Homological Commutative Algebra and Group Actions in Geometry

University Of Minnesota-Twin Cities, Minneapolis MN

Investigators

Abstract

This project concerns types of symmetry in algebraic structures which arise in a wide range of areas of application, including biology, chemistry, geometry, optimization, and physics. The research involves the development of a host of new tools to express geometric properties algebraically. The education and broader impacts portion of the project integrates with the research through work with graduate students. The PI will also continue involvement in several mentoring initiatives, including the Enhancing Diversity in Graduate Education (EDGE) Program, Math Alliance, and a professional development seminar for mathematics graduate students in conjunction with advising 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) complexes corresponding to line bundle resolutions of sheaves on smooth toric varieties, (2) free resolutions of equivariant ideals in an infinite setting, for instance, the case of a symmetric group action on a countably infinite-dimensional space, and (3) a D-module variant of Koszul homology over spherical varieties. The specific parts include (respectively): (1) developing and applying analogues for smooth toric varieties of foundational homological results for projective space, (2) determining how to track and compute invariant syzygies in infinite settings with a suitable monoid or group action, and (3) generalizing a homological framework for hypergeometric systems of PDEs from the setting of a torus action to that of a reductive group. The project will yield new sets of tools for shedding light on the underlying geometry and group actions present, by aiding in the computation of important algebro-geometric and PDE invariants.