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。 (1)我将研究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，也许一个聚合物分子中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，这些模型无论在理论上还是在计算上都很难分析。我的目标是通过关注某些离散模型的某些方面来提高对这些模型的严格数学理解，在这些模型中，聚合物的灵活形状必须遵循三维网格中的线。对物理预测的数学确认可以提高人们对这些模型的更广泛推论的信心。 (2)排列仅仅是一组数字或对象的重新排列。在对称性发挥作用的任何地方，排列都会出现：在物理学、计算机科学、生物信息学以及几乎所有的数学领域。我将研究具有某些限制的排列，这些限制称为“模式避免”。事实证明，强制模式避免极大地限制了一组(大量)对象的有效排列，并且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进行视觉观察。我的目标是开发方法来表征和严格分析这些结构的属性，特别是那些适用于给定的模式避免排列集合中的大多数(但可能不是所有)成员的结构。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)疾病的数学模型已成为公共卫生的重要组成部分。例如，面对有限数量的流感疫苗，将重点放在儿童身上还是老年人身上更好？我们能预测即将到来的流感季节对医院资源的需求高峰吗？哪些疫苗将产生足够的益处，值得政府为其买单？我将研究一类传染病传播模型的数学性质，该模型在模型中具有明显的随机性。具体地说，每个感染者会感染随机数量的其他人，每个人都会随机存活一段时间。大多数针对大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寿命长度的真实概率模型。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次的模式，即使是对于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.