Our working seminar takes place every Monday at 13:00 - 15:00 in HG G 19.2 (ETH).
Further details - including titles, abstracts, and recordings (where available), can be found below. The seminar is structured as seperate minicourses, which should be introductory and aimed at a broad audience within our group.
Recordings can be found here, and should be available shortly after the talks.
Topic 1 - Mixing and Ratner's Theorems
Abstract: We start by proving higher order mixing of the $\operatorname{SL}_2(\mathbb{R})$-action on homogeneous spaces $\operatorname{SL}_2(\mathbb{R})/\Gamma$ for lattices $\Gamma \leq \operatorname{SL}_2(\mathbb{R})$. Revisiting Mautner's phenomenon and the Howe-Moore Theorem we first deduce mixing of the action and then go through an argument to deduce higher-order mixing. Time permitting we will use Margulis' "banana trick" to show unique ergodicity of the horocycle flow for cocompact lattices.
Next we discuss the counting method of Eskin-McMullen. Counting problems for lattice points on affine symmetric varieties are related to the equidistribution problems for translated orbits in homogeneous spaces. We will outline how to deduce such equidistribution problems from mixing.
| Date | Speaker | Topic | Notes |
|---|---|---|---|
| 07 Oct | Konstantin/Wooyeon | Higher order mixing and counting problems | |
| 14 Oct | Wooyeon/Manfred | Higher order mixing and counting problems (contd.), introduction to Ratner's Theorems | |
| 21 Oct | Segev/Manfred | Main ideas in the proof of measure classification | Notes |
| 25 Nov | Elias/Claire | Elkies-McMullen equidsitribuiton of $\sqrt{n}$ (mod $1$) |
Topic 2 - KAM Theory
Main reference: Introduction to KAM theory
| Date | Speaker | Topic |
|---|---|---|
| 28 Oct | Andrea/Hao | Introduction to KAM Theory |
| 4 Nov | Andrea/Hao | Arnold's Theorem about analytic circle diffeomorphisms |
| 11 Nov | Andrea/Hao | Moser's Theorem about $C^k$ circle diffeomorphisms |
| 18 Nov | Baptiste | KAM Theory in the Hamiltonian formalism |
Topic 3 - Complex Dynamics
Abstract: In the last three meetings of the working seminar, we will give a brief (but hopefully accessible) introduction to complex dynamics.
The subject studies iterations of holomorphic maps. The resulting dynamics naturally decomposes the phase space into two disjoint sets: the Fatou set (where the dynamics is stable) and the Julia set (where the dynamics is chaotic). For our crash course, we will mainly focus on iterations of rational maps on the Riemann sphere. Our goal is to give a complete classification of the dynamics on the Fatou set in this case.
In the first meeting, we will start by recalling some facts we need from complex analysis. Then we will give the definitions of the Fatou and Julia set. Afterwards, we will spend the remaining time stating and proving some interesting facts about Julia sets.
The only prerequisite needed for this mini-course is some basic complex analysis.
Main reference: John Milnor. Dynamics in one complex variable.
Complementary references: Alan F Beardon. Iteration of rational functions.