Commutative Algebra and Algebraic Geometry

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Algebraic Geometry is the study of systems of polynomial equations. As such, it has broad applications not only within mathematics but also in many fields of science from medicine to physics. The subject has developed for over 200 years, but there are many fresh problems and new directions. Commutative algebra is a subject that lies at the intersection of algebraic geometry and number theory. Since the advent of powerful and easily available computing resources, the possibilities for experimentation within commutative algebra, algebraic geometry, and their applications has multiplied. A byproduct of this project will be the further development of these computational tools. Cohen-Macaulay modules and sheaves play a role in commutative algebra and algebraic geometry that is a natural analogue of the role of finite dimensional representations in the case of finite dimensional algebras. The Principal Investigator will work on projects in commutative algebra, algebraic geometry, and computational methods that center around the theory of Cohen-Macaulay modules over particularly interesting classes of varieties: Toric varieties, complete intersections, and residual intersections. The PI will also continue to train graduate students in related research fields and actively be involved in several highly recognized outreach activities.The Principal Investigator will work on Ulrich modules and Clifford Algebras. Cohen-Macaulay and Ulrich modules over quadratic hypersurfaces are well-understood from work of Knoerrer (over algebraically closed fields) and Buchweitz-Eisenbud-Schreyer over arbitrary fields. The PI will investigate deeper questions about Ulrich modules and other maximal Cohen-Macaulay modules on complete intersections of two quadrics using Clifford algebra techniques, extending Miles Reid's thesis, and making explicit work of Bondal-Orlov and Kapranov, as well as the theory of maximal Cohen-Macaulay modules over complete intersections developed by the proposer with Irena Peeva. Finally the PI will work on the cohomology of sheaves on toric varieties, extending techniques from exterior algebra algebras introduced in joint work with Daniel Erman and Frank-Olaf Schreyer for cohomology of sheaves on projective space and successfully extended to products of projective space to all toric varieties.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.