Kansas State University Mathematics

Analysis Seminars—Spring 2006

All talks at 3:30 p.m. in Cardwell 131 unless otherwise noted

Wednesday, January 25, 2006

Speaker: Chuck Moore

Title: Probabilistic behavior of  lacunary series

Wednesday, February 1, 2006

Speaker:  Chuck Moore

Title:  Probabilistic behavior of lacunary series (continued)

Wednesday, February 8, 2006

Speaker:  Chuck Moore

Title:  Probabilistic behavior of lacunary series (last talk)

Wednesday, February 15, 2006

Speaker:  Alexander Fish, The Hebrew University of Jerusalem.

Title: WM sets - definitions, Ramsey properties and open problems.

Abstract: The notion of WM sets was introduced by H. Furstenberg and it generalizes the notion of normal sets (subsets of the natural numbers with a statistics of normal infinite binary sequences). The definition uses the notions of geneicity and weak-mixing from ergodic theory. We expect that WM sets behave like random sets. By Ramsey theory of WM sets we mean that many algebraic patterns must intersect with every WM set. In this spirit we characterize all linear Diophantine systems which are solvable in every WM set. In the proof we use Host-Kra theory of nilmanifolds and convergence of ergodic averages along discrete cubes and also we inductively use Van der Corput Bergelson's argument. To provide a necessity direction we use a probabilistic method to build counterexamples. If time  permits we will review open problems related to Ramsey theory of WM sets.

Wednesday, February 22, 2006

Speaker: Xiang Fang

Title: Hardy space and its cousins

Monday, February 27, 2006   (in Cardwell 120, 3:30 p.m.)

Speaker: Almut Bruchard, University of Toronto Sponsored by  an ADVANCE department initiative grant

Title: Rearrangement inequalities for multiple integrals    

Abstract:   Rearrangements are classical tools for solving symmetric  variational problems. One attractive property is that (when they apply) they often allow to determine extremals  without  assuming regularity to begin with. My  talk  will  focus on inequalities for multiple integrals that extend the Riesz rearrangement  inequality  in different directions. One recent such extension is to inequalities for integrals over integrands of the form F(u_1,... , u_m), where  F satisfies appropriate higher-order monotonicity conditions.  I will explain the geometric meaning of these inequalities and some  tools  that enter into their proofs.  At the end, I will describe a "layer-cake" technique developed in recent joint work with H. Hajaiej.

Wednesday, March 8, 2006

Speaker:  Bob Burckel

Title:  Polynomial and Rational Approximation of Holomorphic Functions -- Runge Theory

Friday, March 17, 2006

Speaker: Penny Smith, Lehigh University Sponsored by an ADVANCE department initiative grant

Title: Comparison Principles for Nonlinear Hyperbolic equations and Systems.

Abstract: We explain comparison principles for super and subsolutions ( including semicontinuous viscosity type) for semilinear hyperbolic
equations and Quasilinear Symmetric Hyperbolic Systems.

Thursday, March 30, 2006  (in Cardwell 122, 3:30 p.m.)

Speaker:  Bob Burckel

Title:  Lusin's example

Wednesday, April 12, 2006

Speaker: Jeanne Clelland, University of Colorado, Boulder Sponsored by an ADVANCE department initiative grant

Title: Conservation laws for second order evolution equations

Abstract: Conservation laws are important in the study of partial differential equations because they can be used to obtain useful information about the solutions of a PDE system.  However, finding conservation laws for a given PDE system - or even deciding whether or not a system has any conservation laws - can be very difficult.  In this talk I will describe pioneering work of Bryant and Griffiths in which the theory of exterior differential systems is used to systematize this process, in effect providing a "geometric theory" for conservation laws.  In particular, I will show how this theory  applies to second-order evolution equations in two and three independent variables.

Monday, April 17, 2006

Speaker: Konstantin I. Oskolkov, Department of Mathematics, University of South Carolina, Columbia,
Title: The Talbot phenomenon and the Gauss' sums.

Abstract: The fractal properties of the time-dependent probability density function will be discussed, for the ``free quantum particle in a box". This density is the squared magnitude of the solution of the Cauchy initial value problem for Schroedinger equation with the zero potential, and the``compressed"  periodic initial data. The focus will be on the relations with the optical diffraction phenomenon discovered in 1836 by W.H.F. Talbot, the British inventor of photography.

Wednesday, May 3,  2006
Speaker:  David Kinderlehrer, Carnegie Mellon University  (in CW 120 at 9:30 a.m.)
Title: Multiple state molecular motors

Abstract: In the general lecture we introduced a system of equations which allegedly portrays the behavior of a collection of multiple state molecular motors.  This system involves various species undergoing conformational change and moving under the influence of potentials. We examine why there must be collaboration between these two and how it can occur.  More precisely, we shall prove that under appropriate collaborative conditions, the simple coin toss paradigm introduced in the general lecture may be verified. This is joint work, over several years, with Michel Chipot, Stuart Hastings, Michael Kowalczyk, and Bryce McLeod.


Friday,  May  5 , 2006 (in CW 131 at 3:30 p.m.)

Speaker:  Paul Hagelstein, Baylor University

Title: Almost Everywhere Convergence of Fourier Series and Operators of Restricted Weak Type. 

Abstract: One of the best known conjectures in harmonic analysis is that the Fourier series of any function in L log L on the circle converges almost everywhere. We will consider how this conjecture relates to current open problems regarding general functional analytic properties of translation invariant operators of restricted weak type as well as recent progress made on these problems.

TBA- sometime during the week of May 22
Speaker: Monica Torres, Purdue University, Sponsored by an ADVANCE department initiative grant
Title:Gauss-Green formula for divergence-measure fields over sets of finite perimeter and applications to Cauchy fluxes and balance laws.

Abstract: We obtain the normal trace of  bounded divergence-measure fields on the boundary of any set of finite perimeter E as the limit of the normal traces of the vector field F on smooth surfaces that approximate perimeter E essentially from the inside of  E  with respect to the measure div F. We apply our results to a general
framework for Cauchy fluxes and to the derivation of systems of balance laws.

The analysis seminar is organized by Chuck Moore.

Kansas State University Mathematics Department