Colloquium

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.

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

Khintchine's Theorem provides a zero-one law describing the approximability of typical points by rational ones. In 1984, Mahler asked whether a similar law holds for Cantor’s middle thirds set. His question fits into a long studied line of research aiming at determining conditions under which Diophantine properties of Euclidean space are inherited by its various subsets of interest.

Over the course of two lectures, we will discuss recent joint work with Manuel Luethi yielding the first complete analogue of Khintchine’s Theorem for certain self-similar fractal measures. The key ingredient in the proof is an effective equidistribution theorem for fractal measures on the space of unimodular lattices. To prove the latter, we associate to such fractals certain p-adic Markov operators, reminiscent of the classical Hecke operators, and leverage their spectral properties. We will also discuss some unexpected difficulties in the deduction of the divergence part of the analogue of Khintchine's Theorem as manifestations of certain subconvexity problems.

Large deviations for random hives and the spectrum of the sum of two random matrices

Asymptotic Length Saturation for Zariski Dense Surfaces

The lengths of closed geodesics on a hyperbolic manifold are determined by the traces of its fundamental group. When the latter is a Zariski dense subgroup of an arithmetic group, the trace set is contained in the ring of integers of a number field, and may have some local obstructions. We say that the surface's length set "saturates" (resp. "asymptotically saturates") if every (resp. almost every) sufficiently large admissible trace appears. In joint work with Xin Zhang, we prove the first instance of asymptotic length saturation for punctured covers of the modular surface, in the full range of critical exponent exceeding one-half (below which saturation is impossible).

On Moduli Spaces of Cut-And-Project Quasicrystals III

Translation surfaces and their connections

A translation surface is a Riemann surface equipped with a metric that is locally Euclidean away from finitely many points, which are cone points with angle an integer multiple of 2 pi. This talk will present connections that translations surfaces have with billiards in polygons, smooth flows on surfaces and geometric topology.

On Moduli Spaces of Cut-And-Project Quasicrystals II

Spectrum of the laplacian on self-similar groups and holomorphic dynamics

In this talk based on a joint work with Rostislav Grigorchuk and Mikhail Lyubich, I will explain how the calculation of the spectrum of the laplacian on the Grigorchuk group and the Hanoi group is related to the iteration of particular rational maps in two complex variables and their ergodic properties.

On Moduli Spaces of Cut-And-Project Quasicrystals I

A cut-and-project quasicrystal is a point set obtained from projecting a higher-dimensional lattice of R^(d+m) on a lower-dimensional space R^d.

I will talk about results obtained with Yotam Smilansky and Barak Weiss, in which we give a classification of measures on the space of quasicrystals invariant under the SLd(R) action.

For d>2 we also obtain Roger-type moment formulas and apply those to counting "patterns" arising in this non-aperiodic setting. I promise nice pictures and some details of the proofs.