Minimal Free Resolutions and Syzygies

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A core goal in the mathematical areas Algebraic Geometry and Commutative Algebra deals with understanding the solutions of a system of polynomial equations, possibly in a large number of variables and with a large number of equations. The solution set forms a geometric object. The main idea is to study the rich and beautiful interplay between its geometric and algebraic properties. Closely related to this is the concept of a free resolution, which was introduced by the famous mathematician David Hilbert in two papers in 1890 and 1893. Constructing a free resolution amounts to repeatedly solving systems of polynomial equations. The study of these objects flourished in the second half of the twentieth century, and has seen spectacular progress recently. The field is very broad, with strong connections and applications to other mathematical areas and string theory. Recent computational methods have made it possible to compute some free resolutions by computer. The main research goal in this project is to make significant progress in understanding the structure of free resolutions and their numerical invariants. The main research topics are:(1) The structure of minimal free resolutions of binomial edge ideals with a quadratic initial ideal in a polynomial ring. The project is inspired by a conjecture of Ene, Herzog, and Hibi, that the graded Betti numbers of such an ideal coincide with the graded Betti numbers of its quadratic initial ideal (with respect to the lex order). (2) The structure of infinite minimal free resolutions of lexicographic ideals in Clements-Lindstrom quotient rings, which will be studied using the methods recently introduced by Eisenbud and Peeva in their research monograph "Minimal free resolutions over complete intersections”.(3) The PI plans to explore a new area of research: minimal free resolutions of binomial edge ideals over exterior algebras. The PI plans to train graduate students in her research field.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.