We start by relating a result on the reducibility of
induced representations of a reductive group through local
Langlands correspondence to Artin L-functions. Such a result is of
interest in the study of the arithmetic theory of
intertwining...
New kinds of vortex sheets with vorticity confined to the
boundary layer are proposed and investigated in detail. Exact
solutions of the steady Navier-Stokes equations for a planar vortex
sheet in arbitrary background strain are found in terms of...
Given a finite graph, the arboreal gas is the measure on forests
(subgraphs without cycles) in which each edge is weighted by a
parameter β>0. Equivalently this model is bond percolation
conditioned to be a forest, the independent sets of the...
For the Bargmann--Fock field on Rd with d>2, we prove that
the critical level lc(d) of the percolation model formed by the
excursion sets {f≥l} is strictly positive. This implies that for
every l sufficiently close to 0 (in particular for the nodal...
In this talk I will describe a topological approach to some
problems about algebraic functions due to Klein and Hilbert. As a
sample application of these methods, I will explain the solution to
the following problem of Felix Klein: Let $\Phi_{g,n}$...
Given a polynomial in one variable, what is the simplest formula
for the roots in terms of the coefficients? Hilbert conjectured
that for polynomials of degree 6,7 and 8, any formula must involve
functions of at least 2, 3 and 4 variables...
I will describe the problem of finding the homotopy type of the
space of quantum Hamiltonians in dimension d under certain
constraints. The common assumptions are that the interactions have
finite range and that the ground state is separated from...
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...
