Self-Interacting Discrete Models

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This proposal consist of mathematical research in three rather different areas, ranging from the pure to the applied.  (1) I will examine mathematical models of polymer molecules.  Polymers are very large molecules made of many smaller units called monomers, perhaps many thousands of identical monomers in one polymer molecule.  Examples include polyethylene, DNA, and proteins.  Physicists and chemists have developed many mathematical models to help explain and predict the (frequently surprising) physical properties of polymers.  However these models are difficult to analyze, both theoretically and computationally.  My goal is to improve the rigorous mathematical understanding of these models by focusing on aspects of certain discrete models, in which the polymer's flexible shape must follow the lines in a three-dimensional grid.  Mathematical confirmation of physical predictions can lead to improved confidence in the broader inferences from these models. (2)  A permutation is simply a rearrangement of a set of numbers or objects.  Permutations arise wherever symmetry plays a role:  in physics, computer science, bioinformatics, and almost all areas of mathematics.  I will look at permutations with certain restrictions called "pattern avoidance".  It turns out that imposing pattern avoidance greatly constrains the set of valid permutations of a (large) set of objects, and the permutations satisfying such a restriction turn out to have surprising structures that can be observed visually on a simple scatterplot.  My goal is to develop methods for characterizing and rigorously analyzing properties of these structures, particularly those that hold for most (but maybe not all) members of a given collection of pattern-avoiding permutations.  This is mainly a project in theoretical combinatorics with a probability angle, and whose main applications are within pure mathematics itself. (3) Mathematical modelling of disease has become a crucial part of public health.  For example, when faced with a limited amount of flu vaccine, is it better to focus efforts on children or on the elderly?  Can we predict the peak demand on hospital resources for a coming flu season?  Which vaccines will produce enough benefit to merit a government paying for them?   I will investigate mathematical properties of a class of models for the spread of an infectious disease that has randomness explicitly in the model.  Specifically, each infected person will infect a random number of other people, and each person lives for a random length of time.  Most models for large populations are essentially deterministic, or else have unrealistic simplifying assumptions about individual lifetimes.  I am particularly interested in mathematically analyzing models with realistic probabilities for lengths of lifetimes.  Simulations can tell us some things about specific models, but mathematics can find deeper patterns that hold even for models that have not yet been simulated.