Colloquium 10.18.18 Mohammadi

Title: Effective results in homogeneous dynamics

SpeakerAmir 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.

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Colloquium 2.23.17 Rodriguez Hertz

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.

Colloquium 11.3.16 DeMarco

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.

Colloquium 5.12.16 Pollicott

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.

Colloquium Sharp 4.2.15

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.  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.

Colloquium 4.3.14 Eskin

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.

Colloquium 3.13.14 Fisher

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.

Colloquium 12.6.12

Gap Distribution and Homogeneous Dynamics


Dec 6 2012 – 2:30pm – 3:30 pm




Jayadev Athreya (University of Illinois, Urbana-Champaign)


The Farey sequence F(Q) is the collection of fractions between [0,1] whose denominator (when written in lowest terms) is at most Q. As Q grows, these points become uniformly distributed in the interval, so in some sense, look `random’. However, when you look at the gaps between them, they do not behave like those for uniformly distributed random variables, but instead follow an unusual law known as Hall’s Distribution. We will explain a proof of this result that uses horocycle flow on the space of lattices SL(2,R)/SL(2, Z), and discuss how this picture can be generalized to explicitly computing the gap distribution between directions of saddle connections on Veech surfaces, which we had used computer experiments to approximate. This talk will include elements from joint work with Y. Cheung, joint work with J. Chaika, and joint work with J. Chaika and S. Lelievre.

Colloquium 9.12.13

Ergodicity of the Weil-Petersson geodesic flow


Sep 12 2013 – 4:30pm – 5:30 pm




Keith Burns (Northwestern)


The lecture will describe the Weil-Petersson metric on the moduli
space of a surface and (at least some of ) the ideas that go into the
proof that its geodesic flow is ergodic. This is joint work with Howard Masur and Amie Wilkinson.

Colloquium 11.7.13

A pressure metric for the Hitchin component


Nov 7 2013 – 4:30pm – 5:30 pm


CH 240


Richard Canary (University of Michigan)


If S is a closed surface, its Teichmuller space is the space of all (marked) hyperbolic structures on S. Hitchin showed that there is a component of the space of (conjugacy classes of) representatations of the fundamental group of a closed surface S into PSL(n,R) which is homeomorphic to an open ball. This component contains a copy of the Teichmuller space of S which we call the Fuchsian locus. We will begin by surveying basic facts about Teichmuller space and the Hitchin component. The pressure metric is a analytic Riemannian metric on the Hitchin component which is invariant under the action of the mapping class group and whose restriction to the Fuchsian locus is a multiple of the classical Weil-Petersson metric. We associate to each Hitchin representation a geodesic flow which is a Holder repameterization of the geodesic flow on S. We then use tools from the thermodynamic formalism to construct our metric. (This talk describes joint work with Bridgeman, Labourie and Sambarino.)