Seminars and Colloquia by Series

Shape Optimization - an introduction

Series
Research Horizons Seminar
Time
Wednesday, January 28, 2009 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Antoine HenrotUniversity of Nancy, France
In this talk, we give an insight into the mathematical topic of shape optimization. First, we give several examples of problems, some of them are purely academic and some have an industrial origin. Then, we look at the different mathematical questions arising in shape optimization. To prove the existence of a solution, we need some topology on the set of domains, together with good compactness and continuity properties. Studying the regularity and the geometric properties of a minimizer requires tools from classical analysis, like symmetrization. To be able to define the optimality conditions, we introduce the notion of derivative with respect to the domain. At last, we give some ideas of the different numerical methods used to compute a possible solution.

Couple of variational models for multi-phase segmentation

Series
PDE Seminar
Time
Tuesday, January 27, 2009 - 15:05 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Sung Ha KangSchool of Mathematics, Georgia Tech
Image segmentation has been widely studied, specially since Mumford-Shah functional was been proposed. Many theoretical works as well as numerous extensions have been studied rough out the years. In this talk, I will focus on couple of variational models for multi-phase segmentation. For the first model, we propose a model built upon the phase transition model of Modica and Mortola in material sciences and a properly synchronized fitting term that complements it. For the second model, we propose a variational functional for an unsupervised multiphase segmentation, by adding scale information of each phase. This model is able to deal with the instability issue associated with choosing the number of phases for multiphase segmentation.

A global analysis of a multistrain viral model with mutations

Series
CDSNS Colloquium
Time
Monday, January 26, 2009 - 16:30 for 2 hours
Location
Skiles 255
Speaker
Sergei PilyuginUniversity of Florida
I will present a generalization of a classical within-host model of a viral infection that includes multiple strains of the virus. The strains are allowed to mutate into each other. In the absence of mutations, the fittest strain drives all other strains to extinction. Treating mutations as a small perturbation, I will present a global stability result of the perturbed equilibrium. Whether a particular strain survives is determined by the connectivity of the graph describing all possible mutations.

Sparse Solutions of Underdetermined Linear Systems

Series
Applied and Computational Mathematics Seminar
Time
Monday, January 26, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Ming-Jun LaiUniversity of Georgia
I will first explain why we want to find the sparse solutions of underdetermined linear systems. Then I will explain how to solve the systems using \ell_1, OGA, and \ell_q approaches. There are some sufficient conditions to ensure that these solutions are the sparse one, e.g., some conditions based on restricted isometry property (RIP) by Candes, Romberg, and Tao'06 and Candes'08. These conditions are improved recently in Foucart and Lai'08. Furthermore, usually, Gaussian random matrices satisfy the RIP. I shall explain random matrices with strictly sub-Gaussian random variables also satisfy the RIP.

Introduction to the h-principle

Series
Other Talks
Time
Friday, January 23, 2009 - 15:00 for 2 hours
Location
Skiles 269
Speaker
Mohammad GhomiSchool of Mathematics, Georgia Tech
h-Principle consists of a powerful collection of tools developed by Gromov and others to solve underdetermined partial differential equations or relations which arise in differential geometry and topology. In these talks I will describe the Holonomic approximation theorem of Eliashberg-Mishachev, and discuss some of its applications including the sphere eversion theorem of Smale. Further I will discuss the method of convex integration and its application to proving the C^1 isometric embedding theorem of Nash.

Elliptic curves and chip-firing games on wheel graphs

Series
Combinatorics Seminar
Time
Friday, January 23, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Gregg MusikerMIT
In this talk, I will discuss chip-firing games on graphs, and the related Jacobian groups. Additionally, I will describe elliptic curves over finite fields, and how such objects also have group structures. For a family of graphs obtained by deforming the sequence of wheel graphs, the cardinalities of the Jacobian groups satisfy a nice reciprocal relationship with the orders of elliptic curves as we consider field extensions. I will finish by discussing other surprising ways that these group structures are analogous. Some of this research was completed as part of my dissertation work at the University of California, San Diego under Adriano Garsia's guidance.

Introduction to the h-principle

Series
Geometry Topology Working Seminar
Time
Friday, January 23, 2009 - 15:00 for 2 hours
Location
Skiles 269
Speaker
Mohammad GhomiGa Tech
$h$-Principle consists of a powerful collection of tools developed by Gromov and others to solve underdetermined partial differential equations or relations which arise in differential geometry and topology. In these talks I will describe the Holonomic approximation theorem of Eliashberg-Mishachev, and discuss some of its applications including the sphere eversion theorem of Smale. Further I will discuss the method of convex integration and its application to proving the $C^1$ isometric embedding theorem of Nash.

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