UW-PIMS Mathematics Colloquium

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 co-dimensional 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(z1, . . . , zn) be a holomorphic function defined on an open subset P ∈ U ⊂ Cn. 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 Kähler-Einstein 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 non-deterministic 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 well-posedness 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, 2014 at 2:30pm
Mark Rudelson
University of Michigan
Title and abstract forthcoming

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