Grant: Discovery Grants Program - Individual

I plan to work on several topics in applied probability and combinatorics. 1. Lattice models of polymers: A long flexible polymer molecule (such as polyethylene or DNA) can be modeled as a "self-avoiding walk"---a path in space (or in a three-dimensional grid) that does not cross itself. Other polymers have elaborate branching structure, and are modeled by "lattice animals"---connected clusters of line segments in space (or in the grid). I will investigate questions about these models from a rigorous mathematical viewpoint, including: What can we say about the occurrence of knots on a long polymer, or about linking in a network of branched polymers? If there is strong self-attraction between the different parts of the polymer, will a large polymer collapse into a ball? 2. Markov chain Monte Carlo: This is a common method of computer simulation used in statistics, physics, and other fields. In this method, it is crucial to run the simulation long enough for it to settle into equilibrium. I shall obtain rigorous methods to help researchers decide in advance how long a run will be required for a simulation to achieve equilibrium. 3. Pattern-avoding permutations: This is a problem in pure combinatorics. A permutation is an arbitrary ordering of the numbers 1 to N (for a given number N). How many permutations "avoid the pattern 4231"---i.e. how many of them do not contain numbers w,x,y,z (appearing in that order) such that wxyz is in the same relative order as 4231 (w largest, z smallest, and x<y)? The answer is approximately some number L raised to the power N, but the value of L is unknown. I will investigate rigorous and experimental (simulation) bounds on L. I willl also look at properties that are shared by "most" 4231-avoiding permutations.