Seminars and Colloquia by Series

Wednesday, September 30, 2009 - 12:00 , Location: Skiles 171 , Brett Wick , School of Mathematics, Georgia Tech , , Organizer:
 In the last 10 years there has been a resurgence of interest in questions about certain spaces of analytic functions. In this talk we will discuss various advances in the study of these spaces of functions and highlight questions of current interest in analytic function theory. We will give an overview of recent advances in the Corona Problem, bilinear forms on spaces of analytic functions, and highlight some methods to studying these questions that use more discrete techniques. 
Wednesday, September 30, 2009 - 11:00 , Location: Skiles 269 , Jan Medlock , Clemson University , , Organizer:
The recent emergence of the influenza strain (the "swine flu") and delays in production of vaccine against it illustrate the importance of optimizing vaccine allocation.  Using an age-dependent model parametrized with data from the 1957 and 1918 influenza pandemics, which had dramatically different mortality patterns, we determined optimal vaccination strategies with regard to five outcome measures: deaths, infections, years of life lost, contingent valuation and economic costs.  In general, there is a balance between vaccinating children who transmit most and older individuals at greatest risk of mortality, however, we found that when at least a moderate amount of an effective vaccine is available supply, all outcome measures prioritized vaccinating schoolchildren.  This is vaccinating those most responsible for transmission to indirectly protect those most at risk of mortality and other disease complications.  When vaccine availability or effectiveness is reduced, the balance is shifted toward prioritizing those at greatest risk for some outcome measures. The amount of vaccine needed for vaccinating schoolchildren to be optimal depends on the general transmissibility of the influenza strain (R_0).  We also compared the previous and new recommendations of the CDC and its Advisory Committee on Immunization Practices are below optimum for all outcome measures. In addition, I will discuss some recent results using mortality and hospitalization data from the novel H1N1 "swine flu" and implications of the delay in vaccine availability.
Series: PDE Seminar
Tuesday, September 29, 2009 - 15:05 , Location: Skiles 255 , Stephen Pankavich , University of Texas, Arlington , Organizer: Zhiwu Lin
We formulate a plasma model in which negative ions tend to a fixed, spatially-homogeneous background of positive charge. Instead of solutions with compact spatial support, we must consider those that tend to the background as x tends to infinity. As opposed to the traditional Vlasov-Poisson system, the total charge and energy are thus infinite, and energy conservation (which is an essential component of global existence for the traditional problem) cannot provide bounds for a priori estimates. Instead, a conserved quantity related to the energy is used to bound particle velocities and prove the existence of a unique, global-in-time, classical solution. The proof combines these energy estimates with a crucial argument which establishes spatial decay of the charge density and electric field.
Monday, September 28, 2009 - 14:00 , Location: Skiles 269 , Vera Vertesi , MSRI , Organizer: John Etnyre
Legendrian knots are knots that can be described only by their projections(without having to separately keep track of the over-under crossinginformation): The third coordinate is given as the slope of theprojections. Every knot can be put in Legendrian position in many ways. Inthis talk we present an ongoing project (with Etnyre and Ng) of thecomplete classification of Legendrian representations of twist knots.
Monday, September 28, 2009 - 13:00 , Location: Skiles 255 , Chad Topaz , Macalester College , Organizer:
Biological aggregations such as insect swarms, bird flocks, and fish schools are arguably some of the most common and least understood patterns in nature. In this talk, I will discuss recent work on swarming models, focusing on the connection between inter-organism social interactions and properties of macroscopic swarm patterns. The first model is a conservation-type partial integrodifferential equation (PIDE). Social interactions of incompressible form lead to vortex-like swarms. The second model is a high-dimensional ODE description of locust groups. The statistical-mechanical properties of the attractive-repulsive social interaction potential control whether or not individuals form a rolling migratory swarm pattern similar to those observed in nature. For the third model, we again return to a conservation-type PIDE and, via long- and short-wave analysis, determine general conditions that social interactions must satisfy for the population to asymptotically spread, contract, or reach steady state.
Monday, September 28, 2009 - 11:00 , Location: Skiles 269 , Federico Bonetto , School of Mathematics, Georgia Tech , Organizer: Yingfei Yi

This talk continues from last week's colloquium.

