| 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 p-harmonic world ABSTRACT : pdf file |
John Lewis |
| February 10 | Professor Min-Chun Hong, University of Queensland, Australia TITLE: Global Existence of the Siberg-Witten 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 sub-Riemannian Heisenberg group can be constructed by means of a Riemannian approximation scheme, as limit of Riemannian minimal surfaces. We study the regularity of Lipschitz, non-characteristic minimal surfaces which arise as such limits. Our main results (joint with Citti and Manfredini) are a-priori estimates on the solutions of the approximating Riemannian PDE and the ensuing $C^ {\infty}$ regularity of the sub-Riemannian 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 Korteweg-de 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 nano-structures. In applications, such systems are often approximated by the underlying one-dimensional graph structure, a so-called "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 (Hayden-Howard 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 tug-of-war game approximates the infinity harmonic functions. In addition, Peres and Sheffield studied a connection between the p-harmonic functions and the random turn tug-of-war 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 Pseudo-Euclidean setting are just Monge-Ampere 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 one-to-one on the space of continuous functions $K$ on the boundary of the billiard. This allows us to obtain spectral rigidity of Laplace-Beltrami 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,Tai-Chia National University of Taiwan/IMA TITLE: Skyrmions in Gross-Pitaevskii functionals ABSTRACT : Recently, Skyrmions with integer topological charges have been observed numerically but have not yet been shown rigorously on two-component systems of Gross-Pitaevskii equations (GPEs) describing a binary mixture of Bose-Einstein condensates (BEC). Here we construct skyrmions by studying critical points of Gross-Pitaevskii functionals with two-component wave functions. Using localized energy method, we rigorously prove the existence and configuration of skyrmions in BEC. On the other hand, half-Skyrmions characterized by half-integer 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 half-Skyrmions which may come from a new type of soliton solutions called spike-vortex solutions of two-component systems of GPEs. |
Changyou Wang |