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Autumn Semester 2024

Date / Time Speaker Title Location
25 September 2024
13:30-14:30 		Prof. Dr. Adam Kanigowski
University of Maryland
Details

Ergodic theory and dynamical systems seminar

Title Horocycle flows at semi-prime times
Speaker, Affiliation Prof. Dr. Adam Kanigowski, University of Maryland
Date, Time 25 September 2024, 13:30-14:30
Location Y27 H 28
Abstract In this joint work with G. Forni and M. Radziwill we study the orbits of horocycle flows when sampled at semi-primes (integers with at most two prime factors). We show that if the lattice is arithmetic then such orbits equidistribute towards Haar measure for every non-periodic point. During the talk I will outline a new criterion for semi-primes, and describe main ideas of the proof.
Horocycle flows at semi-prime timesread_more
Y27 H 28
2 October 2024
13:45-14:45 		Prof. Dr. Roman Sauer
Karlsruhe Institute for Technology
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Ergodic theory and dynamical systems seminar

Title The Kazhdan property donates an extra dimension
Speaker, Affiliation Prof. Dr. Roman Sauer, Karlsruhe Institute for Technology
Date, Time 2 October 2024, 13:45-14:45
Location Y27 H 28
Abstract The waist inequality for the sphere by Gromov is wonderful result of geometric measure theory. Formulated appropriately for families of spaces, a (uniform) waist inequality for a family of Riemannian manifolds is the Riemannian analog of higher dimensional expanders. We formulate a conjectural picture for locally symmetric spaces. Finally, we show that the Kazhdan property alone gives rise not only to expanders, which is classical, but also to 2-dimensional expanders. An extra dimension for free! This is joint work with Uri Bader.
The Kazhdan property donates an extra dimensionread_more
Y27 H 28
9 October 2024
13:30-14:30 		Dr. Pouya Honaryar
Universität Zürich
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Ergodic theory and dynamical systems seminar

Title Central limit theorem for homology of simple closed curves
Speaker, Affiliation Dr. Pouya Honaryar, Universität Zürich
Date, Time 9 October 2024, 13:30-14:30
Location Y27 H 28
Abstract Fix a hyperbolic surface $X$, and for $L > 0$, let $\mathcal{S}_L(X)$ denote the set of simple closed geodesics of length at most $L$ on $X$. Fixing a norm on $H_1(X, \mathbb{R})$, we may ask what is the statistics of the norm of homology class of $\alpha$, denoted by $[\alpha]$, when $\alpha$ is chosen randomly uniformly from $\mathcal{S}_L(X)$, as $L \rightarrow \infty$? For example, does one expect the norm of $[\alpha]$ to be of order $L$ or smaller? We answer this question by proving a CLT-type result for the norm of homology of a randomly chosen curve in $\mathcal{S}_L(X)$. We discuss the main steps to reduce the desired CLT to a CLT for the Kontsevich-Zorich cocycle obtained by Forni-Saqban. (Joint work with F. Arana-Herrera)
Central limit theorem for homology of simple closed curvesread_more
Y27 H 28
16 October 2024
13:30-14:30 		Dr. Katy Loyd
University of Maryland
Details

Ergodic theory and dynamical systems seminar

Title Pointwise Ergodic Averages along Sequences of Slow Growth
Speaker, Affiliation Dr. Katy Loyd, University of Maryland
Date, Time 16 October 2024, 13:30-14:30
Location Y27 H 28
Abstract Following Birkhoff's proof of the Pointwise Ergodic Theorem, it is natural to consider whether convergence still holds along various subsequences of the integers. In 2020, Bergelson and Richter showed that in uniquely ergodic systems, pointwise convergence holds along the number theoretic sequence $\Omega(n)$, where $\Omega(n)$ denotes the number of prime factors of $n$, with multiplicities. In this talk, we will see that by removing this assumption, a pointwise ergodic theorem does not hold along $\Omega(n)$. In fact, $\Omega(n)$ satisfies a notion of non-convergence called the strong sweeping out property. We then further classify the strength of this non-convergence behavior by considering weaker notions of averaging. Time permitting, we will introduce a more general criterion for identifying slow growing sequences with the strong sweeping out property (based on joint work with S. Mondal).
Pointwise Ergodic Averages along Sequences of Slow Growthread_more
Y27 H 28
23 October 2024
13:30-14:30 		Prof. Dr. Sobhan Seyfaddini
ETH Zürich
Details

