Non-Commutative Desingularizations and Representation Theory

在几何和代数之间有一本经典的词典，至少可以追溯到17世纪的笛卡尔和费马，这本词典得到了发展和改进，可以用被称为交换环的代数对象来详细研究几何空间。(交换性意味着乘法的顺序并不重要：x乘以y等于y乘以x。)然而，随着我们对物理世界的理解的增长，在量子力学、弦理论和基本粒子研究等领域，我们开始理解宇宙的精细结构本质上是非对易的。为了理解几何和非对易世界之间的联系，人们正在建立新的词典。这项研究项目涉及一种称为奇点分解的几何运算的非对易类似，该运算将一个收缩或折痕的几何图形“展开”为光滑的几何图形。研究人员将研究交换代数、代数几何和非交换代数几何的交集问题。研究计划包括应用局部环、Artin代数和代数群的表示理论中的工具和技巧，来构造奇点分解的非交换类似，研究它们的结构，并应用它们的存在来进一步研究表示论中的问题。非交换决议理论的基础仍在积极发展中，这一提议将有助于该理论的成熟。此外，研究者还提出了NCR的新实例的构建，这将扩大我们对该理论的局限性的理解。其中一些新的例子将依赖于齐次簇上的几何倾斜理论的新构造。

There is a classical dictionary between geometry and algebra, dating back at least to Descartes and Fermat in the seventeenth century, which has been developed and sharpened to allow the detailed study of geometric spaces in terms of algebraic objects known as commutative rings. (Commutativity means that order of multiplication doesn't matter: x times y is equal to y times x.) As our understanding of the physical world has grown, however, in such areas as quantum mechanics, string theory, and the study of fundamental particles, we have come to understand that the fine structure of the universe is essentially non-commutative. New dictionaries are being built to understand the connections between geometry and the non-commutative world. This research project concerns non-commutative analogues of a geometric operation called resolution of singularities, which "unfolds" a pinched or creased geometry to replace it with a smooth one.The investigator will work on problems at the intersection of commutative algebra, algebraic geometry, and non-commutative algebraic geometry. The research plan involves applying tools and techniques from the representation theory of local rings, Artin algebras, and algebraic groups, to the problems of constructing non-commutative analogues of resolutions of singularities, studying their structure, and applying their existence to study further problems in representation theory. The foundations of the theory of non-commutative resolutions (NCRs) are still in active development, and this proposal would contribute to the maturation of the theory. Furthermore the investigator proposes constructions of new examples of NCRs, which would expand our understanding of the limits of the theory. Some of these new examples would rely on new constructions in geometric tilting theory over homogeneous varieties.