RTG: Partial Differential Equations on Manifolds

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Partial differential equations are ubiquitous in models of physical phenomena. Discovering the statistics of the sets where quantum particles are least likely to exist, describing the behavior of light waves near black holes, explaining the existence of bands on the planet Jupiter by analyzing the equations of fluid dynamics on a rotating sphere, and establishing the existence of solutions to systems modeling ferromagnetism or plasmas require an analysis of partial differential equations in the presence of background geometry. All of these are fundamental topics that members of the team of principal investigators have explored, and for which they are ideally situated to pass on their expertise. The purpose of this project is to create a sustainable program that enhances and modernizes the curriculum, enriches the training, and improves recruitment. This will create larger and better prepared cohorts of mathematicians to advance the understanding of the interplay between geometry and solutions to partial differential equations.The associated faculty will thoughtfully mentor postdoctoral, graduate student, undergraduate student, and high school research trainees. Graduate courses that may be taken asynchronously will be developed to broaden the curriculum. A regular workshop will be formed to get new graduate students to engage with research early. Developmental training groups will be designed to improve trainees’ technical writing and to guide them on aspects of academic life such as seeking funding. An online topics course collaborative will be established to promote sharing across academic institutions of specialized courses on cutting-edge material. An online undergraduate research seminar will be created for the dissemination of results from a newly formalized undergraduate research program at the host institution and for the recruitment of new graduate students who are conducting interesting research at other colleges and universities. The host institution’s partial differential equations mini-school program, which allows for trainees to interact with a principal speaker and their students, postdocs, and co-authors in an intimate environment that permits in-depth presentations, will be revived. Outreach activities such as a program to allow for interested local high school students to engage with mathematical research and the Girls Talk Math summer program will be supported. Many of the concrete needs that are addressed are not unique to the host institution, and these initiatives offer great benefit to a broad community of trainees.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.