Colloquium

Inhomogeneous quadratic forms

An inhomogeneous quadratic form is a quadratic form along with a shift. These forms arise in a variety of situations in number theory, dynamics, and in quantum chaos. I will explain these connections and then discuss some recent progress on understanding the values taken by such forms at integer points, using a variety of ergodic, geometric and analytic tools.

Deformation space analogies between Kleinian reflection groups and rational maps

We will describe an explicit correspondence between certain Kleinian reflection groups and certain anti-holomorphic rational maps acting on the Riemann sphere. This correspondence has several dynamical and parameter space consequences. To illustrate some of these, we will discuss many striking similarities between the deformation spaces of these two classes of conformal dynamical systems, including an analogue of Thurston’s compactness theorem for anti-holomorphic rational maps and relations between the global topology of the corresponding deformation spaces.

Totally geodesic submanifolds of real and complex hyperbolic manifolds

After some history and motivation, I will discuss recent works with Bader, Miller and Stover in which we prove finiteness of maximal totally geodesic submanifolds in real and complex hyperbolic spaces. 

Commensurators and arithmeticity of hyperbolic manifolds

The commensurator of a Riemannian manifold M encodes symmetries between all the finite covers of M, and lifts to a subgroup of isometries of the universal cover of M. In case M is an (irreducible) finite volume locally symmetric space, the commensurator is thus a subgroup of a simple Lie group G. Margulis proved that if the commensurator is dense in G, then M is arithmetic. Shalom asked if the same is true for infinite volume M? I will report on recent progress on this question when M regularly covers a finite volume hyperbolic manifold. This is joint work with D. Fisher and M. Mj.

Multifractal analysis of Birkhoff averages for non-uniformly expanding Markov interval maps

Anosov representations, Hodge theory, and Lyapunov exponents

Discrete subgroups of semisimple Lie groups arise in a variety of contexts, sometimes "in nature" as monodromy groups of families of algebraic manifolds, and other times in relation to geometric structures and associated dynamical systems. I will discuss a class of such discrete subgroups that arise from certain variations of Hodge structure and lead to Anosov representations, thus relating algebraic and dynamical situations. Among many consequences of these relations, I will explain Torelli theorems for certain families of Calabi-Yau manifolds, uniformization results for domains of discontinuity of the associated discrete groups, and also a proof of a conjecture of Eskin, Kontsevich, Moller, and Zorich on Lyapunov exponents. The necessary context and background will be explained.

Two problems on homogenization in geometry

In this talk, I show that a random quasiconformal mapping is close to an affine mapping, while a circle packing of a random Delauney triangulation is close to a conformal map. This is joint work with Vlad Marković.

Generalized Hecke Operators and Mahler’s Problem in Diophantine Approximation III

Generalized Hecke Operators and Mahler’s Problem in Diophantine Approximation II

Lower tail bounds beyond Chernoff for very small deviations from the mean

Concentration bounds for sums of independent random variables are ubiquitous in probabilistic setups. The most famous ones of these, the Chernoff-Hoeffding bounds provide guarantees of the following type: given a binomial random variable with expectation k, what is the probability that the variable deviates by a factor of more than (1+y) from k? The (nearly tight) answer is approx exp(-ky^2). However, what if we are interested in small additive deviations from the expectation k? In particular, what is the probability that the variable takes a value strictly less than k? Now, for large k, the Chernoff-Hoeffding bounds will give only the vacuous bound of 1. However, Jogdeo and Samuels showed in 1968 that the right answer is actually 1/2. In this talk, I will describe this result and a helpful result of Hoeffding which shows that the worst case is the Poisson case. I will finally talk about our result from this year (with Nikhil Bansal, Majid Farhadi and Prasad Tetali) which extends the bound of Jogdeo and Samuels for the case of small deviations from k/t for t close to and at least 1.