Geometry, Syzygies and Combinatorics of Algebraic Varieties

Suny At Stony Brook, Stony Brook NY

Investigators

Abstract

The principal investigator and his collaborators will address several groups of problems in areas of algebraic geometry, combinatorics, and computational algebraic geometry, with emphasis on syzygies and their relevance in geometry and combinatorics. Algebraic geometry at large deals with geometric objects described by systems of polynomial equations. The area is central to mathematics since such geometric objects include certain fundamental examples in the most diverse disciplines of mathematics.Its importance in the applications of mathematics comes from the fact that such algebro-geometric objects provide models often used when representing fundamental geometric forms via equations and also on a computer. The investigator will study the geometry and algebra of polynomial ideals in close relation to their complexity, will investigate several combinatorial aspects related to his proposed complexity bounds, and will extend the use of exterior algebra methods to projective geometry. The principal investigator's activity will also include the development of various fundamental algorithms in computational algebraic geometry and their implementation in the Macaulay2 package written by Grayson and Stillman.