Asymptotic Commutative Algebra and Multigraded Syzygies

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This research project concerns commutative algebra, which provides the foundation for a broad range of mathematics, including algebraic geometry and algebraic number theory. Commutative algebra finds application in many fields of science and engineering, including computer science, cryptography, coding theory, robotics, pattern recognition, and theoretical physics. Part of the work in this project aims to connect the Kakeya needle problem, a classical analysis problem about the amount of space needed to turn a needle in a full circle, with modern ideas from commutative algebra. Another part of the project will use commutative algebra to design new algorithms for performing geometric computations about a class of shapes with extraordinary symmetries, among other applications. This research will build new frontiers between commutative algebra and other fields. The first project connects commutative algebra with harmonic analysis. The Kakeya conjecture is a central problem in harmonic analysis that has spawned a parallel literature over finite fields. The project aims to expand this parallel to the p-adic integers, yielding closer connections with the original analytic questions. The second project develops homological algebra methods for new geometric settings. For a variety embedded in something other than projective space, free resolutions often fail to provide sharp connections with geometry; this project will develop homological machinery better suited to toric geometry. This multifaceted project offers an array of potential applications: splitting theorems for vector bundles on toric varieties, rationality proofs for Hurwitz spaces, and new sheaf cohomology algorithms.