CAREER: Free Resolutions

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DMS-0347342Irena PeevaThe proposed research deals with the structure of freeresolutions and their applications in Commutative Algebra,Algebraic Geometry, and Topological Combinatorics. Resolutions will be studied over polynomial rings and theirquotients. In essence constructing a resolution over a ring Rconsists 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 freeresolutions, 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 fosterresearch. This CAREER project also aims to aid undergraduate andgraduate students, and postdoctoral fellows to discover the beautyin mathematical ideas, to inspire them, and lead them to exploreexciting open problems and conjectures. Peeva is also writing a book on resolutions.