Novel homological structures in commutative algebra and their applications

Building on linear algebra--a subject introduced to undergraduate STEM majors--the field of algebraic geometry studies geometric objects defined by systems of polynomial equations, providing a powerful framework for understanding solution spaces of nonlinear equations. When these geometric objects become sufficiently intricate, a central approach is to study them via maps to and from simpler spaces. A key driver in the development of these ideas has been the introduction of homological techniques originating in the mathematical field of topology. These ideas have applications in physics, computer science, engineering, and other areas of mathematics. The project aims to advance these ideas by developing and applying homological and homotopical methods, including tools from noncommutative geometry. Beyond its scientific contributions, the project will strengthen the mathematical community of the southern United States and help train the next generation of researchers by supporting a research symposium for undergraduate mathematics students and by carrying out K-12 STEM outreach. This research follows four interrelated directions concerning categorical aspects of algebraic varieties and free resolutions. The first direction will develop a new reconstruction theorem, in the spirit of the classical Bondal-Orlov theorem, showing how a projective variety can be recovered from the derived category of an associated derived scheme. The second will extend Orlov's Landau-Ginzburg/Calabi-Yau correspondence to complete intersections embedded in smooth projective toric varieties, establishing new relationships between the derived category of a toric variety and the singularity categories of its associated multigraded algebras. The third direction aims to reinterpret the theory of higher matrix factorizations for complete intersections, introduced by Eisenbud and Peeva, as an instance of A-infinity Koszul duality. Working out the case of exterior algebras as a proof of concept, this approach will yield a new method for constructing higher matrix factorizations over complete intersections. The final direction will produce an explicit version of the Bernstein-Gelfand-Gelfand correspondence by extending constructions of Eisenbud, Fløystad, and Schreyer to the homotopy category of matrix factorizations over projective space, with applications to computations in noncommutative algebraic geometry. The project employs differential graded, multigraded, and A-infinity methods, together with Koszul duality, while developing new constructions of duality theories, Serre functors, and integral transforms. 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.