Video Lectures

We study the complexity of computing some basic arithmetic operations over GF(2^n), namely computing q-th root and q-th residuosity, by constant depth arithmetic circuits over GF(2) (also known as AC^0(parity)). Our main result is that these...

The Johnson-Lindenstrauss lemma states that for any n points in Euclidean space and error parameter 0

All known proofs of the lemma construct a distribution over linear mappings so that a random such mapping suffices with high probability. In this talk, I will present various proofs of the JL lemma satisfying, for example, (1) the support of the distribution is small, so that a random embedding can be selected with few random bits (e.g. O(log n loglog n) bits for constant eps, which is suboptimal by a loglog n factor), and (2) every embedding matrix in the support of the distribution is sparse (only O(eps*k) entries per column are non-zero), to speed up computation. I will also describe some open problems. This talk is based on joint works with Daniel Kane (Harvard).

I will explain the main notions of the microlocal theory of sheaves: the microsupport and its behaviour with respect to the operations, with emphasis on the Morse lemma for sheaves. Then, inspired by the recent work of Tamarkin but with really...

GEOMETRY/DYNAMICAL SYSTEMS

Periodic bounce orbits are generalizations of billiard trajectories in the presence of a potential. Using an approximation technique by Benci-Giannoni we prove existence of periodic bounce orbits of prescribed energy. At...

Boolean Threshold Functions (BTF) arise in many contexts, ranging from computer science and learning theory to theoretical neurobiology. In this talk, I will present non-rigorous approaches developed in the statistical physics of disordered...

In this talk we will discuss recent progresses meant as a contribution to the GLS-project, the second generation proof of the Classification of Finite Simple Groups (jointly with R. Lyons, R. Solomon, Ch. Parker).

GEOMETRY/DYNAMICAL SYSTEMS

In this talk we shall discuss the Cartan geometry of the rotating Kepler problem. The rotating Kepler problem appears as the limit of the restricted planar three-body body when one of the masses goes to zero. As such...

Given a cuspidal automorphic representation of GL(n) which is regular algebraic and conjugate self-dual, one can associate to it a Galois representation. This Galois representation is known in most cases to be compatible with local Langlands...