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

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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.