Non-commutative Algebra and Geometry

主要研究人员：S·保罗·史密斯建议编号：0245724机构：华盛顿大学标题：非对易代数和几何摘要：主要研究人员提出的研究位于非对易代数和代数几何之间的边界上。特别是，主要研究人员继续他对非交换代数几何的研究，调查特定的非交换空间，并使用从检查这些特定例子中获得的直觉来同时发展该学科的基础。在所研究的特殊例子中，有二次超曲面，更广泛地说，在射影n-空间的非交换类似中，二次超曲面的完全相交。另一类特殊的例子是Etingof和Ginzburg的有理Cherednik代数，它们被认为是重要的和自然发生的商奇点变形的(可能的)非对易解。M.Van den Bergh最近的工作表明，在许多情况下，某些非交换环提供了对应于它们的中心的奇异簇的“解”：例如，非交换环的派生范畴等价于中心的预解的派生范畴。主要研究人员的研究将这一思想发展并扩展到其他变种，特别是弦理论家在几篇论文中出现的变种。非交换代数源于解方程的需要，其中的解是矩阵，或更一般的线性算子，而不是简单的数字。这样的方程式出现在整个科学界。代数几何的核心是代数和几何之间奇妙而卓有成效的相互作用，这在一定程度上促使人们对发展一种与非交换代数相对应的几何产生了极大的兴趣。这一主题，在代数和分析版本中，已经发展了十多年，取得了一定的活力和成功。弦理论的发展提供了越来越多的证据，表明我们生活的宇宙在一定程度上受到一种温和的非对易几何的支配。这项拟议的研究可以被认为是对这个宇宙的玩具模型的探索，也可以被认为是开发理解非对易空间的基本工具和概念。

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.