School of Mathematics

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...

According to Langlands, pure motives are related to a certain class of automorphic representations.

Can one see mixed motives in the automorphic set-up? For examples, can one see periods of mixed motives in entirely automorphic terms? The goal of...

Joint work with Daniel Kane (UCSD) and Shachar Lovett (UCSD)

We construct near optimal linear decision trees for a variety of decision problems in combinatorics and discrete geometry.

For example, for any constant $k$, we construct linear decision...

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}...

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.

Broadening our scope to include the...

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.

The gravitational waves...