PDE Seminar Spring 2016

Fridays, 4:10pm, Stevenson Center 1307

Date: Friday, January 15, 2016

• Speaker: Brian Benson, Kansas State
• Title: The Cheeger Constant and Higher Eigenvalues of the Laplace Spectrum of Riemannian Manifolds.
• Abstract: The Cheeger constant of a Riemannian manifold is a positive real number assigned to the manifold which is related to the isoperimetric problem. For closed manifolds, the work of Cheeger and Buser show that the first positive eigenvalue of the Laplace spectrum of the manifold is bounded above and below, respectively, by quadratics in the Cheeger constant. These are called Cheeger’s and Buser’s inequalities respectively. Work of Buser also shows that no generalization of Cheeger’s inequality to the higher eigenvalues exists without additional information about the manifold.
We prove the $k$-th eigenvalue of the Laplace spectrum of the manifold is bounded above by the $\lceil (k-1)/2 \rceil$-th eigenvalue of a Sturm-Liouville problem, where the Sturm-Liouville problem depends on exactly the same information about the manifold as Buser’s inequality. This is a generalization of Buser’s inequality for all higher eigenvalues of the Laplace spectrum of closed Riemannian manifolds. It is also related to unpublished work of Agol concerning a quantitative improvement of Buser’s inequality for the first positive eigenvalue of the Laplace spectrum.

Date: Friday, January 29, 2016

• Speaker: William Holmes, Vanderbilt University (Physics Department)
• Title: The Local Perturbation Analysis: a non‐linear stability technique for detectingspatial responses of complex cell regulatory systems.
• Abstract: How cells make key decisions is an important, if mysterious question that has generated sustained investigation in mathematical modeling, biophysics, and molecular biology. In response to sufficiently large stimuli, cells can respond in a number of manners include “spreading” to form greater contact area with the substratum, “contracting” to avoid contact, “polarizing” to prepare for motility, or generating highly dynamic “waves” of intracellular activity. While understanding these cellular responses is a classical PDE pattern formation problem, the models representing the underlying regulatory systems can be highly complex and the questions of interest are inherently non‐linear in nature. To address this issue I will describe a new nonlinear perturbation technique, the “Local Perturbation Analysis”, that has proven immensely useful in investigating these responses. This method is capable of producing non‐linear stability information for a common class (in cell regulatory systems) of reaction diffusion systems of arbitrary dimension (i.e. number of variables), while being no more complex to implement than existing linear methods. I will use this technique to investigate a number of biological regulatory systems and I) provide a hypothesis for how a simple, evolutionary conserved biochemical regulatory system can give rise a diverse array of responses and II) show that this function can be preserved as this system becomes increasingly more complex through evolution.

Date: Friday, February 19, 2016

• Speaker: Colin Klaus, Vanderbilt University
• Title: Visual Transduction: A Signaling Paradigm Across Orders of Scale.
• Abstract: Visual Transduction in Rod and Cone photoreceptor cells is one of the best quantified G-protein signaling cascades known to biologists.  Here photons of light are converted by biochemical processes into a system’s response by diffusion of the 2nd messengers cGMP and Ca2+. The morphology of these photoreceptor cells is striking and both highly regulated, highly ordered.  In this talk, I will present on how the partial differential equations’ techniques of Homogenization and Concentrating Capacity may be used at once to capture the effects of this geometry and especially its disparate physical scales.

Date: Friday, Feb 26, 2016

• Speaker: Nem Kosovalic, University of South Alabama.
• Title: Local bifurcation theory for some nonreversible wave equations.
• Abstract: Over the last fifty years a huge effort has been devoted to the study of the local bifurcation of periodic and quasi-periodic solutions for reversible wave equations. Despite this effort, there are gaps in what is currently known about the nonreversible counterpart. Nonreversible wave equations generally include wave equations having either time delay or damping terms. We discuss some results in this direction and some open problems. The work presented is collaborative work with Dr. Brian Pigott and Dr. Chris Lin.

Date: Friday, April 8, 2016

• Speaker: Tilak Bhattacharya, Western Kentucky University
• Title: Some results for viscosity solutions to some doubly nonlinear degenerate parabolic differential equations.