Dynamics Lecture Series
Specialised lecture series on dynamical systems, geometry, and probability.
- 2025-2026 (Term 2): Tobias Hartnick (KIT)
Approximate lattices and transverse dynamical systems
Lecture 1: Tuesday March 3rd, 14:00-15:00 (B3.02) - ETDS Seminar
Lecture 2: Wednesday March 4th, 14:00-15:00 (B3.01)
Lecture 3: Thursday March 5th, 13:30-14:30 (B3.02) - Geo & Top Seminar
Lecture 1: From approximate lattices to transverse dynamical systems
In this lecture we first introduce approximate subgroups; we then define approximate lattices and motivate the definition by various examples from different areas of mathematics. We then sketch some basic results from the structure theory of approximate lattice, due to Machado and Hrushovski, to get a feeling for the class of approximate lattices. The main focus of the talk will then be on the construction of non-homogeneous dynamical systems associated with approximate lattices which generalize the homogeneous dynamical system associated with a lattice and to sketch some first applications. This will lead us to the class of transverse dynamical systems, introduced in work with Bj\”orklund and Karasik, and further studied by Avraham-Re’em, Bj\”orklund, Cullman and with Hughes. The final goal of the talk will be to discuss some of the main dynamical features of such systems.
Lecture 2: Counting approximate lattice points
In this lecture, which is based on joint works with Bj\”orklund and Hughes, we focus on equidistribution phenomena in transverse dynamical systems. More precisely we are going to give conditions under which translates of transverse measures equidistribute. We then apply this theory to the non-homogeneous systems discussed in Lecture 1 in order to obtain counting asymptotic for approximate lattice points in various homogeneous spaces. Our results are analogous to lattice point counting results in the style of Margulis and Eskin-McMullen. We will also obtain some quantitative density results in the spirit of intrinsic diophantine approximation, but in the context of Archimedean places.
Lecture 3: Geometry of approximate subgroups
In this lecture, which is independent of the first two ones, we will explain some of the more geometric aspects of approximate subgroups related to geometric group theory, measure equivalence and bounded cohomology. A key player in this game is a certain ample groupoid constructed from the local patterns of the underlying point set. This is based on joint works with Bj\”orklund, Cordes, Sarti and Toni\’c.
- 2025-2026 (Term 2): Timothée Bénard (CNRS)
Random walks on homogeneous spaces (Lecture 1 = seminar)
Effective equidistribution of random walks on the torus (Lectures 2-3-4-5)
- 2024-2025 (Term 1): Simon Machado (ETH Zurich)
"The Dynamics of point sets: local and global multiplicative structures"