Research in noncommutative and commutative algebra

提案由几个相互关联的部分组成。第一部分是研究颤振的半不变量环和带关系的颤振的半不变量环。研究者建议继续研究颤振半不变量环权的锥壁和这些环权空间的多重性。在特殊情况下，这包括由Klyachko不等式定义的锥。对于带关系颤振的半不变量，主要问题是用半不变量来刻画有限型和驯服带关系颤振。另一个方面是颤振表示与簇代数的联系。首席研究员计划研究具有势能的颤振和相关雅可比代数的突变。他还计划研究某些球面三角形的联系以及与幂零群相关的图的Igusa-Orr理论。第二部分研究了等变变量的理想定义。提出了几种类型的轨道闭包：具有有限多个轨道表示的轨道闭包，最高权向量的切向和割向轨道闭包，以及对应于Grassmannian中的Schubert变体的Dynkin颤抖的轨道闭包。第三部分涉及梯度模Betti表的Boij-Soderberg猜想问题。主要研究齐次空间上向量束的上同表和Boij-Soderberg猜想的等变细化。这个建议涉及到数学的几个分支：颤栗的表示和交换代数。抖动的表示是一种将向量空间数据与一些有向图的顶点相关联的方法。这些边可以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。该提案的第二部分致力于研究定义几何特征对象的多项式方程，如矩阵上的秩条件。这类张量上的各种条件引起了工程师和计算机科学家的兴趣。第三部分研究了多项式环上与模相关的一类数值不变量贝蒂表。

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.