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

该项目的动机是群体遗传学和优化的应用。该项目的第一个目标是预测一个种群的基因特征的变化，并计算基本的生物学量，例如基因组中某个基因类型灭绝和固定之前的预期时间。这项研究计划的成功实施预计将对理解遗传学中的单基因突变和遗传学中的进化树引起的医学疾病产生直接影响。本研究的第二个目标是集中在一个家庭的开放性问题的优化与应用通信网络，航天器控制，经济等。一个合适的框架来分析这样的问题是奇异随机控制理论，其目标是设计一个受控的过程，在随机环境中以最小的成本执行任务。该项目将有助于深入了解最佳执行策略的构建和网络中最小成本的评估。PI将继续让本科生和研究生参与她的研究计划，以促进女性和数学界代表性不足的群体的知名度，第一部分的研究目标是建立一类定义在奇异流形上的退化椭圆算子的正则性理论能够考虑到影响遗传进化的广泛因素。PI的目的是开发新的分析和概率方法，适应这些运营商的特定退化功能和非光滑流形的几何形状，这些问题的制定。她预计，这项研究将有进一步的影响，谐波分析，数学金融，并在概率。该项目的第二个主题旨在发展数学基础，以解决与奇异随机控制问题中的最优执行策略识别相关的问题。PI追求一个程序，以提高我们的理解的正则性理论的广泛的一类二阶Hamilton-Jacobi-Bellman方程的梯度约束。这项研究将需要开发新的技术，锻造成理论的非线性方程和非线性边界条件的斜型，并需要仔细的边界估计适应于出现在这个框架中的非光滑域。

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.