Date  SPEAKER and TITLE  Host 

January 20  Professor Eric Soccorsi, CPT CNRS
Luminy, Marseille, France TITLE:Eigenvalue asymptotics for a twisted waveguide ABSTRACT : pdf file 
Peter Hislop 
January 27  Mr. Justin Taylor, University of Kentucky TITLE: The Dirichlet eigenvalue problem for the Lame system and for elliptic system on perturbed domains ABSTRACT: pdf file 
Russell Brown 
February 3  Dr. Tomasz Adamowicz, University of Cincinnatti TITLE: On the geometry of the pharmonic world ABSTRACT : pdf file 
John Lewis 
February 10  Professor MinChun Hong, University of Queensland, Australia TITLE: Global Existence of the SibergWitten Flow ABSTRACT : pdf file 
Changyou Wang 
February 13  Professor Andrew Lorent, University of Cincinnatti TITLE: Quantitative Liouville Theorems ABSTRACT: pdf file 
John Lewis 
February 17  Professor Vladimir Eyderman, University of Kentucky TITLE: Cartan Type Estimates on Riesz Transform ABSTRACT : Our aim is to give sharp upper bounds for the size of the set of points where the singular Riesz transform of a linear combination of N point masses is large. This size will be measured by the Hausdorff content with various gauge functions. Among other things, we shall characterize all gauge functions for which the estimates do not blow up as N tends to infinity (in this case a routine limiting argument will allow us to extend our bounds to all finite Borel measures). This is a joint work with F.Nazarov and A.Volberg  James Brennan 
February 24  Professor Luca Capogna, University of Arkansas TITLE: Regularity of certain minimal graphs in the sub Riemannian Heisenberg group ABSTRACT: Minimal surfaces in the subRiemannian Heisenberg group can be constructed by means of a Riemannian approximation scheme, as limit of Riemannian minimal surfaces. We study the regularity of Lipschitz, noncharacteristic minimal surfaces which arise as such limits. Our main results (joint with Citti and Manfredini) are apriori estimates on the solutions of the approximating Riemannian PDE and the ensuing $C^ {\infty}$ regularity of the subRiemannian minimal surface along its Legendrian foliation. 
Zhongwei Shen 
March 3  Professor Yuan, Yu, University of Washington/IAS (Postponed) TITLE: tba ABSTRACT : 
Changyou Wang 
March 10  Professor David Adams, University of Kentucky (Cancelled) TITLE: Wolff potentials and solutions to the inhomogeneus wave equation in 3 space dimensions ABSTRACT: tba 
Changyou Wang 
March 12 (Colloquium) 
Professor Vladimir Sverak, University of Minnesota TITLE: tba ABSTRACT : tba 
Changyou Wang 
March 17  No seminar, Spring break week TITLE: ABSTRACT : 
TBA 
March 24  No seminar TITLE : tba ABSTRACT : tba 

March 31  Professor Peter Perry, University of Kentucky TITLE : Inverse Scattering as Nonlinear Fourier Analysis ABSTRACT : A number of nonlinear dispersive equations such as the Kortewegde Vries Equation (KdV) $u_t + u_xxx+ 6uu_x =0$ and the nonlinear Schrodinger equation (NLS) $i u+t + u_xx+u^2 u = 0$ (both for a function u of one space and one time variable) are solvable by the inverse scattering method. In this talk, I will briefly discuss the role these equations play in the theory of nonlinear dispersive waves, then discuss the inverse scattering method for solving the NLS in detail. In summary, there is an invertible nonlinear map $\mathcal{R}$ which maps solutions of the nonlinear equations to solutions of the same equation but with the nonlinearity removed. We'll construct the mapping $\mathcal{R}$ and discuss how it behaves like a "nonlinear Fourier transform". This is a partly expository talk and will allude both to "classical" work in inverse scattering from the '60's and '70's, and also to more recent work by Tao and collaborators. It is also a shameless plug for Math 773, Topics in Analysis  The Nonlinear Schrodinger Equation, to be offered in Fall 2009. 
