Student Seminar

Ratner's theorem on SL(2,R)-invariant measures

Let G be a Lie group, Γ a discrete subgroup of G, and H a connected subgroup of G generated by unipotent elements. A fundamental result by M. Ratner [Ann. of Math.] asserts that every ergodic H-invariant probability measure on G/Γ is the L-invariant volume on a closed orbit Lx of some subgroup L of G containing H. In this talk, we focus on the special case where H is isomorphic to SL(2,R). We will provide motivation and background to introduce key concepts for those not familiar with the area. The proof illustrates some of the important dynamical ideas involved in understanding the general case, but avoids many technical difficulties. This is based on a paper of M. Einsiedler, titled "Ratner's theorem on SL(2, R)-invariant measures".

Introduction to Bennequin's Inequality

A contact structure on a 3-manifold is a nowhere-integrable plane field on the manifold. Of particular interest are contact structures which are “tight”, in which case the geometry of the contact structure is sufficiently complicated. We focus our discussion to a semi-local study of contact structures near embedded surfaces (possibly, with boundary) in the manifold. As a consequence, we deduce the Thurston-Bennequin inequality for Legendrian knots in tight contact 3-manifolds. We discuss some modest applications, and time permitting, indicate several ambitious ones.

Patterson-Sullivan Measure, Bowen-Margulis Measure and CAT(0)-boundaries

A metric space X is said to be a CAT(0) space if it is geodesically connected, and if every geodesic triangle in X is at least as ”thin” as its comparison triangle in the Euclidean plane. Any complete, simply connected Riemannian manifold having nonpositive sectional curvature is an example of a CAT(0) space. In this talk, we will first describe a construction of the Patterson-Sullivan measure on the boundary of a CAT(0)-space X. Later we will talk about a Bowen-Margulis measure on the space of unit-speed parametrized geodesics of the CAT(0)-space X modulo a proper, non-elementary, isometric action by a group G with a rank one element. The later part of this talk is based on the PhD thesis of Russell Ricks.

Margulis's banana trick

In this talk I will motivate and sketch the proof of the equidistribution of expanding translates of horocycles under the geodesic flow for the modular surface. If time permits I will mention the higher dimensional analogues of this result and mention a few number theoretic applications. All the terms in the abstract will be defined in the talk. The talk will only presume a basic working knowledge of the upper half plane model of the hyperbolic plane and some familiarity with Lie theory. 

I will be roughly following the exposition of this article: https://metaphor.ethz.ch/x/2018/fs/401-3370-67L/sc/banana.pdf

An introduction to random walks on homogeneous spaces

In this talk, I will discuss the basic definitions of random walk on homogeneous spaces. I will start discussing it by comparing it with deterministic walk (Ergodic Theory) and discuss the analogy between them. Then I will discuss the forward and backward dynamical systems. This will translate the discussion of the random walk into a deterministic walk. As a consequence, results from Ergodic theory can be applied. The construction will require some discussion on conditional measures and Doob's Martingale theorem. I will conclude the talk by giving a proof of Brieman's Law of Large numbers.

Conformal expanding repellers and Bowen’s dimension formula

A conformal expanding repeller is a dynamical system, in which the underlying phase space is a domain in the complex plane and the dynamics is given by a conformal expanding map. The main goal of dimension theory of dynamical systems is to study the “size” of certain dynamically relevant sets, for example, those on which dynamics is concentrated like repellers (chaotic dynamics) or attractors (tame dynamics). In this talk, we will introduce conformal expanding repellers and illustrate some examples by converting their dynamics to that of sub-shifts on a symbolic space. The objective would be to describe a method, developed by Bowen and Ruelle, to compute the Hausdorff dimension of conformal repellers, using tools from ergodic theory and thermodynamical formalism.

An introduction to Patterson-Sullivan measures for Kleinian groups

A Kleinian group is a discrete group of orientation-preserving isometries of the n-dimensional hyperbolic space H^n, where n is at least 2. In this talk, given a convex, co-compact Kleinian group we will define a family of measures in the same measure class, called Patterson-Sullivan measures, on the boundary at infinity of H^n supported on the limit set of the group. Key to defining these measures will be the study of the Poincare series associated with the group. We will aim to prove that for a convex, co-compact Kleinian group, the Patterson-Sullivan measure is proportional to the Hausdorff measure on the boundary at infinity of H^n of dimension same as the radius of convergence of the Poincare series associated with the group.

Train tracks and automorphisms of free groups

On a surface S, a measured train track is a combinatorial tool that allows us to assign coordinates to isotopy classes of essential simple closed curves. This allows us to see the action of a diffeomorphism on curves in terms of linear algebra on train tracks. In the context of free group automorphisms, this takes the form of finding an efficient representative of an automorphism as a nice homotopy equivalence of graphs, called a train track map. In this talk, I will introduce the idea of train track maps and show how a generalisation of it, called relative train track maps, was used to prove Scott's conjecture: Fixed subgroups of automorphisms of F_n have rank at most n.

Introduction to foliations on 3-manifolds

Foliations are decomposition of a manifold by connected, non empty, immersed submanifolds. They are in a sense, opposite to contact structures where the 2-plane fields are everywhere integrable. In the talk, I will introduce foliations and a few examples. We will also show that every closed oriented 3-manifold admits a codimension-1 foliation. We will also introduce a special class of foliations called taut foliations and some motivation behind studying them. 

JSJ decomposition of one ended hyperbolic group

The idea of JSJ splitting came first in splitting of 3 manifolds. Bowditch generalized this idea in one ended hyperbolic groups. By Stalling's Theorem we can split any infinite ended group over finite subgroups.  So the natural question is "How much we can split a one ended group?"  In this talk I shall emphasize splitting of one ended hyperbolic group over 2 ended subgroups. Then we will talk about generalized JSJ decomposition by Levitt.