Ergodic theory and dynamical systems seminar

Title The closing lemma and Lagrangian submanifolds
Speaker, Affiliation Prof. Dr. Sobhan Seyfaddini, ETH Zürich
Date, Time 23 October 2024, 13:30-14:30
Location Y27 H 28
Abstract We will discuss the smooth closing lemma for Hamiltonian diffeomorphisms with invariant Lagrangians.  Based on joint work with Erman Cineli & Shira Tanny.
The closing lemma and Lagrangian submanifoldsread_more
Y27 H 28
30 October 2024
13:30-14:30 		Prof. Dr. François Maucourant
Université Rennes I
Details

Ergodic theory and dynamical systems seminar

Title Discriminants of periodic geodesics on the modular surface
Speaker, Affiliation Prof. Dr. François Maucourant, Université Rennes I
Date, Time 30 October 2024, 13:30-14:30
Location Y27 H 28
Abstract We'll recall how to attach to a periodic geodesic of H^2/PSL(2,Z) an arithmetical quantity, its discriminant. After discussing how to pick a geodesic at random, we will show that 'most' geodesics have large discriminant when ordered by length, and that 42% of them have a fundamental discriminant.
Discriminants of periodic geodesics on the modular surfaceread_more
Y27 H 28
6 November 2024
13:30-14:30 		Dr. Hamid Al-Saqban
Universität Paderborn
Details

Ergodic theory and dynamical systems seminar

Title A Central Limit Theorem for the Kontsevich-Zorich cocycle
Speaker, Affiliation Dr. Hamid Al-Saqban, Universität Paderborn
Date, Time 6 November 2024, 13:30-14:30
Location Y27 H 28
Abstract The Kontsevich-Zorich (KZ) cocycle is a key dynamical system that is closely related to the derivative cocycle of the Teichmüller geodesic flow. We will state and sketch a proof of a central limit theorem for the KZ cocycle, and explain some of the motivations, especially towards the goal of proving the existence of large fluctuations of the Hodge norm of the parallel transport of vectors along Teichmüller horocycles. Such fluctuations were leveraged by Chaika-Khalil-Smillie in their work on the ergodic measures of the Teichmüller horocycle flow. Our work is joint with Giovanni Forni.
A Central Limit Theorem for the Kontsevich-Zorich cocycleread_more
Y27 H 28
13 November 2024
13:30-14:30 		Prof. Dr. Michael Hochman
The Hebrew University of Jerusalem
Details

Ergodic theory and dynamical systems seminar

Title New results on embedding and intersections of self-similar sets
Speaker, Affiliation Prof. Dr. Michael Hochman, The Hebrew University of Jerusalem
Date, Time 13 November 2024, 13:30-14:30
Location Y27 H 28
Abstract I will discuss the problem of affinely embedding self-similar sets in the line into other such sets. Conjecturally, embedding is precluded when the contraction ratios of the defining maps are incommensurable. This is closely related to conjectures on intersections of fractals, but in the open cases even the embedding problem is challenging. I will describe recent joint work with Amir Algom and Meng Wu in which we confirm the conjecture whenever the contraction ratios are algebraic numbers, and also for a.e. choice of parameters. I will discuss the proof. If time permits, I will explain how this is related to the dimension of alpha-beta sets, and describe recent examples that show that the problem is more delicate than anticipated.
New results on embedding and intersections of self-similar setsread_more
Y27 H 28
20 November 2024
13:30-14:30 		Prof. Dr. Daren Wei
National University of Singapore
Details

Ergodic theory and dynamical systems seminar

Title Time Change Rigidity of Unipotent Flows
Speaker, Affiliation Prof. Dr. Daren Wei, National University of Singapore
Date, Time 20 November 2024, 13:30-14:30
Location Y27 H 28
Abstract Two non-isomorphic ergodic measure preserving flows can become isomorphic if one of the systems undergoes an appropriate time change. In this case we will say that these flows are Kakutani equivalent to each other. We say that an ergodic flow is loosely Kronecker if it is Kakutani equivalent to the straight line flow on (say) a two torus in an irrational direction (the exact direction is immaterial as these are all equivalent to each other). Landmark work of Ratner from the late 70s (that paved the way to her even more famous results on orbit closures and equidistribution of unipotent flows) establishes that 1) the horocycle flow on any finite area surface of constant negative curvature is loosely Kronecker. 2) the product of two such flows is not loosely Kronecker. It remained an open problem whether e.g. products of two horocycle flows are Kakutani equivalent to each other. We show unipotent flows are very rigid under time changes, and indeed unless the flows are loosely Kronecker, two unipotent flows are Kakutani equivalent if and only if they are isomorphic as measure preserving flows. This is a joint work with Elon Lindenstrauss
Time Change Rigidity of Unipotent Flowsread_more
Y27 H 28
27 November 2024
13:30-14:30 		Prof. Dr. Dimitry Dolgopyat
University of Maryland
Details

