## Knotted 2-spheres in 4-space

Introductory talk for general math audience.

## tba

## Fourier coefficients of polyharmonic Maass forms

Lagarias and Rhoades introduced a new class called polyharmonic Maass forms, which are a generalization of harmonic Maass forms. They gave a standard basis for the space of such forms by means of the higher Laurent coefficients of real analytic Eisenstein series. In this talk, we consider a large space of polyharmonic weak Maass forms of integral or half-integral weight, and construct a basis for this space. Furthermore, as an analogue of Zagier’s work on traces of singular moduli, we express the Fourier coefficients in terms of traces of CM-values and cycle integrals.

## Local asymptotic distribution of rational points

The Batyrev-Manin-Peyre principle predicts some uniform distribution of rational points on rational varieties. We propose a local analogue, whose aim is to describe the local behavior neglected by the global consideration.A heuristic interpretation of the main term valid for all known example is proposed in spirit of the geometric Batyrev-Manin principle together with a conjecture of McKinnon. We give a number of illustrations.

## Lagrangian Field Theories: ind-/pro-approach and Linfinity algebra of local observables

## Quantum versions of Deligne categories

## Truncated current Lie algebras with polynomial rings of symmetric invariants

We prove that under mild restrictions on the Lie algebra $q$ having a polynomial ring of symmetric invariants, the truncated current algebras of $q$ also have a polynomial ring of symmetric invariants.

## Bott-periodicity - first proof II

## Stable cohomology of arithmetic groups, algebraic K-groups of number fields and special values of Dedekind zeta functions

## Introduction to Knots and Links

## Local-global properties of word varieties

We consider equations of the form w(x,y)=g where w

is a group word in two letters (= an element of the free

group on two generators), g is a fixed element of a fixed

group G, and solutions are sought among pairs of elements

of G. Our focus is on the case where G is a simple linear

algebraic group. In this talk we consider the case

G=SL(2,K) where K is a number field. We discuss approximation

properties of the corresponding algebraic K-variety.

## Local asymptotic Diophantine approximations

The irrationality measure of a real number characterizes its Diophantine approximation property. The problem of counting the number of approximates to a given real number with respect to certain approximation order has been considered by a number of authors such as S. Lang, W. Adams,... We propose a local version of this problem and show that in certain cases the approximates are uniformly distributed, by using standard tools from analytic number theory.

## The blocks and abelian envelope of Deligne's Rep S_t

## L^2-Betti numbers, II

## Enumeration of meanders and volumes of moduli spaces of quadratic differentials

A meander is a topological configuration of a line and a simple closed curve in the plane (or a pair of simple closed curves on the 2-sphere) intersecting transversely. In physics, meanders provide a model of polymer folding, and their enumeration is directly related to the entropy of the associated dynamical systems.

## Bott-periodicity - first proof

## On the Hasse Principle for Divisibility in commutative algebraic groups, III

## Kaehler groups and surface bundles

A question going back to Serre asks which groups arise as fundamental groups of smooth complex projective varieties, or more generally, compact Kaehler manifolds.

## Spectral invariants for hyperbolic polygons

In 1966, M. Kac published his seminal article ''Can one hear the shape of a drum?'', which deals with the so-called inverse spectral problem: to determine geometric information from spectral data.

## Counting rational points in arithmetic varieties by the determinant

By the slope method in Arakelov geometry, we can construct a

family of hypersurfaces which cover the rational points of bounded height on

an arithmetic variety, but do not contain the generic point of this variety.

By estimating some invariants of Arakelov geometry, we can control the

number and the maximal degree of this family of auxiliary hypersurfaces

explicitly. In this talk, I will explain the method of studying the problem

of counting rational points by using methods from Arakelov geometry.