Collaborative Research: FRGMS: Multigraded Commutative Algebra, Toric Varieties, and Homological Mirror Symmetry

This project explores exciting new interactions between two central areas of mathematics - algebra and geometry - and their unexpected connections through physics. Algebra and geometry are foundational tools in mathematics, widely used in numerous scientific and engineering applications, such as computer science, data analysis, robotics, and theoretical physics. Historically, the interplay between algebraic equations and geometric shapes has led to powerful methods and profound insights, shaping much of modern mathematics and technology. In recent decades, researchers discovered surprising connections linking algebraic geometry, which studies shapes defined by polynomial equations, to symplectic geometry, an area crucial to physics and engineering. This project leverages these emerging connections to develop new mathematical tools that bridge algebra and geometry. Broader impacts of this research include significant training and mentoring activities. The project supports early-career researchers and graduate students, providing extensive professional development through workshops, virtual seminars, public lectures, and the creation of publicly available computational tools. On the technical side, the project aims to advance understanding in multigraded commutative algebra, toric geometry, and symplectic geometry. It addresses long-standing gaps and open questions in commutative algebra and toric geometry by introducing methods inspired by recent advances in homological mirror symmetry into purely algebraic contexts. The P.I.’s will explore new approaches to studying multigraded polynomial rings, aiming to uncover deeper structural properties that parallel classical results for standard graded polynomial rings. The project will develop algebraic analogues of effective symplectic geometry techniques, such as "stop manipulation," adapting these symplectic methods to algebraic settings. The project will also extend foundational results, including Orlov’s Theorem, to multigraded and toric settings, construct novel categorical structures that unify algebraic and geometric perspectives, explore applications to virtual resolutions and other questions involving shortest resolutions, and investigate extensions to broader classes of geometric objects through toric degenerations and natural generalizations from toric varieties. Furthermore, by establishing explicit links between algebraic constructions and Fukaya categories, the project will introduce new computational tools and theoretical approaches in symplectic geometry. 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.