Changyou Wang 
April 7  Professor Richard Laugesen, UIUC (Postponed) TITLE : ABSTRACT : 
Peter Hislop 
April 14  Professor Olaf Post, Humboldt University, Germany TITLE : Quantum graph approximations of thin branching structures ABSTRACT : Many physical systems have branching structure of thin transversal diameter. One can name for instance quantum wire circuits, thin branching waveguides, or carbon nanostructures. In applications, such systems are often approximated by the underlying onedimensional graph structure, a socalled "quantum graph". In this way, many properties of the system like conductance can be calculated easier (sometimes even explicitly). After briefly explaining the notion of a quantum graph, we show that the system with thin transversal diameter converges to a quantum graph. We also identify which vertex couplings (which influence the current through the graph) can be obtained by appropriate engineering of the branching structure.  Peter Hislop 
April 16 (HaydenHoward Lecture) 
Professor Carlos Kenig, University of Chicago TITLE: The global behavior of solutions to critical nonlinear dispersive and wave equations ABSTRACT : see department handout 
Zhongwei Shen 
April 21  Dr. Mikko Parviainen, Helsinki Institute of Technology TITLE: Nonlinear PDEs and stochastic games ABSTRACT : The theory of partial differential equations is closely related to stochastics. For example, the links between the Laplace operator and Brownian motion are well known. Recently, Peres, Schramm, Sheffield, and Wilson showed that a random turn tugofwar game approximates the infinity harmonic functions. In addition, Peres and Sheffield studied a connection between the pharmonic functions and the random turn tugofwar with noise. We study the nonlinear PDEs and stochastic games in a context of asymptotic expansions. This talk is based on a joint work with J.J. Manfredi and J.D. Rossi. 
John Lewis 
April 23 (Colloquium) 
Professor Yuan, Yu, University of Washington/IAS TITLE: Recent results for Special Lagrangian equations ABSTRACT : We survey some recent results on Hessian, gradient estimates, regularity, and global rigidity for special Lagrangian equations with certain convexity. The gradient graphs of the solutions are minimal Lagrangian submanifolds in Euclidean space. The special Lagrangian equations in the PseudoEuclidean setting are just MongeAmpere equations, for which one has the corresponding classic positive results and counterexamples. Part of the work is joint with Warren, some also with Chen. 
Changyou Wang 
April 28  Professor Peter Topalov, Northeastern University TITLE: On the Integral Geometry of Liouville Billiard Tables ABSTRACT : A notion of Radon transform for completely integrable billiard tables is introduced. It will be shown that in the case of Liouville billiard tables of dimension 3 the Radon transform is onetoone on the space of continuous functions $K$ on the boundary of the billiard. This allows us to obtain spectral rigidity of LaplaceBeltrami operator with Robin boundary conditions in certain domains. 
Peter Perry 
May 5  Professor Richard Laugesen, UIUC TITLE : Low eigenvalues of the Neumann Laplacian on triangles ABSTRACT : Szego and Weinberger showed that the fundamental tone of a free membrane is maximal for the ball, among regions of given volume. That is, the first nonzero eigenvalue of the Laplacian under Neumann boundary conditions is maximal for the ball. In the opposite direction, Payne and Weinberger showed the fundamental tone is minimal for the degenerate rectangle, among convex regions of given diameter. We tackle analogous extremal problems for triangles. Neither Szego's conformal maps nor Weinberger's extended trial functions nor Payne and Weinberger's slicing arguments work for triangular domains. Instead we develop linear transplantation, and the "method of the unknown trial function". [Joint work with B. Siudeja, Univ. of Illinois] 
Peter Hislop 
May 12 (Special PDE Seminar) 
Professor Lin,TaiChia National University of Taiwan/IMA TITLE: Skyrmions in GrossPitaevskii functionals ABSTRACT : Recently, Skyrmions with integer topological charges have been observed numerically but have not yet been shown rigorously on twocomponent systems of GrossPitaevskii equations (GPEs) describing a binary mixture of BoseEinstein condensates (BEC). Here we construct skyrmions by studying critical points of GrossPitaevskii functionals with twocomponent wave functions. Using localized energy method, we rigorously prove the existence and configuration of skyrmions in BEC. On the other hand, halfSkyrmions characterized by halfinteger topological charges can also be found in the nonlinear sigma model which is a model of BEC of the Schwinger bosons. Here we also prove rigorously the existence of halfSkyrmions which may come from a new type of soliton solutions called spikevortex solutions of twocomponent systems of GPEs. 
Changyou Wang 