Seminars a.y 2022-2023

The use of functional-analytic methods in ergodic theory, in particular Hilbert space operator theory, is as old as ergodic theory itself (mean ergodic theorem, systems with discrete spectrum vs. weakly mixing systems). However, the classical functional-analytic toolbox is not powerful enough to cover extensions of measure-preserving systems. For this, one needs “relative” or “conditional” versions of the classical functional-analytic objects and results. For example, the conditional version of a Hilbert space is a so-called Kaplansky-Hilbert module. Whereas this notion has long been known to specialists, its relevance for ergodic theory has been recognized only relatively recently, see [1].

Passing to the functional-analytic structures makes the underlying point set dynamics “vanish”. This is the key to liberate the theory from the usual countability restrictions (e.g. countable group actions on standard Borel probability spaces) and allows to prove “uncountable” versions of the classical theorems.

This short lecture series, based on joint work with Nikolai Edeko (Zurich) and Henrik Kreidler (Wuppertal), is comprised of two sessions.

Session 1: The Classical Picture. (Classical measure-preserving systems; von Neumann’s ergodic theorem; Halmos–von Neumann theorem; weak mixing vs. discrete spectrum; the (classical) Furstenberg–Zimmer structure theory.)

Session 2: The Abstract Approach. (Topological Models; Stone Algebras and Kaplansky-Hilbert modules; operator theory (spectral theorem); application to dynamical systems; the (general) Furstenberg–Zimmer structure theory.)

[1] Edeko, N. and Haase, M. and Kreidler, H.: A Decomposition Theorem for Unitary Group Representations on Kaplansky–Hilbert Modules and the Furstenberg–Zimmer Structure Theorem, to appear in: Analysis Mathematica. 

https://arxiv.org/abs/2104.04865

The use of functional-analytic methods in ergodic theory, in particular Hilbert space operator theory, is as old as ergodic theory itself (mean ergodic theorem, systems with discrete spectrum vs. weakly mixing systems). However, the classical functional-analytic toolbox is not powerful enough to cover extensions of measure-preserving systems. For this, one needs “relative” or “conditional” versions of the classical functional-analytic objects and results. For example, the conditional version of a Hilbert space is a so-called Kaplansky-Hilbert module. Whereas this notion has long been known to specialists, its relevance for ergodic theory has been recognized only relatively recently, see [1].

Passing to the functional-analytic structures makes the underlying point set dynamics “vanish”. This is the key to liberate the theory from the usual countability restrictions (e.g. countable group actions on standard Borel probability spaces) and allows to prove “uncountable” versions of the classical theorems.

This short lecture series, based on joint work with Nikolai Edeko (Zurich) and Henrik Kreidler (Wuppertal), is comprised of two sessions.

Session 1: The Classical Picture. (Classical measure-preserving systems; von Neumann’s ergodic theorem; Halmos–von Neumann theorem; weak mixing vs. discrete spectrum; the (classical) Furstenberg–Zimmer structure theory.)

Session 2: The Abstract Approach. (Topological Models; Stone Algebras and Kaplansky-Hilbert modules; operator theory (spectral theorem); application to dynamical systems; the (general) Furstenberg–Zimmer structure theory.)

[1] Edeko, N. and Haase, M. and Kreidler, H.: A Decomposition Theorem for Unitary Group Representations on Kaplansky–Hilbert Modules and the Furstenberg–Zimmer Structure Theorem, to appear in: Analysis Mathematica. 

https://arxiv.org/abs/2104.04865

Abstract: 

The classical line of research that investigates the existence of trajectories to the gravitational N-body problem having prescribed growth at infinity has recently been re-energized by the injection of new methods of analysis of perturbative, variational, geometric and/or analytic functional nature.

This talk will focus on proving, for the N-body problem, the existence of action minimizing half entire expansive solutions with prescribed asymptotic direction and initial configuration of the bodies. We will tackle

the cases of hyperbolic, hyperbolic-parabolic and parabolic arcs from a unitary perspective, using a global variational approach consisting in minimizing a renormalized Lagrangian action on a suitable functional space. The talk is based on a joint work with Susanna Terracini.

Abstract: 

In a celebrated paper published in 1979, Gidas, Ni & Nirenberg proved a symmetry result for a rigidity problem. With minimal hypotheses, the authors showed that positive solutions of semilinear elliptic equations in the unit ball are radial and radially decreasing.

This result had a big impact on the PDE community and stemmed several generalizations. In a recent work in collaboration with G. Ciraolo, M. Cozzi & M. Perugini this problem was investigated from a quantitative viewpoint, starting with the following question: given that the rigidity condition implies symmetry, is it possible to prove that if said condition is almost satisfied the problem is almost symmetrical?

With the employment of the method of moving planes and quantitative maximum principles we are able to give a positive answer to the question, proving approximate radial symmetry and almost monotonicity for positive solutions of the perturbed problem.

Abstract: 

We discuss the existence of rotating spirals for three-component competition-diffusion systems in B₁ ⊂ R², under the Neumann and the non-homogeneous Dirichlet boundary conditions. 

Abstract: 

The integrability of free billiards in classical mechanics was first studied by G. D. Birkhoff, and later by Y. Sinai, their chaotic behavior and ergodicity were investigated. L. Boltzmann proposed planar billiard systems in the presence of a central force and predicted that such a billiard with a straight reflective wall that does not pass through the center would be ergodic. Recently, Gallavotti-Jauslion showed that under the Kepler potential, such billiards are not ergodic, but rather integrable. In this talk, we will show that conformal mappings can be used to relate various integrable billiard systems in the plane, including Boltzmann's integrable billiard system, and we will show that more general integrable billiard systems can be constructed under the Keplerian potential. Furthermore, by projective transformation, it is possible to obtain the corresponding integrable billiard systems on curved surfaces from these integrable billiard systems on the plane. This talk is based on collaborative research with Lei Zhao from the University of Augsburg. 

Abstract: 

The orbit determination (OD) problem for small bodies of the solar system, like asteroids, attracted the attention of famous mathematicians, as C.F. Gauss and P.-S. Laplace. After introducing the problem, their solutions will be briefly presented.

With the progress of technology and the realization of observation instruments more and more efficient, new mathematical challenges are arising. I will describe some recent initial OD methods that can help to process the huge amount of data produced by modern asteroid surveys. In my presentation I will focus especially on some algebraic aspects related to the proposed methods.