Title: Effective results in homogeneous dynamics
Speaker: Amir Mohammadi (University of California San Diego)
Abstract: Rigidity phenomena in homogeneous dynamics have been extensively studied over the past few decades with several striking results and applications. In this talk we will give an overview of the more recent activities which aim at presenting quantitative versions of some of these strong rigidity results.
Colloquium URL: https://web.math.osu.edu/colloquium/
Title: New developments in the theory of smooth actions
Speaker: Federico Rodriguez Hertz (Pennsylvania State University)
Abstract: In recent years several new advances in the theory of lattice actions have been made. In this talk I will present some of the key ingredients to these advances. I plan to keep the talk at an elementary level so only some basic notions of measure theory and differentiation on manifolds should be needed.
Title: Complex dynamics and elliptic curves
Speaker: Laura DeMarco (Northwestern University)
Abstract: In this talk, I will present some connections between recent research in dynamical systems and the classical theory of elliptic curves and rational points. The main goal is to explain the role of dynamical stability and bifurcations in deducing arithmetic finiteness statements. I will focus on three examples: (1) the theorem of Mordell and Weil from the 1920s, presented from a dynamical point of view; (2) a recent result of Masser and Zannier about torsion points on elliptic curves, and (3) features of the Mandelbrot set.
Speaker: Mark Pollicott (Warwick, UK)
Title: Zeta functions and geodesics
Abstract: The Riemann zeta function is an important tool in the theory of prime numbers. A more geometric analogue of this is the Selberg Zeta function where the prime numbers are replaced by closed geodesics on a compact surface of constant negative curvature. However to generalize this to more general settings one needs to take a more dynamical viewpoint. We will discuss old and new results in this direction.
Speaker: Jana Rodriguez Hertz (IMERL, Uruguay)
Title: Partially hyperbolic diffeomorphisms in 3-manifolds
Abstract: We review some advances for partially hyperbolic diffeomorphisms in 3-manifolds, and will focus on three aspects: ergodicity, dynamical coherence and classification.
Speaker: Richard Sharp (Warwick)
Title: Growth and spectra on regular covers
Abstract: Two natural numerical invariants that can be associated to a Riemannian manifold are the bottom of the spectrum of the Laplacian operator and, if the manifold are negatively curved, the exponential growth rate of closed geodesics. Suppose we have a regular cover of a compact manifold. Then, for each of these quantities, we might ask under what circumstances we have equality between the number associated to the cover and the number associated to the base.Â This question becomes non-trivial questions once the cover is infinite. It turns out that the question has a common answer in the two cases and this depends only on the covering group as an abstract group. For the Laplacian, this result was obtained by Robert Brooks in the 1980s, and Rhiannon Dougall and I have recently obtained the analogue for the growth of closed geodesics. I will discuss this work, relating it to random walks and a class of groups introduced by von Neumann in his study of the Banach-Tarski Paradox.
Bio sketch: Richard Sharp is a mathematician at Warwick University, UK. He grew up in London and obtained a BSc in Mathematics from Imperial College, London, in 1987. This was followed by postgraduate study at Warwick, where he was supervised by William Parry, and he obtained his PhD, on the periodic orbit structure of hyperbolic flows, in 1990. After postdocs at IHES, Queen Mary, London, and Oxford, he moved to Manchester University in 1995, before returning to Warwick in 2012. He is mainly interested in ergodic theory and applications to geometry.
Title: The SL(2,R) action on moduli space
Speaker: Alex Eskin, University of Chicago
Abstract: We prove some ergodic-theoretic rigidity properties of the action of SL(2; R) on the moduli space of compact Riemann surfaces. In particular, we show that any ergodic measure invariant under the action of the upper triangular subgroup of SL(2; R) is supported on an invariant a ffine submanifold. The main theorems are inspired by the results of several authors on unipotent flows on homogeneous spaces, and in particular by Ratner’s seminal work. This is joint work with Maryam Mirzakhani and Amir Mohammadi.
Title: Entropy for smooth systems
Speaker: Todd Fisher, Brigham Young University
Abstract: Dynamical systems studies the long-term behavior of systems that evolve in time. It is well known that given an initial state the future behavior of a system is unpredictable, even impossible to describe in many cases. The entropy of a system is a number that quantifies the complexity of the system. In studying entropy, the nicest classes of smooth systems are ones that are structurally stable. Structurally stable systems, for example automorphisms of the torus, are those that do not undergo bifurcations for small perturbations. In this case, the entropy remains constant under perturbation. Outside of the class of structurally stable systems, a perturbation of the original system may undergo bifurcation. However, this is a local phenomenon, and it is unclear when and how the local changes in the system lead to global changes in the complexity of the system. We will state recent results describing how the entropy (complexity) of the system may change under perturbation for systems that are not structurally stable.