Wednesday, January 30, 2013 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Meredith Casey – Georgia Tech
This is an expository talk on the arc complex and translation distance of open book decompositions. We will discuss curve complexes, arc complex, open books, and finally the application to contact manifolds.
Wednesday, January 30, 2013 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Howie Weiss – Georgia Tech, School of Math
After some brief comments about the nature of mathematical modeling in
biology and medicine, we will formulate and analyze the SIR infectious
disease transmission model. The model is a system of three non-linear
differential equations that does not admit a closed form solution.
However, we can apply methods of dynamical systems to understand a great
deal about the nature of solutions. Along the way we will use this
model to develop a theoretical foundation for public health
interventions, and we will observe how the model yields several
fundamental insights (e.g., threshold for infection, herd immunity,
etc.) that could not be obtained any other way. At the end of the talk
we will compare the model predictions with data from actual outbreaks.
Wednesday, January 30, 2013 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles Bld Room 005
Speaker
Andrew Vlasic – Indiana University
For many evolutionary dynamics, within a population there are finitely many types that compete with each other. If we think of a type as a strategy, we may consider this dynamic from a game theoretic perspective. This evolution is frequency dependent, where the fitness of each type is given by the expected payoff for an individual in that subpopulation. Considering the frequencies of the population, the logarithmic growth is given by the difference of the respective fitness and the average fitness of the population as a whole. This dynamic is Darwinian in nature, where Nash Equilibria are fixed points, and Evolutionary Stable Strategies are asymptotically stable. Fudenberg and Harris modified this deterministic dynamic by assuming the fitness of each type are subject to population level shocks, which they model by Brownian motion. The authors characterize the two strategy case, while various other authors considered the arbitrary finite strategy case, as well as different variations of this model. Considering how ecological and social anomalies affect fitness, I expand upon the Fudenberg and Harris model by adding a compensated Poisson term. This type of stochastic differential equation is no longer continuous, which complicates the analysis of the model. We will discuss the approximation of the 2 strategy case, stability of Evolutionary Stable Strategies and extinction of dominated strategies for the arbitrary finite strategy case. Examples of applications are given. Prior knowledge of game theory is not needed for this talk.
Tuesday, January 29, 2013 - 16:30 for 1 hour (actually 50 minutes)
Location
Skiles 06
Speaker
Rafael de la Llave – Georgia Tech
We will present the method of correctly aligned windows and show how it can lead to large scale motions when there are homoclinic orbits to a normally hyperbolic manifold.
Tuesday, January 29, 2013 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Prof. Yachu Li – Shanghai Jiao Tong University – ycli@sjtu.edu.cn
We study the Dirichlet and Neumann type initial-boundary value problems for strongly degenerate parabolic-hyperbolic equations. We suggest the notions of entropy solutions for these problems and establish the uniqueness of entropy solutions. The existence of entropy solutions is also discussed(joint work with Yuxi Hu and Qin Wang).
We present a method for the detection
of stable and unstable fibers of invariant manifolds of periodic
orbits.
We show how to propagate the fibers to prove transversal
intersections of invariant manifolds. The method can be applied using
interval arithmetic
to produce rigorous, computer assisted estimates
for the manifolds. We apply the method to prove transversal
intersections of stable and unstable manifolds of Lyapunov orbits in
the restricted three body problem.
Monday, January 28, 2013 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Adam Knapp – Columbia University
Given any smooth manifold, there is a canonical symplectic structure on its cotangent bundle. A long standing idea of Arnol'd suggests that the symplectic topology of the cotangent bundle should contain a great deal of information about the smooth topology of its base. As a contrast, I show that when X is an open 4-manifold, this symplectic structure on T^*X does not depend on the choice of smooth structure on X. I will also discuss the particular cases of smooth structures on R^4 and once-punctured compact 4-manifolds.
Friday, January 25, 2013 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Ernie Croot – Georgia Tech
This talk will be on an algebraic proof of theSzemeredi-Trotter theorem, as given by Kaplan, Matousek and Sharir.The lecture assumes no prior knowledge of advanced algebra.