# Abstracts for Colloquium

Alternatively have a look at the program.

## What is 3-dimensional hyperbolic geometry?

Posted in
Speaker:
Stavros Garoufalidis
Affiliation:
Georgia Institute of Technology/MPIM
Date:
Thu, 2019-02-21 15:00 - 16:00
Location:
MPIM Lecture Hall
Parent event:
Colloquium
Parent event:
MPI-Oberseminar

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

## Fundamental groups and rational points

Posted in
Speaker:
Alexander Betts
Affiliation:
MPIM
Date:
Thu, 2019-03-21 15:00 - 16:00
Location:
MPIM Lecture Hall
Parent event:
Colloquium
Parent event:
MPI-Oberseminar

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

Posted in
Speaker:
Mark Penney
Affiliation:
MPIM
Date:
Thu, 2019-05-02 15:00 - 16:00
Location:
MPIM Lecture Hall
Parent event:
Colloquium
Parent event:
MPI-Oberseminar

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)

Posted in
Speaker:
Tam Nguyen Phan
Affiliation:
MPIM
Date:
Thu, 2019-05-16 15:00 - 16:00
Location:
MPIM Lecture Hall
Parent event:
Colloquium
Parent event:
MPI-Oberseminar

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

Posted in
Speaker:
Alexander Koldobsky
Affiliation:
University of Missouri, Columbia/MPIM
Date:
Wed, 2019-06-19 15:00 - 16:00
Location:
MPIM Lecture Hall
Parent event:
Colloquium
Parent event:
MPI-Oberseminar

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

Posted in
Speaker:
Noriko Yui
Affiliation:
Queen's University Kingston/MPIM
Date:
Thu, 2019-06-27 15:00 - 16:00
Location:
MPIM Lecture Hall
Parent event:
Colloquium
Parent event:
MPI-Oberseminar

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

Posted in
Speaker:
Filip Misev
Affiliation:
MPIM
Date:
Thu, 2019-08-08 15:00 - 16:00
Location:
MPIM Lecture Hall
Parent event:
Colloquium
Parent event:
MPI-Oberseminar

## Puzzles about trisections of 4-manifolds

Posted in
Speaker:
David Gay
Affiliation:
University of Georgia/MPIM
Date:
Thu, 2019-08-29 15:00 - 16:00
Location:
MPIM Lecture Hall
Parent event:
Colloquium
Parent event:
MPI-Oberseminar

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.

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