Self-Interacting Discrete Models

这一建议包括三个相当不同的领域的数学研究，从纯粹的到应用的。我将检查聚合物分子的数学模型。聚合物是由许多称为单体的较小单元组成的非常大的分子，可能是一个聚合物分子中数千个相同的单体。例如聚乙烯、DNA和蛋白质。许多物理学家和化学家开发了许多数学模型来帮助解释和预测聚合物的(经常令人惊讶的)物理性质。然而，这些模型在理论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方面来提高对这些模型的严格数学理解，其中聚合物的柔性形状必须遵循三维网格中的线条。对物理预测的数学确认可以提高对这些模型更广泛推断的信心。(2)排列只是一组数字或物体的重新排列。在对称性起作用的地方，排列就会出现：在物理学、计算机科学、生物信息学和几乎所有数学领域。我将研究具有某些限制的排列，称为模式避免。事实证明，施加模式避免极大地限制了一组(大)物体的有效排列集合，满足这种限制的排列具有令人惊讶的结构，可以在简单的散点图上观察到。我的目标是开发方法来表征和严格分析这些结构的属性，尤其是那些包含给定的模式避免排列集合的大多数(但可能不是所有)成员的那些。这主要是一个具有概率角度的理论组合学项目，其主要应用范围是纯数学本身。(3)疾病的数学建模已成为公共卫生的重要组成部分。例如，当面临有限的流感疫苗接种时，将重点放在儿童身上还是老年人身上更好？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？哪些疫苗将产生足够的好处，值得政府为其买单？我将研究一类在模型中具有明显随机性的传染病传播模型的数学性质。具体地说，每个感染者将感染随机数量的其他人。而且每个人的寿命都是随机的。大多数针对大量人口的模型本质上是确定性的，或者对个人寿命有不切实际的简化假设。我特别感兴趣的是用数学方法分析具有真实寿命长度概率的模型。模拟可以告诉我们关于特定模型的一些事情，但数学可以找到更深层次的模式，即使是对尚未模拟的模型也是如此。

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.