Georgia Institute of Technology College of Sciences

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.

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.

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.

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.