Somewhat unexpectedly, a near consensus among theoreticians is that cryptographic theorems should be proved in the non-uniform model of complexity, rather than the standard uniform complexity model developed by Alan Turing, the “father of computer science.” In joint work with Alfred Menezes of the University of Waterloo, we have criticized the use of non-uniformity in cryptography, finding that even some of the most distinguished researchers have been led badly astray by their misplaced faith in non-uniformity

Rigid cohomology defined by Berthelot agrees with the formal
cohomology defined Monsky and Washnitzer for smooth affine varieites.
Motivated by this, mimicking the convergent theory of Ogus, we define a
site using weakly completed algebras. We show a certain sheaf cohomology of
this site agrees with Berthelot's rigid cohomology

A connection is a geometric object that allows to parallel transport vectors along a curve in a domain. A natural question that often arises is whether one can recover a connection inside a domain from the knowledge of the parallel transport along a set of special curves running between boundary points of the domain. In this talk I will discuss this geometric inverse problem in various settings including Riemannian manifolds with boundary and Minkowski space. This problem is related to other inverse problems and is tackled with a range of techniques that I will explore during the talk.

A Hadamard matrix of order n is an n\times n, \pm 1 matrix for which every two rows are orthogonal. It is not hard to check that if n is not divisible by 4, then a Hadamard matrix of order n does not exists. On the other hand, it is widely open to prove the existence of Hadamard matrices of order 4n for all n. 

Years of investigation show that Hadamard matrices are very hard to find. In particular, as an easy exercise, one can show that the probability for a random matrix to be Hadamard is at most 2^{-n^2/2+o(n^2)}. 

In this talk I'll convince ourselves that Hadamard matrices are even harder to find! that is, the probability for a random matrix to be Hadamard is at most 2^{(1+\varepsilon)n^2/} for some small (but fixed) epsilon>0. Note that the gain is of the form 2^{-\theta(n^2)}, which, at least according to my knowledge, does not follow from any standard large deviation inequality. 

I'll give a purely combinatorial proof that does not appear on the paper (or elsewhere..) I have with Jain and Zhao on this topic.  This proof was basically our key for giving improved anticoncentration inequalities for classical results by Halasz regarding random sums of vectors (I won't discuss the anticoncentration bounds though...).

We consider N × N symmetric one-dimensional random band matrices with general distribution of the entries and band width $W$.   The localization - delocalization conjecture predicts that there is a phase transition on the behaviors of  eigenvectors and  eigenvalues of the band matrices. It occurs at $W=N^{1/2}$. For wider-band matrix, the eigenvalues satisfied the so called sine-kernal distribution, and the eigenvectors are delocalized. With Bourgade, Yau and Fan, we proved that it holds when $W\gg N^{3/4}$. The previous best work required $W=\Omega(N).$