Analysis Seminar (Spring 2007) - Department of Mathematics

Analysis Seminar (Spring 2008)
Wednesday 4:30 - 5:20 pm Cardwell 144
(except when otherwise noted)

Organizer: Charles Moore
Previous semesters: Spring 06 Fall 06 Spring 07 Fall 07

Wednesday, February 6 - Daniel Alpay, Ben Gurion University
Reproducing kernel spaces, boundary interpolation and rigidity theorems
We review the interpolation theory of functions analytic and bounded by one in the open unit disk (Schur functions) at a boundary point. The case of functions with a finite number of poles in the open unit disk is also considered. We apply these results to generalize a rigidity result of Burns and Krantz. This is joint work with S. Reich (Technion, Haifa, Israel), and D. Shoikhet (Braude College, Karmiel, Israel).

Wednesday, February 13 - Chuck Moore
Fourier series and extrapolation methods
The partial sums of a Fourier series can converge very slowly. Can extrapolation methods speed this process? In this talk we will discuss what happens when a certain nonlinear transform is applied to the partial sums of a slowly converging Fourier series.

Wednesday, February 20 - Chuck Moore
Fourier series and extrapolation methods, continued

Wednesday, February 27- Marianne Korten
The Cauchy problem for the two phase Stefan problem
On the upper half space, we consider the two-phase Stefan problem taken in the sense of conservation laws. We show that the Cauchy problem is solvable for function and measure data which satisfy the proper growth condition at infinity.

Wednesday, March 5 - Nguyen Cong Phuc, Purdue University
Singular quasilinear and Hessian equations and inequalities
Abstract

Wednesday, March 26- Rodolfo Torres, University of Kansas
A T(1)-Theorem for variable coefficient bilinear Hilbert transforms.
Abstract: We will quickly review the (linear) T1-Theorem for Calder\'on-Zygmund singular integral operators and some of the ideas behind it. We will then introduce some results about bilinear operators and some motivation for their study. We will conclude with a new result for bilinear operators which are variable coefficient versions of the bilinear Hilbert transform. The new results are joint work with A. B\'enyi, C. Demeter, A. Nahmod, C. Thiele, and P. Villarroya.

Wednesday, April 2 - Ray Treinen
Floating drops and functions of bounded variation.

Abstract: Consider a volume of fluid resting on the interface between two other fluids. We measure the energy of the configuration and prove that a minimizing configuration exists. We consider first a bounded cylindrical container, then, in order to make the connection to results of Elcrat, Neel, and Siegel, we take a limit of bounded containers and show that the
limiting configuration is also minimizing in a local sense. This is joint work with Alan Elcrat. We will begin with a short review of the basic theorems from the theory of functions of bounded variation.

Wednesday, April 9 - Ray Treinen
Floating drops and functions of bounded variation, part 2

Wednesday, April 16 - Nguyen Hoang
Dynamical systems method for solving linear finite-rank operator equations.
A version of the Dynamical Systems Method (DSM) for solving ill-conditioned linear algebraic systems is studied in this paper. An a priori and a posteriori stopping rules are justified. An iterative scheme is constructed for solving ill-conditioned linear algebraic systems. This is a joint work with Prof. A. G. Ramm.

Wednesday, April 23 - Sarah Reznikoff, University of Victoria and Kansas State University
The Little Shift Equivalence Conjecture
Determining when two shifts of finite type are topologically conjugate is a fundamental question in the field of symbolic dynamics. There are many known invariants, such as periodic point data, entropy, and the dimension group. In 1970 Bob Williams proved that topological conjugacy of two shifts was equivalent to Strong Shift Equivalence, a condition on their corresponding matrices. However, Strong Shift Equivalence is quite difficult to establish. Williams conjectured that a simpler condition, Shift Equivalence, might suffice, but this was disproved by Kim-Roush in 1997. A special case of Williams' conjecture, the Little Shift Equivalence Conjecture, is still open.

In this talk we will go over the basic definitions and background material, and then present some results related to the Little Shift Equivalent Conjecture. The talk should be accessible to anyone with an undergraduate-level understanding of linear algebra.

Wednesday, April 30 - Pekka Pankka, University of Michigan

Quasiconformal frames

Heinonen and Sullivan have used the Cartan-Whitney presentations of the metric gauge to study natural local parametrization of generalized Lipschitz manifolds. Quasiconformal frames can be seen as a counterpart of Cartan-Whitney presentations in the context of quasiconformal geometry. In this talk I will discuss the notion of quasiconformal frames and show how a normalized blow-up process leads to the results of Heinonen-Sullivan type. This is joint work with Juha Heinonen and Kai Rajala.

Wednesday, May 7 - Diego Maldonado

The Jacobian of biLipschitz maps and non-linear elliptic PDEs

Celebrated examples due to Dmitri Burago and Bruce Kleiner (2002); and Curtis T. McMullen (2003) show that there exist bounded, bounded away from zero, uniformly continuous functions f : [0,1]^2 --> R^2 such that no biLipschitz mapping F: [0,1]^2 --> R^2 satisfies det DF(x) = f(x), a.e. x in [0,1]^2. Basic results on geometric function theory allow to replace the word "biLipschitz" by the word "quasi-conformal" in the previous statement. We use non-linear elliptic PDE techniques to show that if K is a subset of R^n is a Lipschitz domain and f : K --> R^n is bounded, bounded away from zero, and Dini continuous, then there are infinitely many biLipschitz mappings F: K --> R^n verifying det DF(x) = f(x) a.e. x in K. Moreover, we show that each one of those mappings F is of the form F=grad(u), where u: K --> R is a strictly convex function. As a consequence, our result also sheds some light on the so-called quasi-conformal Jacobian problem.
This is joint work with Leonid Kovalev from Texas A&M University.