The Combinatorics of Syzygies

One of the oldest problems in mathematics, and a root of modern algebra today, is finding solutions to polynomial equations. To find and analyze solutions to several polynomials at the same time, more tools need to be developed. One such tool used by algebraists is a free resolution of those polynomials: a sequence of maps (essentially between vector spaces) built from the relations between the polynomials (these relations are called the "1st syzygies"), and then the relations between those relations (called the "2nd syzygies") and so on. The smallest such sequence models what is called a "minimal free resolution". The minimal free resolution, its length and the size of each piece are "invariants" of the polynomials we start with. This means that no matter how you compute the minimal resolution - and there are many ways to do so - you always get the same results. A table of integers, called the "Betti table" keeps track of these invariants. The construction and study of free resolutions goes back many decades to the famous mathematician David Hilbert, and has consequences in the geometry of the solutions of the polynomials we start with. Today researchers worldwide work on open problems related to free resolutions of sets of polynomials as well as more abstract related structures, trying to decipher the properties as well as applications to other areas of mathematics. This proposal offers ways to study resolutions using manipulations on geometric shapes, such as graphs and tetrahedra. Many of the questions in this area are challenging and often wide open, in the sense that one needs to develop new methods to deal with them. Researchers often use "homotopy theory" to change the shape of the object supporting the resolution into something simpler while keeping its important features intact. The typical search is for an appealing geometric object which can get us (very close) to the minimal free resolution. I will give a simple outline of the specific topics of my proposal. For a minimal free resolution F of a set (ideal) I of monomials (polynomials with only one term), and a shape S associated to I (there are several options for S, take any one of them): (1) What impact does the Betti table of I have on the shape S? I plan to investigate conjectures about Betti tables that if true, would reveal surprising geometric properties for S. (2) Is there a "nice" bound for the length of F? My hope is that I can detect the nicest bound from the number of unshaded triangles or unfilled tetrahedra in S. (3) If we take products and powers of the polynomials in I, how does F change? How does S change? (4) Is there are good characterization of the polynomials for which F and S are the "best" behaved sequence/shape (in an algebraic/geometric sense)? Answers to these questions will contribute to the discovery of new geometric structures, and also to understanding the often mysterious behaviours of solution sets to polynomials.