January 4, 2024: Jane Wang "Slope gap distributions: from Farey fractions to translation surfaces"
Abstract:
The Farey fractions of order n are an ordered list of reduced fractions between 0 and 1 with denominators at most n. For example, F_4 = {0/1, 1/4, 1/3, 1/2, 2/3, 3/4, 1/1}. A natural question that one could ask is: how random are these sets of fractions? One way to measure their randomness is to ask for a function called their limiting gap distribution. While the gap distribution of the Farey fractions has been known for some time, recent work of Athreya and Cheung rederived the gap distribution using dynamical techniques relating to dynamics on moduli spaces of translation surfaces.
Translation surfaces are geometric surfaces that are locally Euclidean except at finitely many cone points. In this talk, we will explain what the slope gap distribution of a translation surface is and survey some known results about slope gap distributions, including how one can use dynamical properties of the horocycle flow to compute the slope gap distributions of special translation surfaces called Veech surfaces (the Farey fractions are related to the square torus, an example of a Veech surface). We'll then discuss results showing that the slope gap distributions of Veech surfaces have to satisfy some nice properties. This project is joint work with Luis Kumanduri and Anthony Sanchez.
January 18, 2024: Surena Hozoori "Anosov contact metrics, Dirichlet optimization and entropy"
Abstract: In 1984, Chern and Hamilton initiated the study of various energy functionals on contact 3-manifolds. In particular, given an almost complex structure, they define its Dirichlet energy as the L^2-norm of its variations along the associated Reeb flow and ask when the minimum of such energy functional is achieved. We classify all the minimizers of the Dirichlet energy functional, giving a complete answer to the Chern-Hamilton question. Furthermore, in the case of Anosov contact manifolds, we show that the optimization of this functional is closely related to Reeb dynamics and can be computed in terms of its entropy.
January 25, 2024: Mohammad Latifi-Jebelli "An elementary prelude to quantum fields, I"
Abstract: In my two talks on quantum field theory, I aim to explore the mathematical aspects of the theory while keeping it relatable for physicists. During the first talk, I'll discuss the Lagrangian formulation of Euclidean field theories. We'll also dive into Feynman diagrams, exploring their importance from a probabilistic standpoint. At times, we'll consider the idea that quantum field theory might have been discovered through a mathematical lens (much like calculus) with potential applications beyond the realm of physics. In my second talk, I plan to visit measure theory in infinite dimensions and review rigorous constructions of quantum field theories.
February 8, 2024: Tamara Kucherenko "Realization of Anosov Diffeomorphisms on the Torus"
Abstract: We consider area preserving Anosov maps on the two-dimensional torus within a fixed homotopy class. We show that the set of pressure functions for Anosov diffeomorphisms with respect to the geometric potential is equal to the set of pressure functions for the linear Anosov automorphism with respect to H\"{o}lder potentials. We use this result to provide a negative answer to the $C^{1+\alpha}$ version of the question posted in the paper by A. Erchenko: ``if two homotopic smooth area preserving Anosov difeomorphisms have identical pressure functions for their geometric potentials, must they be $C^\infty$ conjugate?" (This is based on joint work with Anthony Quas.)
February 15, 2024: Mohammad Latifi-Jebelli "An elementary prelude to quantum fields, II"
Abstract: In my second talk, I am planning to give an introduction to measure theory in infinite dimensions and review the rigorous constructions of quantum field theories. The infinite-dimensional Gaussian measures play a pivotal role, and they are defined based on finite-dimensional Gaussian projections. It turns out that the representation of infinite-dimensional measures using a density function is problematic, but one can incorporate Fourier domain to establish a rigorous foundation.
February 22, 2024: Ben Call "Mixing Properties of Equilibrium States for Systems Beyond Negative Curvature"
Abstract: I will first introduce some known results about equilibrium states for the geodesic flow in negative curvature. I will then discuss some tools and techniques that can be used to show similar results in settings beyond negative curvature, including rank one nonpositively curved manifolds and translation surfaces. Then, time permitting, I will discuss upcoming work showing the Bernoulli property for a class of equilibrium states, which shows that they are "the most random" they can be in a qualitative sense. This is joint work with Dave Constantine, Alena Erchenko, Noelle Sawyer, and Grace Work.
February 29, 2024: Ian Adelstein "Geometric Manifold Learning"
Abstract: With the advent of high dimensional high throughput data in many fields including biomedicine, social science, physics, finance, etc, there is an increasing need to understand the topology (shape, structure) and geometry (intrinsic distance, curvature) of the data. This process starts with the manifold hypothesis: that high dimensional data actually live on or near a much lower dimensional manifold. One can then study the geometric and topological properties of this manifold to reveal important features of the data. The primary goal of this talk will be to introduce the main bridge between the data manifold and its geometry: the spectral theory for the graph Laplacian. I will also discuss some recent projects that explore the geometry of the data manifold.
