Self-Interacting Discrete Models

这个建议包括从纯粹到应用的三个不同领域的数学研究。我将研究聚合物分子的数学模型。聚合物是由许多更小的单体组成的大分子，一个聚合物分子中可能有数千个相同的单体。例子包括聚乙烯、DNA和蛋白质。物理学家和化学家已经开发了许多数学模型来帮助解释和预测（经常令人惊讶的）聚合物的物理性质。然而，这些模型在理论上和计算上都很难分析。我的目标是通过关注某些离散模型的各个方面来提高对这些模型的严格数学理解，在这些模型中，聚合物的柔性形状必须遵循三维网格中的线条。物理预测的数学证实可以提高对这些模型更广泛推论的信心。排列就是一组数字或物体的重新排列。只要对称发挥作用，排列就会出现：物理学、计算机科学、生物信息学以及几乎所有数学领域。我将着眼于具有一定限制的排列，称为“模式回避”。事实证明，强加模式避免极大地限制了（大量）对象的有效排列集，而满足这种限制的排列结果具有令人惊讶的结构，可以在简单的散点图上直观地观察到。我的目标是开发用于表征和严格分析这些结构属性的方法，特别是那些适用于给定的避免模式排列集合的大多数（但可能不是全部）成员的结构。这主要是一个从概率角度研究理论组合的项目，它的主要应用是纯数学本身。(3)疾病的数学建模已成为公共卫生的重要组成部分。例如，面对数量有限的流感疫苗，是把重点放在儿童身上好，还是放在老年人身上好？我们能否预测即将到来的流感季节对医院资源的高峰需求？哪些疫苗会产生足够的好处，值得政府为其买单？我将研究一类具有明确随机性的传染病传播模型的数学性质。具体来说，每个感染者将感染随机数量的其他人，每个人的寿命随机长度。大多数大规模人口的模型本质上是确定性的，或者对个体寿命有不切实际的简化假设。我特别感兴趣的是在数学上分析具有生命周期长度的现实概率的模型。模拟可以告诉我们一些关于特定模型的事情，但数学可以找到更深层的模式，甚至适用于尚未被模拟的模型。

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.