University of Münster

We study the definability of valuation rings in ordered fields (in the language of ordered rings). We show that any henselian valuation ring that is definable in the language of ordered rings is already definable in the language of rings. However, this does not hold when we drop the assumption of henselianity.

This is joint work with Philip Dittmann, Sebastian Krapp and Salma Kuhlmann.

Since the completion of the Elliott classification programme it is an important question to ask which C*-algebras satisfy the assumptions of the classification theorem. We will ask this question for the case of crossed-product C*-algebras associated to actions of nonamenable groups and focus on two extreme cases: Actions on commutative C*-algebras and actions on simple C*-algebras. It turns out that for a large class of nonamenable groups, classifiability of the crossed product is automatic under the minimal assumptions on the action. This is joint work with E. Gardella, S. Geffen, P. Naryshkin and A. Vaccaro. 

We say that a graph G is Local-to-Global rigid if there exists R>0 such that every other graph whose balls of radius R are isometric to the balls of radius R in G is covered by G. Examples include the Euclidean building of PSLn(Qp). We show that the rigidity of the building goes further by proving that a reconstruction is possible from only a partial local information, called “print”. We use this to prove the rigidity of graphs quasi-isometric to the building among which are the torsion-free lattices of PSLn(Qp).

Consider simple rank-one Lie groups $SO(n, 1)$, $SU(n, 1)$ and $Sp(n ,1)$ ($n>1$). They are the isometry groups of real, complex and quaternionic hyperbolic spaces respectively.

By a result of Kostant, the trivial representation of $Sp(n ,1)$ is isolated in the space of irreducible unitary representations on Hilbert spaces. That is, $Sp(n ,1)$ has Kazhdan’s property (T) which is equivalent to the vanishing of 1st cohomology of the group in all unitary representations. This is in contrast to the case of $SO(n ,1)$ and $SU(n ,1)$ where they have the Haagerup approximation property, a strong negation of property (T).

This dichotomy between $SO(n ,1)$, $SU(n ,1)$ and $Sp(n ,1)$ disappears when we consider so-called uniformly bounded representations on Hilbert spaces. By a result of Cowling in 1980’s, the trivial representation of $Sp(n ,1)$ is no longer isolated in the space of uniformly bounded representations. Moreover, there is a uniformly bounded representation of $Sp(n ,1)$ with non-zero first cohomology group.

The goal of this talk is to describe these facts.

Forcing axioms spell out the dictum that if a statement can be forced, then it is already true. The P_max axiom (*) goes beyond that by claiming that if a statement is consistent, then it is already true. Here, the statement in question needs to come from a resticted class of statements, and "consistent" needs to mean "consistent in a strong sense". It turns out that (*) is actually equivalent to a forcing axiom, and the proof is by showing that the (strong) consistency of certain theories gives rise to a corresponding notion of forcing producing a model of that theory. Our result builds upon earlier work of R. Jensen and (ultimately) Keisler's "consistency properties".

(This is Part I of a two-part talk.)

The Elekes-Szabó's theorem says very roughly that if a complex irreducible subvariety V of X*Y*Z has ''too many'' intersection with cartesian products of finite sets, then V is in correspondence with the graph of multiplication of an algebraic group G. It was noticed by Breuillard and Wang that the algebraic group G must be abelian. There is a constraint for the finite sets witnessing ''many'' intersections with V, namely a condition called in general position, which plays a key role in forcing the group to be abelian.  In this talk, I will present a result which shows that in the case of the graph of complex algebraic groups, with a weaker general position assumption, nilpotent groups will appear. More precisely, for a connected complex algebraic group G the following are equivalent:

1. The graph of G has ''many'' intersections with finite sets in weak general position;

2. G is nilpotent;

3. The ultrapower of G has a pseudofinite coarse approixmate subgroup in weak general position.

Surprisingly, the proof of the direction from 2 to 3 invokes some form of generic Mordell-Lang theorem for commutative complex algebraic groups.

This is joint work with Martin Bays and Jan Dobrowolski.

In January of this year, a solution to Connes' Embedding Problem was announced on arXiv. The paper itself deals firmly in the realm of information theory and relies on a vast network of implications built by many hands over many years to get from an efficient reduction of the so-called Halting problem back to the existence of finite von Neumann algebras that lack nice finite-dimensional approximations. The seminal link in this chain was forged by astonishing results of Kirchberg which showed that Connes' Embedding Problem is equivalent to what is now known as Kirchberg's QWEP Conjecture. In this talk, I aim to introduce Kirchberg's conjecture and to touch on some of the many deep insights in the theory surrounding it.

The Cuntz semigroup is a geometric refinement of K-theory that plays an important role in the structure theory of C*-algebras. It is defined analogously to the Murray-von Neumann semigroup by using equivalence classes of positive elements instead of projections.
Starting with the definition of the Cuntz semigroup of a C*-algebra, we will look at some of its classical applications. I will then talk about the recent breakthroughs in the structure theory of Cuntz semigroups and some of the consequences.

The optimal matching problem is about the rate of convergence
in Wasserstein distance of the empirical measure of iid uniform points
to the Lebesgue measure. We will start by reviewing the macroscopic
behaviour of the matching problem and will then report on recent results
on the mesoscopic behaviour in the thermodynamic regime. These results
rely on a quantitative large-scale linearization of the Monge-Ampere
equation through the Poisson equation. This is based on joint work with
Michael Goldman and Felix Otto.