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Series: CDSNS Colloquium

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.

Monday, January 26, 2009 - 13:00 ,
Location: Skiles 255 ,
Ming-Jun Lai ,
University of Georgia ,
Organizer: Haomin Zhou

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.

Series: Other Talks

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.

Friday, January 23, 2009 - 15:00 ,
Location: Skiles 269 ,
Mohammad Ghomi ,
Ga Tech ,
Organizer: John Etnyre

$h$-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.

Series: Combinatorics Seminar

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.

Series: SIAM Student Seminar

In this talk, I will focus on some interesting examples in the conditional expectation and martingale, for example, gambling system "Martingale", Polya's urn scheme, Galton-Watson process, Wright-Fisher model of population genetics. I will
skip the theorems and properties. Definitions to support the examples will be introduced. The talk will not assume a lot of probability, just some basic measure theory.

Series: School of Mathematics Colloquium

In this talk we will review some of the global asymptotic results obtained during the last two decades in the theory of the classical Painleve equations with the help of the Isomonodromy - Riemann-Hilbert method. The results include the explicit derivation of the asymptotic connection formulae, the explicit description of linear and nonlinear Stokes phenomenon and the explicit evaluation of the distribution of poles. We will also discuss some of the most recent results emerging due to the appearance of Painleve equations in random matrix theory. The Riemann-Hilbert method will be outlined as well.

Series: ACO Colloquium

We describe recent progress in the study of the binary deletion channel and related channels with synchronization errors, including a clear description of many open problems in this area. As an example, while the capacity of the binary symmetric error channel and the binary erasure channel have been known since Shannon, we still do not have a closed-form description of the capacity of the binary deletion channel. We highlight a recent result that shows that the capacity is at least (1-p)/9 when each bit is deleted independently with fixed probability p.

Series: Job Candidate Talk

Numerical algebraic geometry provides a collection of novel methods to treat the solutions of systems of polynomial equations. These hybrid symbolic-numerical methods based on homotopy continuation technique have found a wide range of applications in both pure and applied areas of mathematics. This talk gives an introduction to numerical algebraic geometry and outlines directions in which the area has been developing. Two topics are highlighted: (1) computation of Galois groups of Schubert problems, a recent application of numerical polynomial homotopy continuation algorithms to enumerative algebraic geometry; (2) numerical primary decomposition, the first numerical method that discovers embedded solution components.

Series: PDE Seminar

Living systems are subject to constant evolution through the two processes of mutations and selection, a principle discovered by Darwin. In a very simple, general, and idealized description, their environment can be considered as a nutrient shared by all the population. This allows certain individuals, characterized by a 'phenotypical trait', to expand faster because they are better adapted to the environment. This leads to select the 'best fitted trait' in the population (singular point of the system). On the other hand, the new-born population undergoes small variance on the trait under the effect of genetic mutations. In these circumstances, is it possible to describe the dynamical evolution of the current trait?
We will give a mathematical model of such dynamics, based on parabolic equations, and show that an asymptotic method allows us to formalize precisely the concepts of monomorphic or polymorphic population. Then, we can describe the evolution of the 'best fitted trait' and eventually compute various forms of branching points, which represent the cohabitation of two different populations.
The concepts are based on the asymptotic analysis of the above mentioned parabolic equations, one appropriately rescaled. This leads to concentrations of the solutions and the difficulty is to evaluate the weight and position of the moving Dirac masses that describe the population. We will show that a new type of Hamilton-Jacobi equation, with constraints, naturally describes this asymptotic. Some additional theoretical questions as uniqueness for the limiting H.-J. equation will also be addressed.
This work is based on collaborations with O. Diekmann, P.-E. Jabin, S. Mischler, S. Cuadrado, J. Carrillo, S. Genieys, M. Gauduchon and G. Barles.