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...
Humans have been thinking about polynomial equations over the
integers, or over the rational numbers, for many years. Despite
this, their secrets are tightly locked up and it is hard to know
what to expect, even in simple looking cases. In this talk...
A meandric system of size $n$ is the set of loops formed from
two arc diagrams (non-crossing perfect matchings) on
$\{1,\dots,2n\}$, one drawn above the real line and the other below
the real line. Equivalently, a meandric system is a coupled...
A subset of a group is said to be product free if it does not
contain the product of two elements in it. We consider how large
can a product free subset of the alternating group An be?
In the talk we will completely solve the problem by...
