Syzygies and Geometry

Suny At Stony Brook, Stony Brook NY

Investigators

Abstract

The PI will pursue research in areas of algebraic geometry, commutative algebra, combinatorics, and their computational aspects, with emphasis on syzygies and their relevance to geometry. The PI will study (1) Syzygies of Lawrence toric ideals and their relation to chromatic numbers of graphs, (2) Syzygies, the rank variety and the complexity of Orlik-Solomon algebras, (3) Syzygies of symmetric sheaves, and corresponding obstruction classes in derived Witt groups to the existence of symmetric locally free resolutions of such sheaves, (4) Natural compactifications of configurations of points in projective space and their applications to enumerative problems. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. The main thrust is towards discovering connections between the geometry of sets defined by the vanishing of polynomials and the algebraic invariants derived from the defining equations of these sets, for instance syzygies. Results in algebraic geometry and combinatorics have also found uses in communications, security and robotics.