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.

Folding papers and turbulent flows

Camillo De Lellis
University of Zürich
February 21, 2017
In the fifties John Nash astonished the geometers with his celebrated isometric embedding theorems. A folkloristic explanation of his first theorem is that you should be able to put any piece of paper in your pocket without crumpling or folding it, no matter how large it is. A couple of decades later Gromov showed how Nash's ideas can be used to reinterpret other known counterintuitive facts in geometry and to discover many new ones. Ten years ago László Székelyhidi and I discovered unexpected similarities with the behavior of some classical equations in fluid dynamics.

Nodal sets of random spherical harmonics

Mikhail Sodin
Tel Aviv University
February 17, 2017
Abstract: In the talk I will describe what is known and (mostly) unknown about asymptotic statistical topology and geometry of zero sets of random spherical harmonics of large degree. I plan to discuss (a) several provoking open questions and (b) the first non-trivial lower bound recently obtained with Fedor Nazarov (work in progress) for the variance of the number of connected components of the zero set.

Log lower bound on the number of nodal domains on some surfaces of negative curvature

Steven Zelditch
Northwestern University
February 17, 2017
Abstract: An open problem is to prove that for any (or at least any generic) Riemannian metric, there is some sequence of eigenfunctions of the Laplacian for which the number of nodal domains tends to infinity. It sounds easy but as yet there are almost no examples of such metrics except for separation of variables situations, and the results of Ghosh-Reznikov-Sarnak and of Junehyuk Jung and myself. This talk is about a quantitative improvement in which we give a log lower bound for the number of nodal domains for negatively curved `real Riemann surfaces'.

Equidistribution of random waves on shrinking balls

Melissa Tacy
Australian National University
February 17, 2017
Abstract: In the 1970s Berry conjectured that the behavior of high energy, quantum-chaotic billiard systems could be well modeled by random waves. That is random combinations of the plane waves e^{ik ·x}. On manifolds it is more natural to randomize over the eigenfunctions of the Laplace-Beltrami operator. In this talk I will present results showing that such random waves equidistribute on balls that shrink with the eigenvalue. This is joint work with Xiaolong Han.

$C^\infty$ closing lemma for three-dimensional Reeb flows via embedded contact homology

Kei Irie
Kyoto University
February 16, 2017
$C^r$ closing lemma is an important statement in the theory of dynamical systems, which implies that for a $C^r$ generic system the union of periodic orbits is dense in the nonwondering domain. $C^1$ closing lemma is proved in many classes of dynamical systems, however $C^r$ closing lemma with $r > 1$ is proved only for few cases. In this talk, I'll prove $C^\infty$ closing lemma for Reeb flows on closed contact three-manifolds. The proof uses recent developments in quantitative aspects of embedded contact homology (ECH).