Joint work with Amir Mohammadi, Zhiren Wang, and Lei Yang

Let Q be an indefinite ternary quadratic form. In the 1980s Margulis proved the longstanding Oppenheim Conjecture, stating that unless Q is proportional to an integral form, the set of values Q attains at the integer points is dense in R. We give quantitative results to that effect.
In particular, if the coefficients of Q are algebraic, but Q is not proportional to an integral form, and if $(\alpha,\beta)$ a fixed interval, the number of integer points v in a ball of radius R for which $\alpha<Q(v)<\beta$ is asymptotic to $c(Q,\alpha,\beta)R$ as $R\to\infty$ with an effective power saving error term (but $c(Q,\alpha,\beta)$ might not be what you expect!).

Our work is based on a quantitative equidistribution result for unipotent flows, as well as upper bound estimates by Eskin-Margulis-Mozes and Wooyeon Kim. 

Abstract: Pour a drop of milk into a cup of coffee. We know from experience that after a few moments of stirring, the mixture becomes homogeneous. In this talk, we study the advection of a passive scalar (the concentration of milk) under a random fluid (coffee) velocity on the torus. We prove, when the velocity solves the 2d stochastic Navier–Stokes equations on the torus, that the scalar converges exponentially to its mean. Our result applies even when only finitely many (as few as 4) Fourier modes are randomly forced. Joint with Keefer Rowan (EPFL). 

We have impromptu and (sometimes) scheduled talks, on topics in probability, combinatorics, geometry, and dynamics.

Everyone is welcome!