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}...
I will discuss a recent result in collaboration with J. Szeftel
concerning the nonlinear stability of the Schwarzschild spacetime
under axially symmetric, polarized perturbations.
The gravitational waves detected recently by LIGO were produced
in...
"Beyond endoscopy", broadly interpreted, is the idea that
functoriality should be realized as a comparison between stable
trace formulas. The nature of this comparison, however, remains
completely unclear.
I will discuss in some detail the main difficulties of the
problem of nonlinear stability of black holes and the recent
advances on the related issue of linear stability.
The gravitational waves detected recently by LIGO were produced
in the final...
On the reality of black holes. I will give a quick introduction
to the initial value problem in GR and overview of the problems of
Rigidity, Stability and Collapse and how they fit with regard to
the Final State Conjecture.
