Fridays 4:10pm, Stevenson Center 1307
Date: Wednesday, September 30, 2009
- Speaker: Roland Schnaubelt, University of Karlsruhe, Germany
- Title: Invariant manifolds and stability for the Stefan problem with surface tension
- Abstract: We study quasilinear parabolic systems with fully nonlinear dynamical boundary conditions, which arise e.g. after a transformation from problems with moving boundaries such as the Stefan problem with Gibbs-Thompson correction. We concentrate on the qualitative behavior near an equilibrium. Depending on the spectrum of the linearization, one obtains local stable, center and unstable invariant manifolds. We discuss their properties and the connection to the stability of the equilibrium. The proofs use maximal regularity results for inhomogeneous initial-boundary value problems, techniques from dynamical systems and semigroup theory.
Date: Friday, October 9, 2009
- Speaker: Juraj Foldes, Vanderbilt University
- Title: Asymptotic symmetry for positive solutions of parabolic problems
- Abstract: In this talk I will discuss qualitative properties of nonlinear parabolic problems. The basic question to be addressed is convergence of positive solutions to the space of symmetric functions. Sufficient conditions are formulated for such convergence in the case of a single equation and cooperative systems. The methods are based on the maximum principle, the Harnack inequality, and the method of moving hyperplanes.
Date: Friday, October 16, 2009
- Speaker: Juraj Foldes, Vanderbilt University
- Title: Asymptotic symmetry for positive solutions of parabolic problems, II
- Abstract: This talk is a continuation of the one given on October 9 and is devoted to the more technical aspects of the topic.
Date: Friday, October 30, 2009
- Speaker: Ugo Gianazza, Universita di Pavia, Italy
- Title: A new regularity approach for weak solutions of degenerate parabolic equations
- Abstract: In order to prove the Hoelder regularity of weak solutions to quasilinear degenerate parabolic equations, I use the same approach originally introduced in recent papers by DiBenedetto-Gianazza-Vespri to obtain Harnack inequalities for nonnegative solutions to these same equations. The new approach gives a more geometric and intuitive proof to the regularity and avoids covering and alternative arguments. This is joint work with M. Surnachev (U. of Swansea) and V. Vespri (U. of Florence).
Date: Friday, November 6, 2009
- Speaker: Zhian Wang, Vanderbilt University
- Title: Micro and macro models for chemotaxis
- Abstract: This talk is focused on two questions of chemotaxis modeling. One is how to establish the communications between microscopic and macroscopic chemotaxis models. The other is how information in the microscopic model is passed to the macroscopic model. For the first question, I use a novel approach to derive the macroscopic limits and express the microscopic quantities in terms of macroscopic quantities with the preservation of energy law. For the second question, I investigate the traveling waves of both microscopic and macroscopic models from which we see how traveling waves in the microscopic model are retained, lost or created during the transition from the microscopic to macroscopic models. Biological implications will be discussed along the talk.
Date: Friday, November 13, 2009
- Speaker: Leonardo Marazzi, Western Kentucky University
- Title: Scattering and inverse scattering on some classes of conformally compact manifolds
- Abstract: We study scattering theory on Asymptotically Hyperbolic (AH) manifolds and its generalizations to conformally compact manifolds and AH Einstein manifolds. Some examples of AH manifolds are the de Sitter-Schwarzschild model of the exterior of a black hole, which can be viewed as an AH manifold with two ends; and the Schwarzschild model of the exterior of a black hole, for which one of the two ends is a AH manifold, and the other end is an Asymptotically Euclidean manifold. Other examples of AH manifolds are given by quotients of the hyperbolic space by particular groups of motion. We also discuss some open problems in this area.
Date: Friday, November 20, 2009
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Date: Friday, December 4, 2009
- Speaker: Joanna Pressley, Vanderbilt University
- Title: Complementary responses to mean and variance modulations in the perfect integrate-and-fire model
- Abstract: In the perfect integrate-and-fire model (PIF), the membrane voltage is proportional to the integral of the input current since the time of the previous spike. It has been shown that the firing rate within a noise free ensemble of PIF neurons responds instantaneously to dynamic changes in the input current, whereas in the presence of white noise, model neurons preferentially pass low frequency modulations of the mean current. Here, we prove that when the input variance is perturbed while holding the mean current constant, the PIF responds preferentially to high frequency modulations. Moreover, the linear filters for mean and variance modulations are complementary, adding exactly to one. Since changes in the rate of Poisson distributed inputs lead to proportional changes in the mean and variance, these results imply that an ensemble of PIF neurons transmits a perfect replica of the time-varying input rate for Poisson distributed input. A more general argument shows that this property holds for any signal leading to proportional changes in the mean and variance of the input current.
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