Analysis of Partial Differential Equations Arising in Population Genetics and Singular Stochastic Control

该项目的动机是应用于种群遗传学和优化。该项目的第一个目标是预测一个种群的基因特征的变化，并计算基本的生物量，例如在基因组中某一基因类型灭绝和固定之前的预期时间。这一研究计划的成功实施有望对理解遗传学中的单基因突变引起的医学疾病和系统发育中的进化树产生直接影响。这项研究的第二个目标集中在通信网络、航天器控制和经济学等领域的优化应用中的一系列公开问题。分析这类问题的一个合适的框架是奇异随机控制理论，其目标是设计一个在随机环境中以最小成本执行任务的受控过程。该项目将有助于深入了解最佳执行策略的构建和评估网络中的最低成本。PI将继续让本科生和研究生参与她的研究计划，促进女性和未被充分代表的群体在数学界的可见度，并在明尼苏达大学和美国数学学会的会议上组织会议。研究计划的第一部分的目标是为定义在奇异流形上的一类退化椭圆算子建立一个全面的正则性理论，能够考虑影响遗传进化的广泛因素。PI的目的是发展新的解析和概率方法，以适应这些算子的特殊退化特征和非光滑流形的几何，其中此类问题是公式化的。她预计，这项研究将对调和分析、数学金融学和概率论产生进一步的影响。该项目的第二个主题寻求发展数学基础，以解决与识别奇异随机控制问题中的最优执行策略相关的感兴趣的问题。PI致力于推进我们对一大类具有梯度约束的二阶Hamilton-Jacobi-Bellman方程正则性理论的理解。这项研究将需要发展新的技术，以形成非线性方程和斜型非线性边界条件的理论，并需要仔细的边界估计，以适应在这个框架中出现的非光滑区域。

The project is motivated by applications to population genetics and optimization. The first objective of the project is to predict the changes of the genic characteristics of a population, and to compute fundamental biological quantities, such as the expected time before extinction and fixation of a gene type in the genome. The successful implementation of this research program is expected to have a direct impact on understanding medical disorders caused by single-gene mutations in genetics and of evolutionary trees in phylogenetics. The second objective of this research is centered on a family of open problems in optimization with applications to communication networks, spacecraft control, and economics, among others. A suitable framework to analyze such questions is the theory of singular stochastic control, where the goal is to design a controlled process that performs a task with minimum cost in a random environment. This project will help to gain insight into the construction of optimal execution policies and in the evaluation of minimal costs in networks. The PI will continue to involve both undergraduate and graduate students in her research program, to promote the visibility of women and underrepresented groups in the mathematical community, and to organize conferences at University of Minnesota and American Mathematical Society meetings.The goal of the first part of the research program is to build a comprehensive regularity theory for a class of degenerate elliptic operators defined on singular manifolds capable of taking into account a wide range of factors that impact the genetic evolution. The PI aims to develop novel analytic and probabilistic methods adapted to the particular degenerate features of these operators and the geometry of the non-smooth manifolds, where such problems are formulated. She expects that this research will have further implications in harmonic analysis, mathematical finance, and in probability. The second topic of the project seeks to develop the mathematical foundations to address the questions of interest related to the identification of optimal execution policies in singular stochastic control problems. The PI pursues a program to advance our understanding of the regularity theory of a wide class of second order Hamilton-Jacobi-Bellman equations with gradient constraints. This research will entail the development of novel techniques that forge into the theories of nonlinear equations and nonlinear boundary conditions of oblique type, and require careful boundary estimates adapted to the non-smooth domains appearing in this framework.