{"id":350,"date":"2015-11-28T16:47:34","date_gmt":"2015-11-28T21:47:34","guid":{"rendered":"https:\/\/my.vanderbilt.edu\/pderesearch\/?page_id=350"},"modified":"2016-04-15T10:42:28","modified_gmt":"2016-04-15T15:42:28","slug":"pde-seminar-spring-2016","status":"publish","type":"page","link":"https:\/\/my.vanderbilt.edu\/pderesearch\/pde-seminar-spring-2016\/","title":{"rendered":"PDE Seminar Spring 2016"},"content":{"rendered":"<h3>Fridays, 4:10pm, Stevenson Center 1307<\/h3>\n<p><span style=\"font-size: 13px\">Date: <\/span><strong>Friday, January 15, 2016<\/strong><\/p>\n<ul>\n<li>Speaker:<strong> Brian Benson, Kansas State<\/strong><\/li>\n<li>Title:\u00a0The Cheeger Constant and Higher Eigenvalues of the Laplace Spectrum of Riemannian Manifolds.<\/li>\n<\/ul>\n<ul>\n<li>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&#8217;s and Buser&#8217;s inequalities respectively. Work of Buser also shows that no generalization of Cheeger&#8217;s inequality to the higher eigenvalues exists without additional information about the manifold.<br \/>\nWe 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&#8217;s inequality. This is a generalization of Buser&#8217;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 It is also related to unpublished work of Agol concerning a quantitative improvement of Buser&#8217;s inequality for the first positive eigenvalue of the Laplace spectrum.<\/li>\n<\/ul>\n<p><span style=\"font-size: 13px\">Date: <\/span><strong>Friday, January 29, 2016<\/strong><\/p>\n<ul>\n<li>Speaker:<strong> William Holmes, Vanderbilt University (Physics Department)<\/strong><\/li>\n<li>Title:\u00a0The Local Perturbation Analysis: a non\u2010linear stability technique for detectingspatial responses of complex cell regulatory systems.<\/li>\n<\/ul>\n<ul>\n<li>Abstract:\u00a0How 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 \u201cspreading\u201d to form greater contact area with the substratum, \u201ccontracting\u201d to avoid contact, \u201cpolarizing\u201d to prepare for motility, or generating highly dynamic \u201cwaves\u201d 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\u2010linear in nature. To address this issue I will describe a new nonlinear perturbation technique, the \u201cLocal Perturbation Analysis\u201d, that has proven immensely useful in investigating these responses. This method is capable of producing non\u2010linear 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.<\/li>\n<\/ul>\n<p>Date:\u00a0<strong>Friday, February 19, 2016<\/strong><\/p>\n<ul>\n<li>Speaker:<strong> Colin Klaus, Vanderbilt University<\/strong><\/li>\n<li>Title:\u00a0Visual Transduction: A Signaling Paradigm Across Orders of Scale.<\/li>\n<\/ul>\n<ul>\n<li>Abstract:\u00a0Visual Transduction in Rod and Cone photoreceptor cells is one of the best quantified G-protein signaling cascades known to biologists. \u00a0Here photons of light are converted by biochemical processes into a system&#8217;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. \u00a0In this talk, I will present on how the partial differential equations&#8217; techniques of Homogenization and Concentrating Capacity may be used at once to capture the effects of this geometry and especially its disparate physical scales.<\/li>\n<\/ul>\n<p>Date:\u00a0<strong>Friday, Feb 26, 2016<\/strong><\/p>\n<ul>\n<li>Speaker:<strong> Nem Kosovalic, University of South Alabama.<\/strong><\/li>\n<li>Title:\u00a0Local bifurcation theory for some nonreversible wave equations.<\/li>\n<\/ul>\n<ul>\n<li>Abstract:\u00a0<span style=\"font-size: 13px\">Over the last fifty years a huge effort has been devoted to\u00a0the study of the local bifurcation of periodic and quasi-periodic\u00a0solutions for reversible wave equations. Despite this effort, there\u00a0are gaps in what is currently known about the nonreversible\u00a0counterpart. Nonreversible wave equations generally include wave\u00a0equations having either time delay or damping terms. We discuss some\u00a0results in this direction and some open problems. The work presented\u00a0is collaborative work with Dr. Brian Pigott and Dr. Chris Lin.<\/span><\/li>\n<\/ul>\n<p><span style=\"font-size: 13px\">Date: <\/span><strong>Friday, April 8, 2016<\/strong><\/p>\n<ul>\n<li>Speaker:<strong> Tilak Bhattacharya, Western Kentucky University<\/strong><\/li>\n<li>Title: Some results for viscosity solutions to some doubly nonlinear degenerate parabolic differential equations.<\/li>\n<\/ul>\n<ul>\n<li>Abstract: Click <a href=\"https:\/\/cdn.vanderbilt.edu\/t2-my\/my-prd\/wp-content\/uploads\/sites\/1330\/2016\/03\/Bhattacharya-Vand16.pdf\">here<\/a> for an abstract.<\/li>\n<\/ul>\n<p><span style=\"font-size: 13px\">Date: <\/span><strong>Friday, April 22, 2016<\/strong><\/p>\n<ul>\n<li>Speaker:<strong> Philip Crooke, Vanderbilt University<\/strong><\/li>\n<li>Title:\u00a0Some Modeling Problems for Retirement.<\/li>\n<\/ul>\n<ul>\n<li>Abstract:\u00a0I will discuss some some modeling projects that I plan to pursue after I retire. \u00a0The first project deals with a computational model to predict breast cancer in menopausal women. \u00a0For me, this is an old project, but I believe that there are a few new wrinkles which I hope to incorporate. The second project deals with predicting the population of patients with kidney disease. It is a multi-compartment PDE model with the compartments being the glomerular filtration rate (GFR) categories. If time permits, I will briefly discuss a Monte Carlo model that relates the population of dialysis treatment centers by zip codes. Lastly, I will outline what I have been doing and will continue to do with personalized medicine, in particular, using gene expression data to predict autoimmune diseases. This involves high throughput sequencing and the search for coding regions of the DNA which are associated with certain diseases.<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Fridays, 4:10pm, Stevenson Center 1307 Date: Friday, January 15, 2016 Speaker: Brian Benson, Kansas State Title:\u00a0The 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,&#8230;<\/p>\n","protected":false},"author":2740,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[],"class_list":["post-350","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/my.vanderbilt.edu\/pderesearch\/wp-json\/wp\/v2\/pages\/350","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/my.vanderbilt.edu\/pderesearch\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/my.vanderbilt.edu\/pderesearch\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/my.vanderbilt.edu\/pderesearch\/wp-json\/wp\/v2\/users\/2740"}],"replies":[{"embeddable":true,"href":"https:\/\/my.vanderbilt.edu\/pderesearch\/wp-json\/wp\/v2\/comments?post=350"}],"version-history":[{"count":22,"href":"https:\/\/my.vanderbilt.edu\/pderesearch\/wp-json\/wp\/v2\/pages\/350\/revisions"}],"predecessor-version":[{"id":353,"href":"https:\/\/my.vanderbilt.edu\/pderesearch\/wp-json\/wp\/v2\/pages\/350\/revisions\/353"}],"wp:attachment":[{"href":"https:\/\/my.vanderbilt.edu\/pderesearch\/wp-json\/wp\/v2\/media?parent=350"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/my.vanderbilt.edu\/pderesearch\/wp-json\/wp\/v2\/tags?post=350"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}