Alternatively have a look at the program.

## What is 3-dimensional hyperbolic geometry?

3-dimensional hyperbolic geometry is a classical subject.

We will give an introduction emphasizing concrete questions,

computations and answers, as well as hints to quantum aspects

of this beautiful theory. Please bring concrete questions and you may be

lucky and get some answers.

## Fundamental groups and rational points

One powerful tool in the study of rational solutions to equations comes from various theories of fundamental groups of varieties, most notably the etale fundamental group developed by Alexander Grothendieck. In this talk, we will outline how one can apply such algebro-topological tools to the determination of solutions to equations, both in terms of Grothendieck's famous *Section Conjecture* and (time permitting) the non-abelian Chabauty method of Minhyong Kim.

## Generalized Dijkgraaf-Witten invariants in low-dimensional topology

The aim of this talk is to introduce a family of invariants of 2-knots which generalize the Dijkgraaf-Witten (DW) knot invariants. I will begin with a casual review of the DW knot invariants, making connections to Fox n-colourings in the process. The naive generalization to 2-knots yields much weaker invariants and so I will discuss a homotopy-theoretic

generalization which has the potential to yield finer results.

## Flat cycles in the homology of congruence covers of SL(n,Z)\SL(n,R)/SO(n)

The locally symmetric space SL(n,Z)\SL(n,R)/SO(n), or the space of flat n-tori of unit volume, has immersed, totally geodesic, flat tori of dimension (n-1). These tori are natural candidates for nontrivial homology cycles of manifold covers of SL(n,Z)\SL(n,R)/SO(n). In joint work with Grigori Avramidi, we show that some of these (n-1)-dim tori give nontrivial rational homology cycles in congruence covers of the locally symmetric space SL(n,Z) \SL(n,R)/SO(n).

## Polynomially integrable convex bodies

## Siegel modularity of certain Calabi--Yau threefolds over $Q$

We will consider a number of examples of Calabi--Yau threefolds

defined over $Q$ having the Hodge numbers $h^{p,q}=1$ for

all pairs $p,q$ with $p+q=3$. Two of these Calabi--Yau

threefolds are equipped with real multiplication by some

real quadratic fields $K=Q(\sqrt{d})$ with square-free integers

$d>1$, and satisfy the Hilbert modularity over $K$.

Starting with the Hilbert modularity over $K$,

we will establish the Siegel modularity over $Q$

of such Calabi--Yau threefolds that their (cohomological)

## Fibre surfaces, knots, and elastic strings

## Puzzles about trisections of 4-manifolds

A trisection of a smooth 4-manifold is a very natural kind of decomposition into three elementary pieces which I will describe. Trisections are a natural 4-dimensional analogue of Heegaard splittings of 3-manifolds, a class of decompositions into two pieces that have yielded tremendous insight into 3-dimensional topology, so the philosophy is that trisections should give a way to port 3-dimensional techniques, questions and results to dimension four.

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