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...
In joint work with Buryak, Pandharipande and Tessler (in
preparation), we define equivariant stationary descendent integrals
on the moduli of stable maps from surfaces with boundary to
$(\mathbb{CP}^1,\mathbb{RP}^1)$. For stable maps of the disk...
We will give a brief overview of the classical topics, problems and
results in Algebraic Combinatorics. Emerging from the
representation theory of $S_n$ and $GL_n$, they took a life on
their own via the theory of symmetric functions and Young...
A martingale is a sequence of random variables that maintain
their future expected value conditioned on the past. A
$[0,1]$-bounded martingale is said to polarize if it converges in
the limit to either $0$ or $1$ with probability $1$. A
martingale...
$p$-adic period spaces have been introduced by Rapoport and Zink as
a generalization of Drinfeld upper half spaces and Lubin-Tate
spaces. Those are open subsets of a rigid analytic $p$-adic flag
manifold. An approximation of this open subset is the...
