Video Lectures

ANALYSIS/MATHEMATICAL PHYSICS SEMINAR

For many elegant mathematical examples, one can 1) find theories behind them, 2) understand why they exist in the first place, 3) explore the consequences in math and physics. If one takes the Euler number as...

GALOIS REPRESENTATIONS AND AUTOMORPHIC FORMS SEMINAR

GEOMETRY AND CELL COMPLEXES

Polycrystalline materials, such as metals, ceramics and geological materials, are aggregates of single-crystal grains that are held together by highly defective boundaries. The structure of grain boundaries is...

SPECIAL LECTURE IN GEOMETRY/TOPOLOGY

https://www.ias.edu/math/files/seminars/Schapira.pdf

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...