## Lamplighter Groups

## Workshop on "$\infty$-categories and their applications", August 17 - 21, 2020

The quest for solid foundations for $\infty$-categories started several decades ago, but from the mid-2000s there have been spectacular developments which excitingly accelerated the pace of progress. This workshop will focus on these objects, both in their foundations and in the applications to different fields of mathematics. There will be minicourses on $\infty$-categories, and also on $\infty$-operads, on algebraic K-theory and on parametrized homotopy theory, all three of which have benefited substantially from the $\infty$-categorical viewpoint.

## Some faces of the Poincaré homology sphere

## IMPRS seminar on various topics: Knot theory

## Higher Geometric Structures along the Lower Rhine XIV, March 12 - 13, 2020

This is the eleventh of a series of short workshops jointly organized by the Geometry/Topology groups in Bonn, Nijmegen, and Utrecht, all situated along the Lower Rhine. The focus lies on the development and application of new structures in geometry and topology such as Lie groupoids, differentiable stacks, Lie algebroids, generalized complex geometry, topological quantum field theories, higher categories, homotopy algebraic structures, higher operads, derived categories, and related topics.

## tba

## New guests at the MPIM

Wouter Van Limbeek: Symmetry and self-similarity in geometry

## Symplectic bordism: something old and something new, II

I will begin by reviewing the history of the stable homotopy theory approach to understanding (co)bordism groups via the Pontrjagin-Thom isomorphism and analysis of the homotopy type of the associated Thom spectra. This proved remarkably successful and led to the determination of many important classical examples of bordism groups such as unoriented, oriented, unitary, special unitary, spin and spin^c. The two outstanding classical cases which are not completely understood are framed bordism (aka stable homotopy groups of spheres) and symplectic bordism.

## Symplectic bordism: something old and something new

I will begin by reviewing the history of the stable homotopy theory approach to understanding (co)bordism groups via the Pontrjagin-Thom isomorphism and analysis of the homotopy type of the associated Thom spectra. This proved remarkably successful and led to the determination of many important classical examples of bordism groups such as unoriented, oriented, unitary, special unitary, spin and spin^c. The two outstanding classical cases which are not completely understood are framed bordism (aka stable homotopy groups of spheres) and symplectic bordism.

## The Hitchin System

Using the Dolbeault picture of the moduli space of Higgs bundles, we will construct a function from the moduli space to an affine space.This function will be the Hamiltonian of an integrable system called the Hitchin system. This by definition gives a foliation of the moduli space generically by open subsets of abelian varieties. Time permitting, we will see some applications and how this leads to Higgs bundles offering new, and some times unexpected, geometric insight.

## Quantum group symmetry in CFT

I discuss applications of a hidden $U_q({sl}_2)$ symmetry in CFT with central charge $c \leq 1$ (focusing on the generic, semisimple case, withcirrational). This symmetry provides a systematic method for solving Belavin-Polyakov-Zamolodchikov PDE systems, and in partic-ular for explicit calculation of the asymptotics and monodromy properties of the solutions.Using a quantum Schur-Weyl duality, one can understand solution spaces of such PDE systems in a detailed way.

## 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.

## 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.

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