Wednesday, February 13, 2013 - 12:05 , Location: Skiles 005 , Michael Lacey , Georgia Tech, School of Math , Organizer: Robert Krone
I'll introduce the Hilbert transform in a natural way justifying it as a canonical operation. In fact, it is such a basic operation, that it arises naturally in a range of settings, with the important complication that the measure spaces need not be Lebesge, but rather a pair of potentially exotic measures. Does the Hilbert transform map L^2 of one measure into L^2 of the other? The full characterization has only just been found. I'll illustrate the difficulties with a charming example using uniform measure on the standard 1/3 Cantor set.
From microscopic to macroscopic: some consideration on a simple model for a gas in or out of equilibriumWednesday, February 6, 2013 - 12:05 , Location: Skiles 005 , Federico Bonetto , Georgia Tech, School of Math , Organizer: Robert Krone
The derivation of the properties of macroscopic systems (e.g. the air in a room) from the motions and interactions of their microscopic constituents is the principal goal of Statistical Mechanics. I will introduce a simplified model of a gas (the Kac model). After discussing its relation with more realistic models, I'll present some known results and possible extension.
Wednesday, January 30, 2013 - 12:05 , Location: Skiles 005 , Howie Weiss , Georgia Tech, School of Math , Organizer: Robert Krone
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 23, 2013 - 12:05 , Location: Skiles 005 , Doug Ulmer , Georgia Tech, School of Math , Organizer: Robert Krone
I will review a little bit of the theory of algebric curves, which essentialy amounts to studying the zero set of a two-variable polynomial. There are several amazing facts about the number of points on a curve when the ground field is finite. (This particular case has many applications to cryptography and coding theory.) An open problem in this area is whether there exist "supersingular" curves of every genus. (I'll explain the terminology, which has something to do with having many points or few points.) A new project I have just started should go some way toward resolving this question.
Wednesday, January 16, 2013 - 12:05 , Location: Skiles 005 , Hao Min Zhou , Georgia Tech, School of Math , Organizer: Robert Krone
In this talk, I will use the shortest path problem as an example to illustrate how one can use optimization, stochastic differential equations and partial differential equations together to solve some challenging real world problems. On the other end, I will show what new and challenging mathematical problems can be raised from those applications. The talk is based on a joint work with Shui-Nee Chow and Jun Lu. And it is intended for graduate students.
Tuesday, January 15, 2013 - 12:00 , Location: Skiles 005 , Dr. Joseph Rabinoff , School of Mathematics , email@example.com , Organizer:
Wednesday, December 5, 2012 - 12:05 , Location: Skiles 005 , Heinrich Matzinger , Georgia Tech, School of Math , Organizer: Robert Krone
The question of the asymptotic order of magnitude of the fluctuation of the Optimal Alignment Score of two random sequences of length n has been open for decades. We prove a relation between that order and the limit of the rescaled optimal alignment score considered as a function of the substitution matrix. This allows us to determine the asymptotic order of the fluctuation for many realistic situations up to a high confidence level.
Wednesday, November 28, 2012 - 12:05 , Location: Skiles 005 , Guillermo Goldsztein , Georgia Tech, School of Math , Organizer: Robert Krone
I will describe a class of mathematical models of composites and polycrystals. The problems I will describe two research projects that are well suited for graduate student interested in learning more about this area of research.
Wednesday, November 14, 2012 - 12:05 , Location: Skiles 005 , Matt Baker , Georgia Tech, School of Math , Organizer: Robert Krone
I will discuss how one can solve certain concrete problems in number theory, for example the Diophantine equation 2x^2 + 1 = 3^m, using p-adic analysis. No previous knowledge of p-adic numbers will be assumed. If time permits, I will discuss how similar p-adic analytic methods can be used to prove the famous Skolem-Mahler-Lech theorem: If a_n is a sequence of complex numbers satisfying some finite-order linear recurrence, then for any complex number b there are only finitely many n for which a_n = b.
Wednesday, November 7, 2012 - 12:05 , Location: Skiles 005 , Santosh Vempala , Georgia Tech, College of Computing , Organizer: Robert Krone
The hyperplane conjecture of Kannan, Lovasz and Simonovits asserts that the isoperimetric constant of a logconcave measure (minimum surface to volume ratio over all subsets of measure at most half) is approximated by a halfspace to within an absolute constant factor. I will describe the motivation, implications and some developments around the conjecture and an approach to resolving it (which does not seem entirely ridiculous).