It is classically understood how to learn the parameters of a
Gaussian even in high dimensions from independent samples. However,
estimators like the sample mean are very fragile to noise. In
particular, a single corrupted sample can arbitrarily...
The classical affine cubic surface of Markoff has a well-known
interpretation as a moduli space for local systems on the
once-punctured torus. We show that the analogous moduli spaces for
general topological surfaces form a rich family of log Calabi...
For integer parameters $n \geq 3$, $a \geq 1$, and $k \geq 0$ the
Markoff-Hurwitz equation is the diophantine equation \[ x_1^2 +
x_2^2 + \cdots + x_n^2 = ax_1x_2 \cdots x_n + k.\] In this talk, we
establish an asymptotic count for the number of...
We report on some recent work with Peter Sarnak. For integers $k$,
we consider the affine cubic surfaces $V_k$ given by $M(x) = x_1^2
+ x_2 + x_3^2 − x_1 x_2 x_3 = k$. Then for almost all $k$, the
Hasse Principle holds, namely that $V_k(Z)$ is non...
Markoff triples are integer solutions of the equation $x^2+y^2+z^2
= 3xyz$ which arose in Markoff's spectacular and fundamental work
(1879) on diophantine approximation and has been henceforth
ubiquitous in a tremendous variety of different fields...
We will explain how the circle method can be used in the setting of
thin orbits, by sketching the proof (joint with Bourgain) of the
asymptotic local-global principle for Apollonian circle packings.
We will mention extensions of this method due to...
In spin systems, the existence of a spectral gap has far-reaching
consequences. So-called "frustration-free" spin systems form a
subclass that is special enough to make the spectral gap problem
amenable and, at the same time, broad enough to include...
We consider the coherent cohomology of toroidal compactifications
of Shimura varieties with coefficients in the canonical extensions
of automorphic vector bundles and show that they can be computed as
relative Lie algebra cohomology of automorphic...
Proof complexity studies the problem computer scientists and
mathematicians face every day: given a statement, how can we prove
it? A natural and well-studied question in proof complexity is to
find upper and lower bounds on the length of the...
