Sponsored by the UW Department of Mathematics and the Pacific Institute for the Mathematical Sciences.
October 17, 2014 at 2:30pm SIG 225 

Marianna Csörnyei The University of Chicago 
Geometry of Null Sets 
We show how product decompositions of measures can detect directionality in sets. In order to show this we prove that sets of small measure are always contained in a "small" number of Lipschitz surfaces. We also discuss the higher codimensional versions of our statements. This is a joint work with Peter Jones. 
October 31, 2014 at 2:30pm SIG 225 

Christopher D. Hacon The University of Utah 
Which Powers Of A Holomorphic Function Are Integrable? 
Let f = f(z_{1}, . . . , z_{n}) be a holomorphic function defined on an open subset P ∈ U ⊂ C^{n}. The log canonical threshold of f at P is the largest s ∈ R such that f^{s} is locally integrable at P. This invariant gives a sophisticated measure of the singularities of the set defined by the zero locus of f which is of importance in a variety of contexts (such as the minimal model program and the existence of KählerEinstein metrics in the negatively curved case). In this talk we will discuss recent results on the remarkable structure enjoyed by these invariants. 
November 21, 2014 at 2:30pm SIG 225 

Andrea R. Nahmod University of Massachusetts Amherst 
Randomization and long time dynamics in nonlinear evolution PDE 
In the last two decades, there has been substantial progress in the study of nonlinear dispersive PDE thanks to the influx of ideas and tools from nonlinear Fourier and harmonic analysis, geometry and analytic number theory, to the existing functional analytic methods. This body of work has primarily focused on deterministic aspects of wave phenomena and answered important questions related to existence and long time behavior of solutions in various regimes. Yet there remain important obstacles and open questions. A natural approach to tackle some of them, and one which has recently seen a growing interest, is to consider certain evolution equations from a nondeterministic point of view (e.g. the random data Cauchy problem, invariant measures, etc) and incorporate to the deterministic toolbox, powerful but still classical tools from probability as well. Such approach goes back to seminal work by Bourgain in the mid 90's where global wellposedness of certain periodic Hamiltonian PDEs was studied in the almost sure sense via the existence and invariance of their associated Gibbs measures. In this talk we will explain these ideas, describe some recent work and future directions. with an emphasis on the interplay of deterministic and probabilistic approaches. 
February 20, 2015 at 2:30pm MEB 246 

Mark Rudelson University of Michigan, Ann Arbor 
Nonasymptotic approach in random matrix theory 
Random matrix theory studies the asymptotics of the spectral distributions of families of random matrices, as the sizes of the matrices tend to infinity. Derivation such asymptotics frequently requires analyzing the spectral properties of random matrices of a large fixed size, especially of their singular values. 
March 6, 2015 at 2:30pm MEB 246 

Lauren Williams University of California, Berkeley 
The positive Grassmannian 
The positive Grassmannian is a remarkable subset of the real Grassmannian which has recently arisen in diverse contexts such as integrable systems, scattering amplitudes, and free probability. I will provide an elementary introduction to the positive Grassmannian, focusing on a few intriguing aspects: what it "looks like," and its connection to shallow water waves (via the KP hierarchy). 
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