Thursday, December 5, 2013, 1:30 - 2:30 PM EST
Hybrid Subconvexity Bounds for Rankin-Selberg ConvolutionsAbstract: We'll discuss several recent results concerning the subconvexity problem for L-functions. In all of the cases we shall consider, one benefits analytically from the particular size or form of the L-function's conductor. The results presented are joint with either Ritabrata Munshi, Nicolas Templier or Zhi Qi.
Thursday, December 5, 2013, 3:30 - 4:30 PM EST
Thursday, December 5, 2013, 4:30 - 5:30 PM EST
Mathematical Analysis of Hydrodynamic Flow for Nematic Liquid CrystalsAbstract: In this talk, I will first briefly review the Ericksen-Leslie equation modeling the hydrodynamic motion of nematic liquid crystals. I will then present some of my recent results on a simplified Ericksen-Leslie system such as the global existence and uniqueness of partially smooth weak solutions in dimension two, and various well-posedness for rough initial data in higher dimensions. Finally I will also describe some works on the general Ericksen-Leslie system and mention a few challenging problems. This is based on joint works with Fanghua Lin, Tao Huang, Jay Hineman, and other collaborators.
Thursday, December 5, 2013, 4:30 - 5:30 PM EST
Log Ramification via CurvesAbstract: Recently, Esnault-Kerz conjectured that Abbes-T. Saito's conductor is `measured' by sub-curves and we prove their expectation in rank 1. As an application, we use results of Kerz-S. Saito to deduce an existence theorem in higher class field theory for schemes that captures `log' ramification. Finally, we also mention work in progress on modulus conditions (with respect to a divisor) for Bloch's higher Chow groups.
Computational and Applied Mathematics Seminar, Professor Dongwoo Sheen, Texas A&M University, REC 315
Friday, December 6, 2013, 3:30 - 4:30 PM EST
Recent Progress on Nonconforming Finite ElementsAbstract: Finite element methods are efficient tools for the analysis of the mechanics of fluid flows and solid structures in a variety of fields. The governing equations for these mechanics are usually described by vector-valued functions, and the mixed behaviors of variables requires some kind of careful mixed combination. Failure in the preservation of these rules leads to numerical instabil- ity. The use of nonconforming elements to solve fluid and fluid mechanical problems has shown several advantages over the use of conforming counterparts. Replacing conforming elements by nonconforming ones is simple and guarantees numerical stability in many cases. Indeed, we will raise a fundamental question why conforming elements have dominated over nonconforming ones. We will further claim that nonconforming elements are more intrinsic to finite elements than conforming elements. The mixed finite elements, discontinuous Galerkin elements are in some sence in the direction to support the claim. We give a brief review on the recent development in quadrilateral nonconforming finite elements in 2D and 3D. We will then discuss several new interesting observations about these quadrilateral nonconforming elements. Error estimates and numerical results will be presented.
Friday, December 6, 2013, 4:30 - 5:30 PM EST
Free Boundaries in Random Domains, from An Invariance Principle to the Homogenization of A Free Boundary ProblemAbstract: Free boundary problems are models where an unknown physical field is coupled with an unknown submanifold of the underlying physical space (the "free boundary"), i.e. the temperature around a melting crystal, which interacts naturally with its solid/liquid interface. The analysis of such problems combines ideas from geometric measure theory and harmonic analysis. This talk deals with these ideas in a random setting, specifically, the analysis of a (one-phase) Hele-Shaw type model set in a domain with many (random) microscopic obstacles. The main result is that the free boundaries converge (in the macroscopic limit) to the solution of an effective, deterministic problem. This is made possible by new pointwise estimates for linear elliptic equations in perforated domains which are used to control the geometry of the interface. Many of the ideas and tools have a parallel in statistical mechanics, in fact, at the linear level, the proof leads to an invariance principle and estimates for the transition probabilities of reflected Brownian motion on perforated domains.
Computational and Applied Mathematics Seminar, Professor Bob aka RS Eisenberg, Rush Medical Center, Chicago, REC 316
Friday, January 17, 2014, 3:30 - 4:30 PM EST
Mathematics of Ions in Channels and Solutions: Stochastic Derivations, Direct, Variational and Inverse Solutions that fit DataAbstract: Ion channels have a role in biology like the channels of field effect transistors in computers: both are valves for electricity controlling nearly everything. Ion channels are proteins with a hole down their middle that catalyze the movement of sodium, potassium, calcium and chloride ions across the otherwise impermeable membranes that define cells. Once a channel opens, it has a single structure on the biological time scale slower than say 2 microseconds. The ions present around every cell and molecule in biology are hard spheres and so the calculation of how hard spheres go through a channel of one structure is a central problem in a wide range of biology. Literally thousands of biologists study the properties of channels in experiments every day. My collaborators and I have shown how the relevant equations can be derived (almost) from stochastic differential equations, and how they can be solved in inverse, variational, and direct problems using models that describe a wide range of biological situations with only a handful of parameters that do not change even when concentrations change by a factor of 10^7. Variational methods hold particular promise as a way to solve problems outstanding for more than a century because they describe interactions of 'everything with everything' else that characterise ions crowded into channels.
Friday, January 24, 2014, 11:30 - 12:30 PM EST
Friday, January 24, 2014, 3:30 - 4:30 PM EST
Computational and Applied Mathematics Seminar, Professor Euan Spence, University of Bath, UK, REC 316
Friday, January 31, 2014, 3:30 - 4:30 PM EST
Friday, February 7, 2014, 3:30 - 4:30 PM EST
Friday, February 28, 2014, 3:30 - 4:30 PM EST
Friday, March 7, 2014, 11:30 - 12:30 PM EST