Mathematics Colloquia and Seminars

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Many applications of commutative algebra to combinatorics arise by encoding combinatorial problems as toric ideals, and studying their numerical invariants. Grobner bases have been a computational mainstay in this area, but new algorithms are emerging which instead use ideas from convexity and elementary combinatorial topology.

A theorem with Bernd Sturmfels gives a universal cover construction which unfolds the lattice action acting on a toric ideal, allowing one to view any toric ideal as an infinite periodic monomial ideal. This allows the study of toric ideals by monomial methods. In particular, chains of syzygies can be often be described by the cells of naturally arising cell complexes.

We give several applications. As one application, graph colorings can be studied as lattice points not contained in any hyperplane of an infinite periodic hyperplane arrangement whose vertices form a lattice. We obtain a toric ideal whose minimal chain of syzygies is cellular and supported on the arrangement.