Research in noncommutative and commutative algebra

非交换代数和交换代数研究

• 批准号: 0901185
• 负责人: Gordana Todorov
• 金额: $ 39.54万
• 依托单位: Northeastern University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901185&HistoricalAwards=false
• 关键词: Research; noncommutative; commutative; algebra

The proposal consists of several interrelated parts. The first part is to study the rings of semi-invariants of quivers and of quivers with relations. The investigator proposes to continue study the walls of cones of weights of rings of semi-invariants of quivers and multiplicities of weight spaces for these rings. In particular case this includes the cones defined by Klyachko inequalities. For the semi-invariants of quivers with relations the main problem is to characterize the finite type and tame quivers with relations in terms of semi-invariants. Another aspect is the connection of quiver representations and cluster algebras. The principal investigator plans to study quivers with potential and the mutations of related Jacobian algebras. He also plans to study the connection of certain sphere triangulations and the Igusa-Orr theory of pictures related to nilpotent groups. The second part is devoted to studying defining ideals of equivariant varieties. Several types of varieties are proposed: orbit closures for representations with finitely many orbits, tangential and secant varieties of orbits of highest weight vectors, and orbit closures for Dynkin quivers corresponding to Schubert varieties in the Grassmannian. The third part involves problems related to Boij-Soderberg conjectures for Betti tables of graded modules. The principal investigator proposes to study cohomology tables of vector bundles on homogeneous spaces and equivariant refinements of Boij-Soderberg conjectures. This proposal is related to several branches of mathematics: representations of quivers and commutative algebra. A representation of a quiver is a way to associate vector space data to the vertices of some oriented graph. The edges can be viewed as relations between these data. Abstract algebra allows to study such objects systematically and the results of research might lead to better algorithms dealing with linear algebra problems. Commutative algebra studies polynomial functions of geometric objects. The second part of the proposal is devoted to studying polynomial equations defining objects characterized geometrically such as rank conditions on matrices. Various conditions of this type on tensors are of interest for engineers and computer scientists. The third part of the proposal studies Betti tables: certain family of numerical invariants associated to modules over a polynomial ring.

该提案由几个相互关联的部分组成。第一部分是研究箭图的半不变量环和带关系的箭图的半不变量环。研究人员建议继续研究壁锥的重量环的半不变量的箭和多重权空间的这些环。在特殊情况下，这包括由Klyachko不等式定义的锥。对于带关系的箭图的半不变量，主要问题是用半不变量刻画有限型和驯服带关系的箭图。另一个方面是将群表示与簇代数联系起来。主要研究者计划研究具有势的箭图和相关雅可比代数的突变。他还计划研究连接某些领域的三角和伊格萨奥尔理论的图片有关幂零群体。第二部分研究了等变簇的定义理想。提出了几种类型的品种：轨道封闭的代表与mundemmany轨道，切线和割线品种的轨道的最高权重向量，和轨道封闭Dynkin箭图对应舒伯特品种在格拉斯曼。第三部分涉及分次模的Betti表的Boij-Soderberg构造问题。 主要研究者建议研究齐性空间上向量丛的上同调表和Boij-Soderberg拓扑的等变加细。这个提议与数学的几个分支有关：箭图的表示和交换代数。图的表示是一种将向量空间数据与某个有向图的顶点相关联的方法。边可以被看作是这些数据之间的关系。 抽象代数允许系统地研究这些对象，研究结果可能会导致更好的算法处理线性代数问题。 交换代数研究几何对象的多项式函数。该提案的第二部分是专门研究多项式方程定义的几何特征，如矩阵的秩条件的对象。 张量上的这种类型的各种条件是工程师和计算机科学家感兴趣的。该提案的第三部分研究贝蒂表：某些家庭的数值不变量相关的模块在一个多项式环。