University of Bonn

The quasi-isometry group QI(X) of a metric space X is a natural group of automorphisms of the space that preserve its large-scale structure. The quasi-isometry groups of most familiar spaces are usually enormous and quite wild. Spaces X for which QI(X) is understood tend to exhibit a sort of rigidity phenomenon: every quasi-isometry of such spaces is close to an isometry. We exploit this phenomenon to address the question of which abstract groups arise as the quasi-isometry groups of metric spaces. This talk is based on joint work with Paula Heim and Joe MacManus.

In this talk I will give a gentle introduction to some aspects of the theory of hypergeometric functions as a natural language for addressing various integrals appearing in quantum field theory (QFT). In particular I will focus on the so-called intersection pairings as well as the differential equations satisfied by the integrals, and I will show how these aspects of the mathematical theory can find a natural interpretation in concrete QFT applications. I will mostly focus on Feynman integrals as paradigmatic example, where the language will shed new light on our most powerful method for computing Feynman integrals as well as their non-local symmetries. I will then give an outlook how these methods could allow us to also learn about integrals appearing in other places in field and string theory, such as Coulomb branch amplitudes, celestial holography and AdS (supergravity and string) amplitudes.

Let F(x,y) be a polynomial over the complex numbers. The Elekes-Ronyai theorem says that if F(x,y) is not essentially addition or multiplication, then F(x,y) exhibits expansion: for any finite subset A, B of complex numbers of size n, the size of F(A,B)={F(a,b):a in A, b in B} will be much larger than n. In fact, it is proved that |F(A,B)|>Cn^{4/3} for some constant C. In this talk, I will present a recent joint work with Martin Bays, which is an asymmetric and higher dimensional version of the Elekes-Rónyai theorem, where A and B can be taken to be of different sizes and y a tuple. This result is achieved via a generalisation of the Elekes-Szabó theorem.

The monadic second-order theory S1S of (ℕ,<) is decidable (it essentially describes ω-automata). Undecidability of the monadic theory of (ℝ,<) was proven by Shelah. Previously, Rabin proved decidability if the monadic quantifier is restricted to Fσ-sets.
We discuss decidability for Borel sets, or even σ-combinations of analytic sets. Moreover, the Boolean combinations of Fσ-sets form an elementary substructure. Under determinacy hypotheses, the proof extends to larger classes of sets.

In this talk, we discuss Sobolev embeddings into rearrangement-invariant function spaces on (regular) domains in $\mathbb{R}^n$ endowed with measures whose decay on balls is dominated by a power $d$ of their radius, called $d$-Frostman measures. We show that these embeddings can be deduced from one-dimensional inequalities for an operator depending on $n$, $d$ and the order $m$ of the Sobolev space. We also point out an interesting feature of this theory - namely that the results take a substantially different form depending on whether the measure is decaying fast ($d\geq n-m$) or slowly ($d<n-m$). This is a
joint work with Andrea Cianchi and Lubos Pick.

The Anderson Hamiltonian is used to model particles moving in
disordered media, it can be thought of as a Schrödiger operator with an
extremely irregular random potential. Using the recently developed theory of
"Paracontrolled Distributions" we are able to define the Anderson
Hamiltonian as a self-adjoint non-positive operator on the 2- and
3-dimensional torus and give an explicit description of its domain.
Then we use these results to solve some semi-linear PDEs whose linear part
is given by the Anderson Hamiltonian, more precisely the multiplicative
stochastic NLS and nonlinear Wave equation.
This is joint work with M. Gubinelli and B. Ugurcan.

Using ideas from paracontrolled calculus, we prove local well-posedness of a renormalized version of the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity forced by an additive space-time white noise on a periodic domain. There are two new ingredients as compared to the parabolic setting. (i) In constructing stochastic objects, we have to carefully exploit dispersion at a multilinear level. (ii) We introduce novel random operators and leverage their regularity to overcome the lack of smoothing of usual paradifferential commutators

I will explain how Lurie‘s approach to L-theory via Poincaré categories can be extended to yield cobordism categories of Poincaré objects à la Ranicki. These categories can be delooped by an iterated Q-construction and the resulting spectrum is a derived version of Grothendieck-Witt-theory.  Its homotopy type can be described in terms of K- and L-theory as conjectured by Hesselholt-Madsen. Furthermore, it has a clean universal property analogous to that of K-theory, localisation sequences in much greater generality than classical Grothendieck-Witt theory, gives a cycle description of Weiss-Williams‘ LA-theory and allows for maps from the geometric cobordism category, refining and unifying various known invariants.

All original material is joint work with B.Calmès, E.Dotto, Y.Harpaz, M.Land, K.Moi, D.Nardin, T.Nikolaus and W.Steimle.

We consider the discrete Bilaplacian on a cube in two and three dimensions with zero boundary data and prove estimates for its Green's function that are sharp up to the boundary. The main tools in the proof are Caccioppoli estimates and a compactness argument which allows one to transfer estimate for continuous PDEs to the discrete setting. One application of these estimates is to understand the so-called membrane model from statistical physics, and we will outline how these estimates can be applied to understand the phenomenon of entropic repulsion. We will also describe some connections to numerical analysis, in particular another approach to these estimates based on convergence estimates for finite difference schemes.

The (kinetic) Langevin equation is an SDE with degenerate noise that describes the motion of a particle in a force field subject to damping and random collisions. It is also closely related to Hamiltonian Monte Carlo methods. An important open question is, why in certain cases kinetic Langevin diffusions seem to approach equilibrium faster than overdamped Langevin diffusions. So far, convergence to equilibrium for kinetic Langevin diffusions has almost exclusively been studied by analytic techniques. In this talk, I present a new probabilistic approach that is based on a specific combination of reflection and synchronous coupling of two solutions of the Langevin equation. The approach yields rather precise bounds for convergence to equilibrium at the borderline between the overdamped and the underdamped regime, and it may help to shed some light on the open question mentioned above.