# Winterter term 2021/22

## February 10, 2022

*13:15 and 14:30 (DS&MP), Online via Zoom*

https://uni-jena-de.zoom.us/j/62215856917

Meeting ID: 622 1585 6917

Passcode: 632172

13:15 **Henrik Kreidler** (Bergische Universität Wuppertal)

**The Furstenberg-Zimmer structure theorem revisited**

*Abstract*:

The Furstenberg-Zimmer theorem is a key structural result in ergodic the- ory. It allows, e.g., to show multiple recurrence for measure-preserving systems which, by Furstenberg's correspondence principle, is equivalent to the theorem of Szemerédi on arithmetic progressions. In this talk we propose a new operator theoretic approach to this classical result. We show that, in essence, the Furstenberg- Zimmer theorem is a consequence of operator theory on so-called Kaplansky- Hilbert modules which are natural relative versions of classical Hilbert spaces.

This functional analytic perspective provides a systematic approach to extensions of measure-preserving systems. In addition, it allows to drop any countability assumptions yielding the structure theorem for actions of arbitrary groups on arbitrary probability spaces. It therefore contributes to a recent endeavour by Asgar Jamneshan and Terence Tao to remove such assumptions from classical results of ergodic theory. This is a joint work with Nikolai Edeko (Zurich) and Markus Haase (Kiel).

14:30 **Yonatan Gutman** (Institute of Mathematics of the Polish Academy of Sciences, Warsaw)

**Maximal pronilfactors and a topological Wiener-Wintner theorem**

*Abstract:*

For strictly ergodic systems, we introduce the class of CF-Nil(*k*) systems: systems for which the maximal measurable and maximal topological *k*-step pronilfactors coincide as measure-preserving systems. Weiss' theorem implies that such systems are abundant in a precise sense.

We show that the CF-Nil(*k*) systems are precisely the class of minimal systems for which the *k*-step nilsequence version of the Wiener-Wintner average converges everywhere. As part of the proof, we establish that pronilsystems are coalescent both in the measurable and topological categories. In addition, we characterize a CF-Nil(*k*) system in terms of its (*k*+1)-th dynamical cubespace. In particular, for* k*=1, this provides for strictly ergodic systems a new condition equivalent to the property that every measurable eigenfunction has a continuous version.

Joint work with Zhengxing Lian.

(Contact: T. Jäger / M. Dafinger)

## December 16, 2021

*14:30 -16:00 (A&G), online via Zoom*

**Prof. Dr. Lorenz Schwachhöfer ** (TU Dortmund)

**What Lie algebras can tell us about Jordan algebras**

*Abstract: You can find the pdf file here.*

(Contact: M. Dafinger / V. Matveev)

## December 9, 2021

*14:30 -16:00 (A&G), online via Zoom*

**Dr. Georgios Moschidis ** (Princeton University, New Jersey)

**Weak turbulence and formation of black holes in general relativity**

*Abstract: *

The AdS instability conjecture provides an example of weak turbulence appearing in the dynamics of the Einstein equations in the presence of a negative cosmological constant. The conjecture claims the existence of arbitrarily small perturbations to the initial data of Anti-de Sitter spacetime which, under evolution by the vacuum Einstein equations with reflecting boundary conditions at conformal infinity, lead to the formation of black holes after sufficiently long time. In this talk, I will first introduce the setup of the initial-boundary value problem for the Einstein equations and discuss how non-trivial geometric features such as black holes can appear dynamically in the evolution of those equations. I will then present a rigorous proof of the AdS instability conjecture in the setting of the spherically symmetric Einstein-scalar field system. If time permits, I will also discuss possible paths for extending these ideas to the vacuum case.

(Contact: M. Dafinger / V. Matveev)

## December 2, 2021

*14:30 -16:00 (A&G), online via Zoom*

**Prof. Thomas Barthelmé **(Queen's University, Kingston, Canada)

**Geometry and flexibility in a fixed conformal class of a surface **

*Abstract: *(joint work with Alena Erchenko) Classical works from the 80s and onwards gave famous inequalities between different geometrical or dynamical invariants for negatively curved metrics on surfaces, as well as rigidity results.

While the proofs often used conformal classes, the setting for the results was the space of negatively (or non-positively) curved metrics on a fixed surface. In particular, the geometry of the set of non-positively curved Riemannian metric in a fixed conformal class on a surface was not studied as such, and a number of fairly basic questions were left open.

In this talk, I will explain how one can get a coarse understanding of the geometry inside that space, giving some new bounds on certain geometrical and dynamical invariants, as well as finding the flexibility of others, in the sense of Katok’s flexibility program.

(Contact: M. Dafinger / V. Matveev)

## November 25, 2021

*14:30 -16:00 (A&G), online via Zoom*

**Prof. Colin Guillarmou** (Universite Paris-Saclay)

**Geodesic stretch and marked length spectrum rigidity problem**

*Abstract: *I will explain recent results obtained with Lefeuvre and with Lefeuvre-Knieper on the question of determination of a Riemannian metric with Anosov flow from its marked length spectrum.

