# Seminars & Events for Analysis Seminar

March 26, 2012
4:00pm - 5:00pm
##### University of North Carolina at Chapel Hill
###### Analysis Seminar

It is well known that on $\reals^n$, the Schrödinger propagator is unitary on $L2$ based spaces, but that locally in space and on average in time there is a $1/2$ derivative smoothing effect. We consider a family of manifolds with trapped geodesics which are degenerately hyperbolic and prove a sharp local smoothing estimate with loss depending on the type of trapping. Further, we construct a microlocal parametrix extended polynomially beyond Ehrenfest time, and as a consequence, we obtain Strichartz estimates with near-sharp loss depending only on the dimension of the trapping. This is partly joint work with J. Wunsch (Northwestern)

Speaker: Hans Cristianson, University of North Carolina at Chapel Hill
Location:
Fine Hall 314
April 4, 2012
4:00pm - 5:00pm
##### The Frobenius Theorem, With Applications to Analysis
###### Analysis Seminar

This talk concerns the classical Frobenius theorem from differential geometry, about involutive distributions. For many problems in harmonic analysis, one needs a quantitative version of the Frobenius theorem. In this talk, we present such a quantitative version, and discuss various applications.

Speaker: Brian Street, University of Wisconsin, Madison
Location:
Fine Hall 314
April 9, 2012
4:00pm - 5:00pm
##### Growth of Sobolev Norms for the Cubic Defocusing Nonlinear Schrodinger Equation in Polynomial Time
###### Analysis Seminar

We consider the cubic defocusing nonlinear Schrodinger equation in the two dimensional torus. Fix $s>1$. Colliander, Keel, Staffilani, Tao and Takaoka (2010) proved existence of solutions with s-Sobolev norm growing in time by any given factor $R$. Refining their methods in several aspects we find solutions with s-Sobolev norm growing in polynomial time in $R$. This is a joint work with V. Kaloshin.

Speaker: Marcel Guardia, Universitat Politecnica de Catalunya & IAS
Location:
Fine Hall 314
April 16, 2012
3:15pm - 4:15pm
##### On the two weight inequality for the Hilbert transform
###### Analysis Seminar

The two weight inequality for the Hilbert transform arises in the settings of analytic function spaces, operator theory, and spectral theory, and what would be most useful is a characterization in the simplest real-variable terms. We show that the $L2$ to $L2$ inequality holds if and only if two $L2$ to weak-$L2$ inequalities hold. This is a corollary to a characterization in terms of a two-weight Poisson inequality, and a pair of testing inequalities on bounded functions. Joint work with Eric Sawyer, Chun-Yun Shen, and Ignacio Uriate-Tuero.

Speaker: Michael Lacey, Georgia Institute of Technology
Location:
Fine Hall 314
April 30, 2012
3:15pm - 4:15pm
##### The Cauchy problem for the Benjamin-Ono equation in L^2 revisited (Joint work with Luc Molinet)
###### Analysis Seminar

The Benjamin-Ono equation models the unidirectional evolution of weakly nonlinear dispersive internal long waves at the interface of a two-layer system, one being infinitely deep. The Cauchy problem associated to this equation presents interesting mathematical difficulties and has been extensively studied in the recent years. In a recent work (2007), Ionescu and Kenig proved well-posedness for real-valued initial data in L^2(R). In this talk, we will give another proof of Ionescu and Kenig's result, which moreover provides stronger uniqueness results. In particular, we prove unconditional well-posedness in H^s(R), for s > 1/4 . Note that our approach also permits to simplify the proof of the global well-posedness in L^2(T) by Molinet (2008) and yields unconditional well-posedness in H^{1/2}(T).

Speaker: Didier Pilod, Universidade Federal do Rio de Janeiro / University of Chicago
Location:
Fine Hall 314
May 7, 2012
3:15pm - 4:15pm
##### BBM: a statiscal point of view
###### Analysis Seminar

After presenting the BBM equation and some of its properties, we will try to understand which kind of statistics have a chance to remain invariant by its flow and produce one of them. The stability of this statistics will be studied then : to do so, we will sketch the parallelism between properties of equations and the statistics whose laws remain invariant by their flow.

Speaker: Anne-Sophie de Suzzoni, Universite de Cergy-Pontoise
Location:
Fine Hall 314