CAREER: Free Resolutions

DMS-0347342 Irena Peeva拟议的研究涉及自由归结的结构及其在交换代数，代数几何和拓扑组合学中的应用。将研究多项式环及其等价环上的分解。环R上的归结式的构造实质上是重复求解R-线性方程组，从另一个角度讲，归结式为研究模的结构提供了一种同调方法。 将一个分解与一个模联系起来的想法是在希尔伯特著名的1890年和1893年的论文中提出的。 这个职业项目特别关注复曲面簇的Castelnuovo-Mumford正则性和Koszulness、Betti数的下界、无限自由归结的渐进结构、幂受限环上的自由归结以及单项归结。Peeva将组织一个研讨会和暑期学校关于决议。她还计划组织其他会议。这些会议为讨论最新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中的美，激励他们，并引导他们探索令人兴奋的开放问题和挑战。Peeva也在写一本关于决心的书。

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.