Research in noncommutative and commutative algebra

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有关系的箭图环。作者建议继续研究环的权的锥壁、箭图的半不变量以及这些环的权空间的多重性。在特殊情况下，这包括由Klyachko不等式定义的圆锥。对于带有关系的箭图的半不变量，主要问题是用半不变量刻画有限类型和驯服的带有关系的箭图。另一个方面是箭图表示和簇代数之间的联系。首席研究员计划研究具有位势的箭图和相关雅可比代数的突变。他还计划研究某些球面三角剖分和与幂零群有关的图画的伊古萨-奥尔理论之间的联系。第二部分研究等变变种的定义理想。提出了几种类型的簇：具有有限多个轨道的表示的轨道闭包，最高权向量的切向和切向轨道变量，以及对应于Grassmanian中Schubert簇的动态箭图的轨道闭包。第三部分涉及分次模的Betti表的BOIJ-Soderberg猜想的有关问题。主要研究者建议研究齐次空间上向量丛的上同调表和BOIJ-Soderberg猜想的等变加细。这一建议与数学的几个分支有关：箭图的表示法和交换代数。箭图的表示是将向量空间数据与某个有向图的顶点相关联的一种方式。这些边可以被视为这些数据之间的关系。抽象代数允许系统地研究这样的对象，研究的结果可能会导致处理线性代数问题的更好的算法。交换代数研究几何对象的多项式函数。该提案的第二部分致力于研究定义几何特征对象的多项式方程，例如矩阵上的秩条件。工程师和计算机科学家对这类张量的各种条件很感兴趣。提案的第三部分研究Betti表：与多项式环上的模相关的某些数值不变量族。

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.