Kansas State University Mathematics

Analysis Seminars, Fall 2006

All talks at 3:30 p.m. in Cardwell 120 unless otherwise noted

Wednesday, September 6,  2006
Speaker:  Diego Maldonado
Title:  Doubling measures and quasi-conformal mappings

Wednesday, September 13, 2006
Speaker:  Diego Maldonado
Title:  Doubling measures and quasi-conformal mappings

Wednesday, September 20, 2006
Speaker:  Ryan Berndt
Title: A global formula for the Fourier transform that improves on known asymptotic results, and generalizes the well known formula for the Fourier transform of 1/x.

Abstract: In this talk we show a formula for the Fourier transform of certain odd, differentiable functions defined on the real line. The formula can be seen as a generalization of the well known formula for the Fourier transform of the kernel of the Hilbert transform. This result improves upon asymptotic results for so called slowly varying functions as given in Zygmunds book "Trigonometrical Series" as well as in Bingham, Goldie, and Teugels book "Regular Variation."

Wednesday, September 27, 2006
Speaker: Virginia Naibo
Title: Dispersive and smoothing properties of solutions to the Schrödinger equation

Wednesday, October 4, 2006
Speaker: Pietro Poggi-Corradini
Title: The Chang-Marshall theorem in space

Wednesday, October 18, 2006
Speaker: Sergey Levendorskiy,  University of Kansas
Title: Free boundary problems and optimal stopping under Levy processes

Abstract: The main goal of the talk is to demonstrate mutually useful interactions between analysis and probability in applications to free boundary problems for pseudo-differential operators and optimal stopping problems under jump-diffusion processes, many important problems in finance and economics including. The analysis of the regularity of solutions allows one to guess a natural candidate for the optimal exercise boundary and show that the smooth pasting principle may fail; the probabilistic (to be more precise,  financial) interpretation allows one to show that for parabolic problems, the free boundary near t=0 may be farther than the standard intuition suggests, and obtain a simple general explicit  formula for the lower bound for the gap between the boundary at t=0 and the limit of the boundary as t goes to 0. The Wiener-Hopf factorization method in a form used in analysis and a novel interpretaion of the factors using the probabilistic  language (rather, language of economics) allows one to develop a new general approach to optimal stopping problems under processes with i.i.d. increments. Explicit results are obtained in 1D case; however, recent developments for regime-switching models can be used to approximate multi-dimensional case discretizing all dimensions but one. The talk is based, mainly, on 4 joint papers with Svetlana Boyarchenko:
1. "American Options in Regime-Switching Models" http://ssrn.com/abstract=929215   2. "Perpetual American Options in Regime-Switching Models" http://ssrn.com/abstract=928474  3. Exit Problems in Regime-Switching Models http://ssrn.com/abstract=906961    4.General option exercise rules, with applications to embedded options and monopolistic expansion" (10/30/05).http://ssrn.com/abstract=838624 and partially, on  5. Levendorskii, Sergei Z, "American and European Options Near Expiry, Under Markov Processes with Jumps http://ssrn.com/abstract=610544

Wednesday, October 25, 2006
Speaker:  Ryan Berndt
Title: New Singularities in singular integrals. Atomic Hardy space theory for unbounded singular integrals.

Friday, November 3, 2006  (in Cardwell 144 at 2:30 p.m.)
Speaker:  Sarah Raynor, Wake Forest University      Sponsored in part by an ADVANCE departmental initiative grant
Title:  Nonvariational methods for semilinear elliptic equations of critical growth    Abstract


Wednesday, November 8, 2006
Speaker:  Tom DeLillo,  Wichita State University
Title: Schwarz-Christoffel Mapping of Bounded and Unbounded Multiply Connected Domains

Abstract: A Schwarz-Christoffel formula for conformal maps from the exterior of a finite number of disks to the exterior of polygonal curves was derived in T. K. DeLillo, A. R. Elcrat, and J. A. Pfaltzgraff, J. d'Analyse Math., 94(2004), pp. 17-47, using the reflection principle.  A similar formula for the bounded case was derived in D. Crowdy, Proc. R. Soc. A, 461 (2005), pp. 2653-2678, using Schottky-Klein prime functions. Both of these formulas express the derivative of the mapping function in terms of infinite products. In this talk, we will review these formulas and the relations between them and various canonical slit maps. for multiply connected domains. We will also discuss some convergence results and the numerical implementation of these formulas.

Monday, November 13, 2006  (in Cardwell 130 at 4:30 p.m.)
Speaker:  James Colliander, University of Toronto    Sponsored in part by ADVANCE Distinguished lecture series
Title: Mass concentration for L^2 critical nonlinear Schrodinger equations

Abstract:  Finite time blowup solutions of the L^2 critical nonlinear Schrodinger equation (e.g. the physically relevant cubic problem in two space dimensions) are known to concentrate a portion of their L^2 norm into a point at blowup time. In this talk, I will review what is known about this phenomenon and describe some recent advances.

Friday, November 17,  2006   (in Cardwell 130 at 9:30 a.m.)
Speaker:  Jill Pipher, Brown University        Sponsored in part by an ADVANCE departmental initiative grant
Title:  On the Garnett-Jones theorem. 

Wednesday, November 29, 2006
Speaker: Atanas Stefanov,  University of Kansas
Title: Pseudodifferential operators with rough symbols and applications

  Classical estimates (like Hormander-Mikhlin, Calderon-Vaillancourt, Coifman-Meyer) for pseudodifferential operators (PDO) require pointwise bounds on certain number of derivatives in both the x and  xi variables. In the applications however, the PDO that arise rarely satisfy such stringent assumptions. In this talk, I will present some recent results for Lp boundedness of PDO, which require only boundedness in the x variable and Sobolev type condition (i.e. L2 integrability of certain derivatives) on the xi variable. The results are sharp up to an endpoint. It turns out that one can even study maximal singular integral operators in this framework. As an application and with simple proofs consisting of essentially checking the conditions, I will show the L2 boundedness of the maximal Bochner-Riesz operator in any dimension, Lp, p>=2 bounds for the maximal spherical cap operator and Lp, p>=2 bounds for (a variant of) the maximal directional Hilbert transform operator. The last one is closely related to the  Kakeya maximal function.

The analysis seminar is organized by Chuck Moore.

Previous semester

Spring 2007

Kansas State University Mathematics Department