Caltech/UCLA joint Analysis seminar

Fall 2019

Past: Winter 2018, Spring 2018, Fall 2018, Winter 2019, Spring 2019
Future: Winter 2020

Friday Oct 4

4:30-5:20 Caltech, Linde 310
Svetlana Jitormirskaya (UC Irvine)
Fractal properties of the Hofstadter's butterfly and singular continuous spectrum of the critical almost Mathieu operator

Harper's operator - the 2D discrete magnetic Laplacian - is the model behind the Hofstadter's butterfly. It reduces to the critical almost Mathieu family, indexed by phase. We discuss the proof of singular continuous spectrum for this family, for all phases, finishing a program with a long history. We also present a result (with I. Krasovsky) that proves one half of the Thouless' one half conjecture from the early 80s: that Hausdorff dimension of the spectrum of Harper's operator is bounded by 1/2 for all irrational fluxes.

5:30-6:20 Caltech, Linde 310
Khang Huynh (UCLA)
Hodge-theoretic analysis on manifolds with boundary, heatable currents, and Onsager's conjecture in fluid dynamics

We use Hodge theory and functional analysis to develop a clean approach to heat flows and Onsager's conjecture on Riemannian manifolds with boundary, where the weak solution lies in the trace-critical Besov space $B_{13}^{3,1}$. We also introduce heatable currents as the natural analogue to tempered distributions and justify their importance in Hodge theory.

Friday Oct 18

4:00-4:50 UCLA, MS 6627
Tarek Elgindi (UCSD)
Singularity formation in incompressible fluids

We will discuss recent results on singularity formation for solutions to the incompressible Euler equation in various settings. We will discuss in some detail a construction of self-similar blow-up for C^{1,a} solutions to the 3D Euler equation as well as a proof of stability that yields finite-energy local self-similar blow-up. Some of the results are joint with I. Jeong, T. Ghoul, and N. Masmoudi.

5:00-5:50 UCLA, MS 6627
Eyvindur Palsson (Virginia Tech)
Falconer type theorems and integral operators

Finding and understanding patterns in data sets is of significant importance in many applications. One example of a simple pattern is the distance between data points, which can be thought of as a 2-point configuration. Two classic questions, the Erdos distinct distance problem, which asks about the least number of distinct distances determined by N points in the plane, and its continuous analog, the Falconer distance problem, explore that simple pattern. Questions similar to the Erdos distinct distance problem and the Falconer distance problem can also be posed for more complicated patterns such as various 3-point configurations. In this talk I will explore such generalizations. The key focus will be integral operators, that arise naturally from such point configuration questions. In addition, a novel group-theoretic viewpoint, which has allowed for much progress recently, will be highlighted.

Friday Nov 1

4:30-5:20 Caltech, Linde 310
TBA

5:30-6:20 Caltech, Linde 310
Danielle Hilhorst (U. Paris Sud)
TBA

Friday Nov 15

4:00-4:50 UCLA, MS 6627
Florian Richter (Northwestern)
TBA

5:00-5:50 UCLA, MS 6627
Susan Friedlander (USC)
TBA

Friday Dec 6

4:30-5:20 Caltech, Linde 310
Vjekoslav Kovac (U. Zagreb/Georgia Tech)
TBA

5:30-6:20 Caltech, Linde 310