CAREER: Free Resolutions

提出的研究涉及自由分辨率的结构及其在交换代数、代数几何和拓扑组合学中的应用。我们将研究多项式环及其商的分辨率。在本质上，构造环r上的分辨率由重复求解r -线性方程组组成。从另一个角度来看，分辨率为研究模块的结构提供了一种同源方法。希尔伯特在1890年和1893年的著名论文中提出了将分辨率与模块联系起来的想法。这个CAREER项目特别关注环型的Castelnuovo-Mumford正则性和Koszulness、Betti数的下界、无限自由分辨率的渐近结构、受限幂环上的自由分辨率和单项式分辨率。Peeva将组织一个关于新年决心的研讨会和暑期学校。她还计划组织其他会议。这些会议为讨论最近的发展和促进研究提供了一个论坛。这个CAREER项目还旨在帮助本科生、研究生和博士后发现数学思想的美，激励他们，并引导他们探索令人兴奋的开放性问题和猜想。Peeva还在写一本关于新年决心的书。

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.