The celebrated product theorem says if A is a generating subset
of a finite simple group of Lie type G, then |AAA| \gg \min \{
|A|^{1+c}, |G| \}. In this talk, I will show that a similar
phenomenon appears in the continuous setting: If A is a
subset...
The talk will consists of a long historical introduction
to the topic of deviation of ergodic averages for locally
Hamiltonian flows on compact surafces as well as some current
results obtained in collaboration with Corinna Ulcigrai and
Minsung...
The theory of graph quasirandomness studies sequences of graphs
that "look like" samples of the Erdős--Rényi random graph. The
upshot of the theory is that several ways of comparing a sequence
with the random graph turn out to be equivalent. For...
Given several real numbers α1,...,αk, how well can you
simultaneously approximate all of them by rationals which each have
the same square number as a denominator? Schmidt gave a clever
iterative argument which showed that this can be done...
Many algorithms and heuristics that work well in practice have
poor performance under the worst-case analysis, due to
delicate pathological instances that one may never encounter. To
bridge this theory-practice gap, Spielman and Teng introduced
the...
Consider the family of automorphic representations on some
unitary group with fixed (possibly non-tempered) cohomological
representation π0 at infinity and level dividing some finite upper
bound. We compute statistics of this family as the level...
Let G be a simply-connected complex semisimple algebraic group
and let C be a smooth projective curve of any genus. Then, the
moduli space of semistable G-bundles on C admits so called
determinant line bundles. E. Verlinde conjectured a
remarkable...
There is a celebrated connection between minimal (or constant
mean curvature) hypersurfaces and Ricci curvature in Riemannian
Geometry, often boiling down to the presence of a Ricci term in the
second variation formula for the area. The first goal...
A famous conjecture of Littlewood states that the Fourier
transform of every set of N integers has l^1 norm at least log(N),
up to a constant multiplicative factor. This was proved
independently by McGehee-Pigno-Smith and Konyagin in the 1980s.
This...
