Support theories for cohomologically poor categories

Support theories are powerful mathematical tools for understanding complex systems by translating algebraic problems into geometric ones. They have transformed areas ranging from representation theory to algebraic geometry, but many important mathematical structures remain beyond the reach of existing methods because the traditional invariants contain too little information. This project will develop new techniques for studying these “cohomologically poor” settings, opening the way to a broader and more flexible geometric framework. The anticipated advances will strengthen mathematics that underlies many areas of modern science. The project will support the next generation of researchers through graduate mentoring, research collaborations, and extensive K-12 outreach. The project includes research opportunities for graduate students, continued leadership of mathematical enrichment programs serving hundreds of school students each year, and the development of innovative educational initiatives that broaden access to advanced mathematics. The research will develop support theories and tensor triangular geometry for representation categories whose structure cannot be explained by existing cohomological methods. It will investigate the derived categories of Schur algebras, representation categories of Lie superalgebras, quotient stacks arising from actions of finite group schemes, and finite tensor categories associated with Hopf algebras and Nichols algebras. The project will seek new methods for constructing and computing geometric invariants, classify tensor ideals in several previously inaccessible settings, and advance the cohomological finite generation program for finite-dimensional Hopf algebras. These investigations combine techniques from representation theory, homological algebra, algebraic geometry, and tensor triangular geometry, with the goal of extending geometric methods to broad new classes of algebraic structures and establishing new connections among these areas. 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.