We explicitly construct an extractor for two independent sources on
$n$ bits, each with min-entropy at least $\log^C n$ for a large
enough constant $C$. Our extractor outputs one bit and has error
$n^{-\Omega(1)}$. The best previous extractor, by...
Starting from the Poisson summation formula, I discuss spectral
summation formulae on GL(2) and GL(3) and present a variety of
applications to automorphic forms, analytic number theory, and
arithmetic.
The Bernoulli convolution with parameter $\lambda$ is the law of
the random variable: $\sum X_i \lambda^i$, where $X_i$ are
independent unbiased $+1/-1$ valued random variables. If $\lambda
1/2$, then the Bernoulli convolution is singular and is...
We investigate the shape of the reduction of certain crystalline
Galois representations of integral slope 1 and of fractional slopes
in (1,2). The proof uses the compatibility between the p-adic and
mod p Local Langlands Correspondences with respect...
Given a variety $Y$ with a rectangular Lefschetz decomposition of
its derived category, I will discuss an interesting relation
between the derived categories of a cyclic cover of $Y$ and its
branch divisor. As examples, I will describe the cases of...