A weight-$t$ halfspace is a Boolean function
$f(x)=\mathrm{sign}(w_1 x_1 + \cdots + w_n x_n - \theta)$ where
each $w_i$ is an integer in $\{-t,\dots,t\}.$ We give an explicit
pseudorandom generator that $\delta$-fools any intersection of $k$
weight...
Let $E$ be a CM elliptic curves over rationals and $p$ an odd prime
ordinary for $E$. If the $\mathbb Z_p$ corank of $p^\infty$ Selmer
group for $E$ equals one, then we show that the analytic rank of
$E$ also equals one. This is joint work with...
We present a new approach to the existence of time quasi-periodic
solutions to nonlinear PDE's. It is based on the method of Anderson
localization, harmonic analysis and algebraic analysis. This can be
viewed as an infinite dimensional analogue of a...
In 1979, O. Heilmann and E.H. Lieb introduced an interacting dimer
model with the goal of proving the emergence of a nematic liquid
crystal phase in it. In such a phase, dimers spontaneously align,
but there is no long range translational order...
The generalised Kato classes of Darmon-Rotger arise as $p$-adic
limits of diagonal cycles on triple products of modular curves, and
in some cases, they are predicted to have a bearing on the
arithmetic of elliptic curves over $Q$ of rank two. In...
Inspired by homological mirror symmetry for non-compact manifolds,
one wonders what functorial properties wrapped Fukaya categories
have as mirror to those for the derived categories of the mirror
varieties, and also whether homological mirror...
We introduce the notion of a "crystallographic sphere packing,"
which generalizes the classical Apollonian circle packing. Tools
from arithmetic groups, hyperbolic geometry, and dynamics are used
to show that, on one hand, there is an infinite zoo...
The approximate degree of a Boolean function $f$ is the least
degree of a real polynomial that approximates $f$ pointwise to
error at most $1/3$. For any constant $\delta > 0$, we exhibit
an AC$^0$ function of approximate degree
$\Omega(n^{1-\delta}...
