however, requires a good understanding of the singularities of the variety in question.

I will talk about methods that can be applied to the study of singular varieties without resolution of singularities, which originate from localization theorems in topology. As an example, I will present some computations involving the torus-equivariant cohomology of the (refined) Hilbert scheme of points on the plane.

## Global Koszul duality

Koszul duality is a phenomenon that shows up in rational homotopy theory, deformation theory and other subfields of algebra and topology. Its modern formulation is due to the works of Hinich, Keller-Lefevre and Positselski. It is a certain correspondence between categories of differential graded (dg) algebras and conilpotent dg coalgebras; there is also a module-comodule level version of it. In this talk I explain what happens if one drops the condition of conilpotency on the coalgebra side; the consequences turn out to be quite dramatic.

## Construction of stable rank 2 sheaves on projective space

## New guests at the MPIM

## Coxeter Groups

## Higgs bundles on Riemann surfaces, I

The relation between representations of the fundamental group of a Riemann surface into the unitary group, flat connections on Hermitian vector bundles, and stable holomorphic bundles on the surface, goes back to the celebrated theorem of Narasimhan and Seshadri. The case of representations into a non-compact reductive Lie group required the introduction of new holomorphic objects on the Riemann surface, called Higgs bundles.

## L-functions and isogenies of abelian varieties

Faltings's isogeny theorem states that two abelian varieties

over a number field are isogenous precisely when the characteristic

polynomials associated to the reductions of the abelian varieties at all

prime ideals are equal. This implies that two abelian varieties defined

over the rational numbers with the same L-function are necessarily

isogenous, but this is false over a general number field.

In order to still use the L-function to determine the underlying field

and abelian variety, we extract more information from the L-function by

© MPI f. Mathematik, Bonn | Impressum & Datenschutz |