Multigraded Methods for Syzygies, Arrangements, and Differential Operators

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At the heart of the research components of this project are homological objects with group actions, which arise in a wide range of areas of application, including biology, chemistry, algebraic topology and geometry, optimization, and physics, among others. The project involves the development of a host of new tools to express geometric properties algebraically, via certain families of matrices with polynomial entries. The education and broader impacts portions integrate with these projects through work with undergraduate and graduate students, as well as postdocs. The PI will also continue involvement in several mentoring initiatives, including the Enhancing Diversity in Graduate Education (EDGE) Program, Math Alliance, and Minnesota's Women in Math program; organization of regional and international conferences organization; and software development and distribution for the open source computer algebra system Macaulay2.The research components of this project seek to establish a foundational framework for each of the following: (1) free complexes corresponding to line bundle resolutions of sheaves on smooth toric varieties, (2) resolutions of certain equivariant sheaves via vector bundles over smooth toric varieties, (3) arrangements of hyperplanes, and (4) rings of differential operators for toric face rings. More specifically the research includes (respectively): (1) constructing an explicit vector bundle resolution of the diagonal for smooth toric varieties, which will yield an analogue of the celebrated Hilbert Syzygy Theorem, (2) resolutions of certain equivariant sheaves via vector bundles over smooth toric varieties, (3) A-hypergeometric polynomials in prime characteristic, and (4) rings of differential operators for toric face rings. The research will yield a host of new tools that will shed light on the underlying geometry and group actions by aiding the computation of important algebro-geometric and PDE invariants.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.