The PI's research aims to provide a deeper understanding of the structure of free resolutions and their use algebra and geometry via several specified projects. One such project will continue the systematic development of Boij-Soederberg theory. This theory was recently developed by Eisenbud and Schreyer, and it provides results for understanding free resolutions in terms of specified atomic building blocks. This is a powerful new theory, and there remain many open questions about both the foundations and the reach of the theory. Another project aims to develop a new framework for homological algebra in the context of smooth toric varieties. The work of David Cox and others suggests the potential for a far-reaching algebraic/geometric dictionary that largely parallels the powerful dictionary between the algebra of the graded polynomial ring and the geometry of projective space. Yet many of the results from homological commutative algebra have no satisfying analogue in the context of toric varieties. A more robust homological theory would rely on connections between multigraded commutative algebra, algebraic geometry, and combinatorics. A third project involves the asymptotic structure of the free resolutions of high degree Veronese subrings. This aims to fill a gap in the literature: whereas the study of the free resolutions of high degree Veronese subrings for curves has been fruitful and widely applied, little is known about the free resolutions of higher dimensional varieties under a very positive embedding.
Free resolutions are built from matrices of polynomials. They arise naturally in many algebraic contexts, with connections to topics in commutative algebra, algebraic geometry, computational algebra, and more. Matrices of polynomials have a richer structure than the matrices of scalars that arise in linear algebra, and so basic questions about free resolutions remain unresolved. The PI's research projects aim to provide overarching structural results about free resolutions, and to apply these new structural insights to open problems in commutative algebra and algebraic geometry. These research projects all offer potential for explicit computations, providing research opportunities for undergraduate and graduate students, as well as leading to new software packages to be developed for Macaulay2. The PI will also continue his outreach projects on K-12 education, which often involve collaborative efforts between research mathematicians, educators, and public or nonprofit groups.