Ergodic theory and dynamical systems seminar

Title Statistical properties of random dynamical systems
Speaker, Affiliation Prof. Dr. Dimitry Dolgopyat, University of Maryland
Date, Time 27 November 2024, 13:30-14:30
Location Y27 H 28
Abstract We survey results about statistical properties of random dynamical systems and describe a number of open questions.
Statistical properties of random dynamical systemsread_more
Y27 H 28
4 December 2024
13:30-14:30 		Prof. Dr. Jens Marklof
University of Bristol
Details

Ergodic theory and dynamical systems seminar

Title The moduli space of twisted Laplacians and random matrix theory
Speaker, Affiliation Prof. Dr. Jens Marklof, University of Bristol
Date, Time 4 December 2024, 13:30-14:30
Location Y27 H 28
Abstract One of the long-standing conjectures in quantum chaos is that the spectral statistics of quantum systems with chaotic classical limit are governed by random matrix theory. Despite convincing heuristics, there is currently not a single example where this phenomenon can be established rigorously. Rudnick recently proved that the spectral number variance for the Laplacian of a large compact hyperbolic surface converges, in a certain scaling limit and when averaged with respect to the Weil-Petersson measure on moduli space, to the number variance of the Gaussian Orthogonal Ensemble of random matrix theory. In this lecture we will review Rudnick's approach and extend it to explain the emergence of the Gaussian Unitary Ensemble for twisted Laplacians (which break time-reversal symmetry) and to the Gaussian Symplectic Ensemble for Dirac operators. This addresses a question of Naud, who obtained analogous results for twisted Laplacians on high genus random covers of a fixed compact surface. This lecture is based on joint work with Laura Monk (Bristol).
The moduli space of twisted Laplacians and random matrix theoryread_more
Y27 H 28
5 December 2024
11:00-12:00 		Max Auer
University of Maryland
Details

Ergodic theory and dynamical systems seminar

Title Trimmed Ergodic Sums for Non-integrable Functions over Irrational Rotations
Speaker, Affiliation Max Auer, University of Maryland
Date, Time 5 December 2024, 11:00-12:00
Location Y34 K 01
Abstract For a probability-preserving ergodic dynamical system (X, T, u) and an integrable function f, the asymptotic almost sure behaviour of the ergodic sums S_N (f ) is described by the Birkhoff Ergodic Theorem. The situation is much more complicated if f is not integrable, a result by Aaronson forbids almost sure Limit Theorems. Instead, the notion of trimming is introduced, by excluding the largest observations from S_N(f) . Trimmed limit Theorems are well-studied for iids. In the dynamical setting, results are only known for systems exhibiting strong mixing behaviour. We study trimming for irrational rotations in and functions with polynomial singularities.
Trimmed Ergodic Sums for Non-integrable Functions over Irrational Rotationsread_more
Y34 K 01
11 December 2024
13:30-14:30 		Dr. Meg Doucette
University of Maryland
Details

Ergodic theory and dynamical systems seminar

Title Smooth Models for Fibered Partially Hyperbolic Systems
Speaker, Affiliation Dr. Meg Doucette, University of Maryland
Date, Time 11 December 2024, 13:30-14:30
Location Y27 H 28
Abstract I will discuss the existence and construction of smooth models for certain fibered partially hyperbolic systems. Fibered partially hyperbolic systems are partially hyperbolic diffeomorphisms that have an integrable center bundle, tangent to a continuous invariant fibration by invariant submanifolds. I will explain how under certain restrictions on the fiber, any fibered partially hyperbolic system over a nilmanifold is leaf conjugate to a smooth model that descends to a hyperbolic nilmanifold automorphism on the base. I will then discuss how the restrictions on the fiber can be replaced by certain dynamical restrictions on the behavior of the fibered system in the center direction. This is part of a joint work in progress with Jon Dewitt and Oliver Wang.
Smooth Models for Fibered Partially Hyperbolic Systemsread_more
Y27 H 28

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