# Seminars & Events for Analysis Seminar

##### University of North Carolina at Chapel Hill

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)

##### The Frobenius Theorem, With Applications to Analysis

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.

##### Growth of Sobolev Norms for the Cubic Defocusing Nonlinear Schrodinger Equation in Polynomial Time

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.

##### On the two weight inequality for the Hilbert transform

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.

##### The Cauchy problem for the Benjamin-Ono equation in L^2 revisited (Joint work with Luc Molinet)

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).

##### BBM: a statiscal point of view

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.