Function Theory Study Seminar (Fall 2007) - Department of Mathematics

Function Theory Study Seminar (Spring 08)
Thursdays 3:30-4:20pm CW144

Organizer: Pietro Poggi-Corradini
Previous semesters: F05, S06, F06, S07, F07

The seminar where people are not allowed to speak about their own research, but instead relate on classic papers, or obscure old papers, or brand new preprints not yet checked ...

January 31 - Xiang Fang
"Random vectors in Hilbert spaces"

Abstract: Let a_n= 1 or -1, then the series a_1 + a_2/2 + a_3/3 + ... + a_n/n + ... is not necessarily convergent, but it "almost always" converges. I will prove a classical convergence theorem for random vectors in Hilbert spaces, which generalizes the above fact.

Then I will give two examples--multipliers of the Dirichlet space from complex analysis and Pisier algebra from harmonic analysis--to illustrate interests in random objects.

February 7 - Kasso Okoudjou, University of Maryland
"Time-frequency estimates and nonlinear Schrödinger equations"

Abstract: In the first part of this talk, we shall present some new time-frequency estimates for a family of unimodular Fourier multipliers. It must be pointed out that, in general, these Fourier multipliers are known to be unbounded on Lebesgue spaces. In the second part of the talk we shall use these time-frequency estimates to obtain improved local well-posedness results for certain nonlinear Schrödinger equations with Cauchy data in the so-called modulation spaces.

The talk will be based on recent joint work with A.Bényi, K.Gröchenig and L.Rogers.

February 14 - Pietro Poggi-Corradini
"On the Hausdorff dimension of harmonic measure in higher dimension"

Abstract: Celebrated results of Makarov, Jones, Wolff, etc.. showed that harmonic measure in the plane has dimension at most 1. I will read from a 1987 Inventiones paper of J. Bourgain where he showed that in n-space with n=3,4,5,...harmonic measure has dimension at most n-c(n) for some constant c(n)>0. I will make an effort to present all the necessary background.

February 21 - Pietro Poggi-Corradini
"On the Hausdorff dimension of harmonic measure in higher dimension", Part II

February 28 - Pietro Poggi-Corradini
"On the Hausdorff dimension of harmonic measure in higher dimension", Part II

March 6 - Mukta Bhandari
"Integral Transforms Controlled by Maximal Functions" by Mircea Martin and Pawel Szeptycki

Abstract: The paper characterizes the kernel functions on R^n with the property that the associated convolution operators are controlled by certain maximal operators.

March 13 - Mukta Bhandari
"Integral Transforms Controlled by Maximal Functions" by Mircea Martin and Pawel Szeptycki. Part II.

March 20 - Spring Break

March 27 - Mukta Bhandari
"Integral Transforms Controlled by Maximal Functions" by Mircea Martin and Pawel Szeptycki. Part III.

April 3 - Dan Volok
"Complete Nevanlinna-Pick Kernels" by Jim Agler and John E. McCarthy, publ. in JFA 175 (2000)
Abstract: In this paper the authors give a representation of all complete Nevanlinna-Pick kernels and show that they are all restrictions of a universal complete Nevanlinna-Pick kernel.

April 10 - Dan Volok
"Complete Nevanlinna-Pick Kernels" by Jim Agler and John E. McCarthy, publ. in JFA 175 (2000). Part II

April 17 - Sharad Silwal
"A note on the Gurov-Reshetnyak condition." by Korenovskyy, A. A.; Lerner, A. K.; Stokolos, A. M.
Abstract: I will present the main result of this paper which establishes an equivalence condition between the Gurov-Reshetnyak (GR(epsilon)) and Muckenhoupt A(infinity) conditions.

April 24 - Adrian Jenkins
"Local Non-Archimedean Dynamics "
Abstract: In the early 80's, Herman and Yoccoz laid the foundation for the study of the dynamics of an analytic map with respect to a non-archimedean norm near its fixed point. This is of course a classical problem in complex analysis. I will give an overview of this subject, and will discuss recent results obtained in joint work with Steven Spallone.

All relevant definitions will be given, and technical details will be kept to a minimum - the talk should be accessible to anyone with an understanding of power series, really.

May 1

May 8