Fourier's Law assert that the heat flow through a point in a solid is proportional to the temperature gradient at that point. Fourier himself thought that this law could not be derived from the mechanical properties of the elementary constituents (atoms and electrons, in modern language) of the solid. On the contrary, we now believe that such a derivation is possible and necessary. At the core of this change of opinion is the introduction of probability in the description. We now see the microscopic state of a system as a probability measure on phase space so that evolution becomes a stochastic process. Macroscopic properties are then obtained through averages. I will introduce some of the models used in this research and discuss their relevance for the physical problem and the mathematical results one can obtain.
Friday, September 25, 2009 - 16:00 , Location: Skiles 154 , Linwei Xin , Georgia Tech , Organizer:
In this talk, we will introduce the classical Cramer's Theorem. The pattern of proof is one of the two most powerful tools in the theory of large deviations. Namely, the upper bound comes from optimizing over a family of Chebychef inequalities; while the lower bound comes from introducing a Radon-Dikodym factor in order to make what was originally "deviant" behavior look like typical behavior. If time permits, we will extend the Cramer's Theorem to a more general setting and discuss the Sanov Theorem. This talk is based on Deuschel and Stroock's .
Friday, September 25, 2009 - 15:00 , Location: Skiles 269 , Anh Tran , Georgia Tech , Organizer:

(This is a 2 hour lecture.)

In this talk I will give a quick review of classical invariants of Legendrian knots in a 3-dimensional contact manifold (the topological knot type, the Thurston-Bennequin invariant and the rotation number). These classical invariants do not completely determine the Legendrian isotopy type of Legendrian knots, therefore we will consider Contact homology (aka Chekanov-Eliashberg DGA), a new invariant that has been defined in recent years. We also discuss the linearization of Contact homology, a method to extract a more computable invariant out of the DGA associated to a Legendrian knot.
Friday, September 25, 2009 - 15:00 , Location: Skiles 255 , Mihyun Kang , Technische Universitat Berlin , Organizer: Prasad Tetali
Since the seminal work of Erdos and Renyi the phase transition of the largest components in random graphs became one of the central topics in random graph theory and discrete probability theory. Of particular interest in recent years are random graphs with constraints (e.g. degree distribution, forbidden substructures) including random planar graphs. Let G(n,M) be a uniform random graph, a graph picked uniformly at random among all graphs on vertex set [n]={1,...,n} with M edges. Let P(n,M) be a uniform random planar graph, a graph picked uniformly at random among all graphs on vertex set [n] with M edges that are embeddable in the plane. Erodos-Renyi, Bollobas, and Janson-Knuth-Luczak-Pittel amongst others studied the critical behaviour of the largest components in G(n,M) when M= n/2+o(n) with scaling window of size n^{2/3}. For example, when M=n/2+s with s=o(n) and s \gg n^{2/3}, a.a.s. (i.e. with probability tending to 1 as n approaches \infty) G(n,M) contains a unique largest component (the giant component) of size (4+o(1))s. In contract to G(n,M) one can observe two critical behaviour in P(n,M), when M=n/2+o(n) with scaling window of size n^{2/3}, and when M=n+o(n) with scaling window of size n^{3/5}. For example, when M=n/2+s with s = o(n) and s \gg n^{2/3}, a.a.s. the largest component in P(n,M) is of size (2+o(1))s, roughly half the size of the largest component in G(n,M), whereas when M=n+t with t = o(n) and t \gg n^{3/5}, a.a.s. the number of vertices outside the giant component is \Theta(n^{3/2}t^{-3/2}). (Joint work with Tomasz Luczak)
Friday, September 25, 2009 - 13:00 , Location: Skiles 255 , Benjamin Webb , School of Mathematics, Georgia Tech , Organizer:
In the study of one dimensional dynamical systems one often assumes that the functions involved have a negative Schwarzian derivative. In this talk we consider a generalization of this condition. Specifically, we consider the interval functions of a real variable having some iterate with a negative Schwarzian derivative and show that many known results generalize to this larger class of functions. The introduction of this class was motivated by some maps arising in neuroscience