analysis math-physics

Singularity formation in incompressible fluids

Tarek Elgindi
Princeton University
February 22, 2017
We discuss the problem of singularity formation for some of the basic equations of incompressible fluid mechanics such as the incompressible Euler equation and the surface quasi-geostrophic (SQG) equation. We begin by going over some of the classical model equations which have been proposed to understand the dynamics of these equations such as the models of Constantin-Lax-Majda and De Gregorio. We then explain our recent proof of singularity formation in De Gregorio's model.

Discrete harmonic analysis and applications to ergodic theory

Mariusz Mirek
University of Bonn; Member, School of Mathematics
February 8, 2017
Given $d, k\in\mathbb N$, let $P_j$ be an integer-valued polynomial of $k$ variables for every $1\le j \le d$. Suppose that $(X, \mathcal{B}, \mu)$ is a $\sigma$-finite measure space with a family of invertible commuting and measure preserving transformations $T_1, T_2,\ldots,T_{d}$ on $X$. For every $N\in\mathbb N$ and $x \in X$ we define the ergodic Radon averaging operators by setting \[ A_N f(x) = \frac{1}{N^{k}}\sum_{m \in [1, N]^k\cap\mathbb Z^k} f\big(T_1^{ P_1(m)}\circ T_2^{ P_2(m)} \circ \ldots \circ T_{d}^{ P_{d}(m)} x\big).

Large coupling asymptotics for the Lyapunov exponent of quasi-periodic Schrödinger operators with analytic potentials

Christoph Marx
Oberlin College
January 25, 2017
In this talk we will quantify the coupling asymptotics for the Lyapunov exponent (LE) of a one-frequency quasi-periodic Schrödinger operator with analytic potential sampling function. By proving an asymptotic formula for the LE valid for all irrational frequencies, our result refines the well-known lower bound by Sorets and Spencer.