March 28, 2024: Thomas Barthelmé "Group actions on bifoliated planes and classification of (pseudo)-Anosov flows in dimension 3"
Abstract: An old problem in dynamical systems is to try to classify Anosov flows up to orbit-equivalence. This question is particularly interesting in dimension 3 where we both have lots of examples and a rich, but still poorly understood, relationships between the dynamics of the flow and the topology of the manifold. By a result of T. Barbot, classifying Anosov flows (or more general pseudo-Anosov flows) in dimension 3 up to orbit equivalence restricts to classifying, up to conjugacy, certain actions of $\pi_1(M)$ on the orbit space, a topological plane with two transverse foliations. In this talk, I will recall the above and discuss a new complete invariant for transitive (pseudo)-Anosov flows which often reduces to just knowing which conjugacy classes in $\pi_1(M)$ are represented by periodic orbits of the flow. If time permits, I’ll talk about some applications with link to contact geometry. This is all joint work with Kathryn Mann, Steven Frankel and Sergio Fenley.
April 23, 2024: Jody Trout "Spectral Kabbalah and the converse functional calculus"
Abstract: Self-adjointness is an important property in functional analysis, differential equations, operator algebras, noncommutative geometry, index theory, spectral geometry and quantum theory. Associated to a self-adjoint (un)bounded operator on a Hilbert space are several structures: spectral measures, Cayley transforms, one-parameter unitary groups, and the C-functional calculus. We show how to organize these into a 2D (and even 3D) “Kabbalahistic” type diagram. In order to complete it, we need to be able to show that there is a converse to the C-functional calculus.
April 25, 2024: Diana Davis "Periodic billiard paths on regular polygons"
Abstract: Mathematicians have long understood periodic trajectories on the square billiard table, which occur when the slope of the trajectory is rational. In this talk, I'll explain my joint work with Samuel Lelièvre on periodic trajectories on other regular polygons, describing their geometry, symbolic dynamics, and group structure. The periodic trajectories are very beautiful, and some of them exhibit a surprising "dense but not equidistributed" behavior.
May 9, 2024: Joel Rosenfeld "Operator decompositions for inverse problems"
Abstract: Dynamic Mode Decomposition is a tool to extract a reduced order model of time series data that has seen a lot of success in the analysis of fluid dynamical data. Originally, DMD began as a matrix decomposition approach, introduced by Schmidt in 2008. Over the past decade, DMD has evolved to involve the Koopman operator as a theoretical underpinning. The Koopman operator exchanges a finite dimensional nonlinear dynamical system for a linear operator over an infinite dimensional function space. We will examine Koopman based DMD from the perspective of kernel functions and as a methodology for resolving a certain class of inverse problems. We will then give a general framework for handling inverse problems through function theoretic operators and kernel spaces. The talk will conclude with a variety of examples, including scattered data approximation and other problems.
May 23, 2024: Daniel Mitsutani "Symmetries of geodesic flows on covers and rigidity"
Abstract: An old result of Bochner proves that closed Riemannian manifolds of negative Ricci curvature admit only finitely many isometries. On the other hand, work beginning with Eberlein, and later extended by Farb and Weinberger, shows that rigidity in the presence of too many isometries still occurs provided one looks at covers of a closed manifold of negative sectional curvature to find “hidden symmetries”: Eberlein proves that a closed Riemannian manifold of negative sectional curvatures admitting infinitely many isometries of its universal cover must be locally symmetric. From the dynamical perspective, hyperbolic dynamical systems also display such a phenomenon: if the centralizer group of a hyperbolic dynamical system is too large often it is conjugate to an algebraic one. In this talk we will consider hidden symmetries of the hyperbolic dynamical system given by the geodesic flow of a manifold of negative sectional curvatures. We will introduce an appropriate notion of a centralizer for the geodesic flow on the universal cover, and prove that when it is not discrete the metric must be locally symmetric.
May 30, 2024: Dave Constantine "Geodesic flow on CAT(-1) and (some) CAT(0) spaces"
Abstract: How do we define geodesic flow on a metric space? Which of the nice dynamical properties of geodesic flow on negatively or nonpositively curved Riemannian manifolds can we prove for their metric space analogues: CAT(-1) and CAT(0) spaces? In this talk I'll discuss some results on the geodesic flow for CAT(-1) spaces obtained jointly with Jean-Francois Lafont and Dan Thompson. Then I will report on some recent work with Ben Call, Alena Erchenko, Noelle Sawyer, and Grace Work on geodesic flow for flat surfaces with large-angle cone point singularities. These spaces include translation surfaces, and provide a first step in the direction of extending our results to the CAT(0) setting.