Self-Interacting Discrete Models

自交互离散模型

• 批准号: RGPIN-2020-06124
• 负责人: Madras, Neal
• 金额: $ 1.31万
• 依托单位: York University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=740705
• 关键词: Self; Interacting; Discrete; Models

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.

这个建议包括三个不同领域的数学研究，从纯数学到应用数学。（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，一个聚合物分子中可能有数千个相同的单体。例如聚乙烯，DNA，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然而，这些模型难以分析，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，其中聚合物的柔性形状必须遵循三条线，物理预测的数学确认可以导致对来自这些模型的更广泛的推断的置信度的提高。(2)置换是一组数字或对象的简单重新排列。置换出现在对称性发挥作用的地方：在物理学、计算机科学、生物信息学和几乎所有的数学领域。我将研究具有某些限制的排列，称为“模式避免”。事实证明，强加模式避免极大地限制了（大）对象集合的有效排列集合，满足这种限制的排列结果是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的性质的方法，特别是那些对大多数人来说（但可能不是所有）给定模式集合的成员-避免排列。这主要是一个具有概率角度的理论组合学项目，其主要应用是纯数学本身。(3)疾病的数学模型已经成为公共卫生的一个重要组成部分。例如，当面临有限数量的流感疫苗时，集中精力在儿童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高峰需求吗？ 哪种疫苗会产生足够的好处，值得政府为它们买单？ 我将研究一类传染病传播模型的数学性质，这些模型在模型中具有显式的随机性。具体来说，每个感染者将感染随机数量的其他人，每个人的寿命是随机的。大多数大人口模型本质上是确定性的，或者对个人的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模型，这些模型具有寿命长度的现实概率。模拟可以告诉我们一些关于数学可以找到更深层次的模式，甚至适用于尚未模拟的模型。