(Contact: M. Dafinger / V. Matveev)

## November 18, 2021

*14:30 -16:00 (A&G), online via Zoom*

**Prof. Dr. Katharina Neusser** (Masaryk University, Brno)

**Cone structures and parabolic geometries (based on joint work with J.-M. Hwang)**

*Abstract: *A cone structure on a complex manifold M is a closed submanifold C of the projectivized tangent bundle of M which is submersive over M. Such structures arise naturally in differential and algebraic geometry, and when they do, they are typically equipped with a conic connection that specifies a distinguished family of curves on M in direction of C. In differential geometry, as we will see, cone structures with conic connections arise from so-called holomorphic parabolic geometries, a classical example of which is the null cone bundle of a holomorphic conformal structure with the conic connection given by the null-geodesics. In algebraic geometry, we have the cone structures consisting of varieties of minimal rational tangents (VMRT) given by minimal rational curves on uniruled projective manifolds. In this talk we will discuss various examples of cone structures and will introduce two important invariants for conic connections. As an application of the study of these invariants, we obtain a local-differential-geometric version of the global algebraic-geometric recognition theorem of Mok and Hong–Hwang, which recognizes certain generalized flag varieties from their VMRT-structures.

(Contact: M. Dafinger / V. Matveev)

## November 4, 2021

*14:30 -16:00 (A&G), online via Zoom*

**Frederic Weber ** (Universität Ulm)

**Non-local Bakry-Émery theory and related functional inequalities**

*Abstract: * The curvature-dimension condition of Bakry and Émery constitutes a powerfull tool to show various functional inequalities for Markov semigroups and their generators. However, a key assumption in the classical Bakry-Émery theory is the validity of certain chain rules, which do not hold in genreal for non-local operators. In this talk we identify a curvature-dimension condition for a large class of non-local operators that implies modified logarithmic Sobolev inequalities under positive curvature bounds, logarithmic entropy-information inequalities under positive curvature and finite dimension bounds and Li-Yau inequalities under non-negative curvature and finite dimension bounds. While we mainly focus on generators of continuous-time Markov chains, we also discuss the fractional Laplacian as an important example.

(Contact: D. Lenz)

## October 28, 2021

*14:30 -16:00 (A&G), online via Zoom*

**Prof. Christos Saroglou** (University of Ioannina, Greece)

**A non-existence result for the *** L*_{p}-Minkowski problem

*L*-Minkowski problem

_{p}*Abstract: You can find the pdf file here.*

(Contact: Th. Wannerer)

## October 21, 2021

*14:15-15:45 (A&G), Online via Zoom*

**Dr. Alexey Bolsinov ** (Loughborough University)

**Frobenius pencils and compatible non-homogeneous Poisson structures**

*Abstract: *This work, joint with A.Konyaev and V.Matveev, is another application of Nijenhuis geometry in the theory of infty-dimensional integrable systems. We study compatible (differential geometric) non-homogeneous Poisson structures of the form B + A, where B and A are homogeneous Darboux-Poisson structures of order 3 and 1 respectively. The problem is reduced to an algebraic problem, namely to classification of pairs of compatible Frobenius algebras. We solve it completely, under some minor genericity conditions, by methods of differential geometry. As an application in mathematical physics we construct new interesting examples of multicomponent integrable (PDE-)dynamical systems

(Contact: M. Dafinger / V. Mateev)

*16:15-17:45 (DS&MP), SR 3517*

**Jobst Ziebell** (FSU Jena, Theoretisch-Physikalischen Institut)

**The Wetterich equation of a real scalar field**

*Abstract: *In the theoretical physics community the Wetterich equation has become an essential tool in the study of renormalisation group flows of quantum field theories. Its cousin, the Polchinski equation, has seen several applications in mathematical physics but so far the mathematical step to Wetterich's equation had not been taken. I will present a simple regularisation scheme that is designed to "keep everything smooth" and allows a rigorous derivation of the equation.

(Contact: D. Hasler)

## October 14, 2021

*14:30 -16:00 (A&G), online via Zoom*

**Prof. Dmitry Faifman **(Tel Aviv University)

**The Funk metric, between convex and projective geometry.**

*Abstract: *The Funk metric in the interior of a convex set is a lesser-known cousin of the Hilbert metric. The latter generalizes the Beltrami-Klein model of hyperbolic geometry, and both have straight segments as geodesics, thus constituting solutions of Hilbert's 4th problem alongside normed spaces. Unlike the Hilbert metric, the Funk metric is not projectively invariant. I will explain how, nevertheless, the Funk metric gives rise to many projective invariants, which moreover enjoy a duality extending results of Holmes-Thompson and Alvarez Paiva on spheres of normed spaces and Gutkin-Tabachnikov on Minkowski billiards. Time permitting, I will also discuss how extremizing the volume of metric balls in Funk geometry yields extensions of the Blaschke-Santalo inequality and Mahler conjecture.

(Contact: Th. Wannerer)