Complex Geometry and Applications

Mathematics advances when seemingly different phenomena turn out to share a common underlying structure. This project investigates how mathematical constraints shape the behavior of mathematical objects across several interconnected areas of pure and applied mathematics. A central theme is rigidity: when does satisfying a natural condition force an object to have a very special form? Discovering such principles deepens basic scientific understanding and creates tools that can be used in other fields. The project serves the national interest by advancing the mathematical sciences, training doctoral students and undergraduates, supporting public outreach, and developing ideas with connections to quantum information, machine learning, and biomedical science. In particular, the applied part of the project seeks better mathematical models of disease progression in Alzheimer's disease and multiple sclerosis, with the long-term aim of improving the design and interpretation of clinical trials. The project develops this theme in four connected directions. The first concerns the complex geometry of the polydisk: the goal is to classify subvarieties whose intrinsic Caratheodory metric agrees with the one inherited from the ambient domain, and to determine how this geometric condition relates to holomorphic extension and Pick interpolation. The second develops a unified theory of complete Pick spaces, a broad class of function spaces that includes the Hardy space as a prototype. The investigator studies factorization, multiplier algebras, Hardy-type function scales, and corona-type problems in this general setting, with the aim of explaining why many classical Hardy space theorems extend far beyond their original context. The third extends classical random matrix theory to tuples of matrices satisfying algebraic constraints, particularly commuting tuples. This opens a new direction with natural connections to quantum information, where commutativity encodes the compatibility of measurements. The fourth applies mathematical analysis to scientific problems, especially the development of improved two-parameter models of disease progression in Alzheimer's disease and multiple sclerosis using brain age derived from MRI data. 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. NSF Award ID: 2554487 | Program: 01002627DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: John McCarthy | Institution: Washington University, SAINT LOUIS, MO | Award Amount: $299,998 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2554487 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2554487.html