Stochastic Systems: Theory and Models

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My research program is focused on probability theory and its applications.  I will work on three different topics, ranging from the pure to the applied. (1) How reliable are simulation results?  I shall study the efficiency of a class of Monte Carlo simulation algorithms known as Markov chain Monte Carlo methods.  These methods have been used to tackle computationally challenging problems in statistical analysis (for example, in a digital picture that has been blurred by random noise, what is the likely true image?) and other fields.  The user must decide how long to run the algorithm so that the results are not biased by the initial conditions.  This decision is not always clear, and it is easy to get misled by the output.  The only guarantees come with theoretical analysis. To that end, I aim to prove upper bounds on the time required to make the effect of the initial bias arbitrarily small (formally, I bound the rates of convergence of the Markov chains to equilibrium) in some simplified models, which I hope can serve as guidelines for the more complex situations that arise in practice. (2) Next, here is problem in pure combinatorics with a probabilistic viewpoint.  A permutation of size N is an arrangement of the numbers from 1 to N (e.g. 86425713 is a permutation of size 8).  We say that a permutation "contains the pattern 4231" if you can find four numbers in the permutation that occur in the same relative order as 4231 (i.e. the first one is largest, the second one is second smallest, the third one is third smallest, and the fourth one is smallest).   E.g., 86425713 contains the pattern 4231 because the numbers 8573 appear in this order.  Consider the set of permutations of size N that do not contain the pattern 4231.  When N is large, mathematicians have tried (with limited success) to determine approximately how big this set is.  I have observed that randomly chosen elements of this set are likely to have some striking structural properties, and a goal of my research is to understand why this happens. (3) Finally, I will work on aspects of mathematical modelling of the immune system using models that incorporate randomness, to account for uncertainties in outcomes. One question concerns mutations, which are intrinsically random events. When fighting a pathogen such as a flu virus, our immune system uses T cells, which are designed to hunt down certain identifiable signs of the virus. But the pathogen's offspring can escape a T cell if they have a mutation in the right place. I would like to estimate the probability that the T cells can destroy all of the pathogen before some offspring accumulates enough mutations to be able to evade all of the T cells. A second question asks for the probability that our "innate" immune system can control an infection of West Nile virus from a mosquito bite rapidly enough that the T cells are not needed.