Non-Commutative Desingularizations and Representation Theory

在几何和代数之间有一本经典的字典，至少可以追溯到17世纪的笛卡尔和费马，这本字典已经得到了发展和完善，允许根据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无关紧要：x乘以y等于y乘以x。）然而，随着我们对物理世界的理解在量子力学、弦理论和基本粒子研究等领域的发展，我们逐渐认识到，宇宙的精细结构本质上是非对易的。 新的字典正在建立，以了解几何和非交换世界之间的联系。本研究课题是关于奇点解析的几何运算的非交换类似物，奇点解析是指将收缩或褶皱的几何体“展开”，用光滑的几何体代替。研究者将研究交换代数、代数几何和非交换代数几何的交叉问题。该研究计划涉及应用工具和技术，从表示理论的地方环，阿廷代数和代数群，以问题的建设非交换类似物的决议奇异性，研究其结构，并应用其存在，以研究进一步的问题表示理论。 非交换归结理论的基础仍在积极的发展中，这一建议将有助于理论的成熟。 此外，研究者提出了新的NCR的例子，这将扩大我们的理解的理论的局限性的建设。其中一些新的例子将依赖于新的建设几何倾斜理论齐次品种。

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.