Non-commutative Algebra and Geometry

主要研究者：S. Paul Smith提案编号：0245724机构：华盛顿大学题目：非交换代数和几何摘要：主要研究者的研究方向是非交换代数和代数几何之间的边界。特别是，首席研究员继续他的研究非交换代数几何，调查特定的非交换空间，并使用从检查这些特定的例子获得的直觉，同时发展该主题的基础。在特定的例子中检查是二次超曲面，更一般地说，完全相交的二次曲面，在非交换类似物的射影n-空间。另一个特殊的例子是埃廷夫和金斯堡的有理切雷德尼克代数，它们被视为重要的和自然发生的商奇点变形的（可能的）非交换解。最近的工作M。货车den Bergh已经证明了许多情况，其中某些非交换环提供了对应于其中心的奇异簇的"分解"：例如，非交换环的导出范畴等价于中心的crepant分解的导出范畴。主要研究者的研究发展并扩展了这个想法到其他的种类，特别是弦理论家的几篇论文中出现的种类。非交换代数产生于解是矩阵的方程的需要，或者更一般的线性算子，而不是简单的数字。这样的方程在科学中随处可见。部分原因是代数和几何之间的互动非常富有成效，这是代数几何的核心，人们对发展一种与非交换代数相对应的几何有很大的兴趣。这个主题，在代数和分析的版本，已经发展了一些活力和成功的十多年了。弦理论的发展提供了越来越多的证据，证明我们所生活的宇宙部分地受一种温和的非对易几何的支配。拟议的研究可以被认为是对这个宇宙的玩具模型的探索，也可以被认为是为理解非对易空间开发基本工具和概念。

Principal Investigator: S. Paul Smith Proposal Number: 0245724Institution: University of WashingtonTitle: Noncommutative algebra and geometryABSTRACT:The principal investigator's proposed research lies on the boundary between non-commutative algebra and algebraic geometry. In particular, the principal investigator continues his research into non-commutative algebraic geometry, investigating particular non-commutative spaces and using the intuition obtained from examination of those particular examples to simultaneously develop the foundations of the subject. Among the particular examples examined are quadric hypersurfaces and, more generally, complete intersections of quadrics, in non-commutative analogues of projective n-space. Another family of particular examples is the rational Cherednik algebras of Etingof and Ginzburg viewed as (possible) non-commutative resolutions of important and naturally occurring deformations of quotient singularities. Recent work of M. Van den Bergh has shown many situations in which certain non-commutative rings provide "resolutions" of the singular variety corresponding to their centers: for example, the derived categories of the non-commutative rings are equivalent to the derived categories of a crepant resolution of the center. The principal investigator's research develops and extends this idea to other varieties, particularly the varieties that arise in several papers by string theorists.Non-commutative algebra arises from the need to solve equations where the solutions are matrices, or more generally linear operators, rather than simply numbers. Such equations arise throughout science. Motivated in part by the wonderfully fruitful interaction between algebra and geometry that lies at the heart of algebraic geometry, there has been great interest in developing a geometry that is an appropriate counterpart to non-commutative algebra. This theme, in both an algebraic and analytic version, has been developed with some vigor and success for more than a decade now. There is growing evidence, provided by developments in string theory, that the universe in which we live is in part governed by a mildly non-commutative geometry. The proposed research can be thought of as an exploration of toy models of this universe, and also as developing basic tools and concepts for understanding non-commutative spaces.