LEAPS-MPS: Combinatorial Aspects of Finite and Infinite Free Resolutions

Polynomials and polynomial systems are fundamental in the study of mathematics and in its applications, with appearances in fields ranging from robotics to medical imaging to biochemical reaction networks. Commutative algebra, along with algebraic geometry, is the study of polynomial systems and their solutions. One powerful tool to obtain information about a polynomial system is a minimal free resolution, which can be thought of as a step-by-step unfolding of a polynomial system into simpler, more linear structures. While for arbitrary polynomials, this process is unpredictable, when the polynomials in question have some extra structure, building and understanding a minimal free resolution can be done with combinatorial methods. Broader impacts will be achieved by mentoring undergraduate and graduate research, leading a STEM faculty writing group, and founding a local Math Circle, which gives middle school students in Huntington, West Virginia and the surrounding area the opportunity to encounter mathematics that is in addition to the standard K-12 curriculum. The proposed projects will both utilize and further develop the bridge between combinatorics and free resolutions. The focus will be free resolutions in three different combinatorial contexts: numerical semigroups, which study nonnegative integer combinations of nonnegative integers, as in the Frobenius coin problem; affine semigroups, which correspond to integer lattices and thus affine toric varieties; and almost complete intersections, generalizations of complete intersections -- defined by "maximally independent" polynomials and thus carrying very predictable structure -- that nonetheless display surprisingly combinatorial numerics. The techniques will include polyhedral geometry, partially ordered sets, Hilbert functions and Hilbert series, and homological tools. 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.