Mathematical Interests
Traffic Modeling
I have been looking at probabilistic interacting particle systems which model traffic flow. The models I've studied are discrete time cellular automaton models which look at the behavior of individual cars on a highway.
Basically, the idea is as follows. We'll take a one-dimensional integer lattice, and call this the set of sites available for cars. Each site is either occupied by a car, or it is vacant. Thus, we are discretizing a one-lane one-directional roadway into sites. At each time step, any car looks at the configuration of cars in nearby sites to determine its movement. For instance, if the car is being immediately followed by another car, then it will move at a certain probability, say $\alpha$, at the next time step. If there is nowhere for the car to move (the next site is already occupied), then the car in question has to wait.
Microscopically, this doesn't give a very realistic behavior of traffic. However, the goal is that these simple update rules will yield some of the more interesting macroscopic phenomena, such at the formation and persistence of traffic jams. Indeed, what I have been looking for is a model in which the traffic state space converges to a mixture of traffic jam state and free flow state.
Branching Annihilating Random Walks
Introduced earlier, the first results came in the mid 1980s by Bramson and Gray. Since then the theory of branching annihilating random walks has received a lot of attention over the last decade, primarily in the physics community. There are several different models, each with its own list of theorems, conjectures, highlights, and problems.
A branching annihilating random walk is a collection of particles which are performing random walks, subject to a few more rules. When two particles collide, they both are immediately removed from the system (annihilated). Also, at some given rate, each particle gives birth to new particles, which then behave just as the parent.
Several variants introduced pertain to the number of children born at a branch time, the placement of the children relative to the parent, and the amount of time it takes for two particle occupying the same site to be annihilated. Work has been done on many of these variants. Perhaps the first essential question for each one is that of survival. When we start with a finite number of particles, will there be particles around for all time? Some answers to this question can be found in my Research statement.
Combinatorial Games
I have recently been looking at the mathematics combinatorial games, as set forth in the series of books Winning Ways for Your Mathematical Plays by Berlekamp, Conway, and Guy. The games described in this setting are "games of no chance", meaning that each player is aware of the other player's options and nothing is left to random chance. I have been working with a student who has been examining one of the simplest of these games, known as hackenbush, and adding a small element of randomness to the game. We have been looking at several questions about this new entity. What can we say about the distributions of values which ensue? How do we add games together? What types of strategy need to be defined, and how much can we say without making any assumptions on strategy at all? Can we characterize the resulting distributions? This is an interesting new area, and it holds bright prospects for future work with students.