CAREER: Free Resolutions

Cornell University, Ithaca NY

Investigators

Abstract

DMS-0347342 Irena Peeva The proposed research deals with the structure of free resolutions and their applications in Commutative Algebra, Algebraic Geometry, and Topological Combinatorics. Resolutions will be studied over polynomial rings and their quotients. In essence constructing a resolution over a ring R consists of repeatedly solving systems of R-linear equations. From another point of view, resolutions provide a homological method for studying the structure of modules. The idea to associate a resolution to a module was introduced in Hilbert's famous 1890 and 1893 papers. This CAREER project focuses especially on Castelnuovo-Mumford regularity and Koszulness of toric varieties, lower bounds on Betti numbers, asymptotic structure of infinite free resolutions, free resolutions over rings with restricted powers, and monomial resolutions. Peeva will organize a workshop and a summer school on resolutions. She plans to organize other conferences as well. Such conferences provide a forum for discussion of recent developments and foster research. This CAREER project also aims to aid undergraduate and graduate students, and postdoctoral fellows to discover the beauty in mathematical ideas, to inspire them, and lead them to explore exciting open problems and conjectures. Peeva is also writing a book on resolutions.