Georgia Institute of Technology College of Sciences

Collective behavior can be seen in many animal species, such as flocking birds, herding mammals, and swarming bacteria. In the continuum limit, these phenomena can be modeled by nonlocal PDEs. In this talk, after discussing some PDE models for collective dynamics, I will focus on the analysis of the Keller-Segel equation, which models bacterial chemotaxis. Mathematically, this equation exhibits an intriguing "critical mass phenomenon": namely, solutions exist globally in time for all initial data whose mass is below some certain constant, whereas finite-time blow-up always happen if the initial mass is above this constant. I will introduce some useful analysis tools that lead to this result, and discuss some active areas of current research.

The estimation of phylogenetic trees from molecular sequences (e.g., DNA, RNA, or amino acid sequences) is a major step in many biological research studies, and is typically approached using heuristics for NP-hard optimization problems. In this talk, I will describe a new approach for computing large trees: constrained exact optimization. In a constrained exact optimization, we implicitly constrain the search space by providing a set X of allowed bipartitions on the species set, and then use dynamic programming to find a globally optimal solution within that constrained space. For many optimization problems, the dynamic programming algorithms can complete in polynomial time in the input size. Simulation studies show that constrained exact optimization also provides highly accurate estimates of the true species tree, and analyses of both biological and simulated datasets shows that constrained exact optimization provides improved solutions to the optimization criteria efficiently. We end with some discussion of future research in this topic. (Refreshments will be served before the talk at 10:30.)

Estimating the Tree of Life is one of the grand computational challenges in Science, and has applications to many areas of science and biomedical research. Despite intensive research over the last several decades, many problems remain inadequately solved. In this talk I will discuss species tree estimation from genome-scale datasets. I will describe the current state of the art for these problems, what is understood about these problems from a mathematical perspective, and identify some of the open problems in this area where mathematical research, drawing from graph theory, combinatorial optimization, and probability and statistics, is needed. This talk will be accessible to mathematicians, computer scientists, probabilists and statisticians, and does not require any knowledge of biology. (Refreshments will be served after the talk.)

A nonlinear filtering problem can be classified as a stochastic Bayesian optimization problem of identifying the state of a stochastic dynamical system based on noisy observations of the system. Well known numerical simulation methods include unscented Kalman filters and particle filters. In this talk, we consider a class of efficient numerical methods based on forward backward stochastic differential equations. The backward SDEs for nonlinear filtering problems are similar to the Fokker-Planck equations for SDEs. We will describe the process of deriving such backward SDEs as well as high order numerical algorithms to solve them, which in turn solve nonlinear filtering problems.