Topological Techniques for Computing with Perverse Sheaves

9971030格林伯格该奖项支持德容教授博士后助理米哈伊尔·格林伯格的研究。倒槽是代数拓扑学的一部分，大约有二十年的历史了。它们已成功地应用于复代数奇点的研究。也许最值得注意的应用是与代数群有关的空间和分层，在表示理论中产生了新的见解。在接下来的两年里，M.Grinberg将致力于四个研究项目，他们的共同目标是开发该学科的计算技术。具体地说，重点是计算层和层的微局部不变量，以及在存在对称性的情况下使用邻近的循环层进行计算。第一个问题是理解复半单李代数中尼尔锥的微局部几何。第二个是研究半单群的包络半群的几何，最近由E.Vinberg引入。第三个项目是探索准圆锥对称性在层结微局部研究中的可能应用。与D·Kazhdan合作的最后一个项目是研究奇异簇中有限余维形式弧的空间。这项工作的共同主题是学习提取在代数群、其表示和不变量的研究中出现的奇异空间和映射的关键拓扑不变量。不严格地说，倒槽理论是一种系统的方法，它利用了这样一个事实，即对于定义在复数上的奇异空间，实方向和虚方向以双重方式促成了拓扑。这一理论是代数拓扑学的一部分，代数拓扑学是数学的一个分支，研究在连续变形下保持的高维空间的性质。代数拓扑学的大多数应用都是基于以下关键思想的：将某种类型的同构对象看作是单个“构型空间”(有时称为“参数空间”或“相空间”)中的点。例如，机械系统的所有可能状态或经济博弈(例如拍卖)中的所有可能情况可以被视为适当配置空间的点。然后，该空间的拓扑属性可用于确定相关对象的定性属性，例如，机械系统是否将向周期性运动演化，或者博弈中的参与者是否具有最优策略。

9971030Grinberg This award supports the research of Mikhail Grinberg, a postdoctoralassociate of Professor deJong. Perverse sheaves are a part of algebraictopology that is about twenty years old. They have been applied with greatsuccess to the study of complex algebraic singularities. Perhaps the mostnotable applications have been to spaces and stratifications related toalgebraic groups, yielding new insights in representation theory. Over thenext two years, M. Grinberg will work on four research projects whosecommon goal is to develop the computational techniques of the subject.Specifically, the focus is on computing microlocal invariants of sheavesand stratifications and on computing with nearby cycles sheaves, in thepresence of symmetry. The first problem is to understand the microlocalgeometry of the nilcone in a complex semisimple Lie algebra. The secondis to study the geometry of the enveloping semigroup of a semisimple group,recently introduced by E. Vinberg. The third project is to explore possibleapplications of quasi-conical symmetry to the microlocal study ofstratifications. The last project, joint with D. Kazhdan, is to study thespace of formal arcs of finite codimension in a singular variety. Thecommon theme throughout this work is to learn to extract the key topologicalinvariants of the singular spaces and maps that arise in the study ofalgebraic groups, their representations, and invariants. Loosely speaking, the theory of perverse sheaves is a systematic wayto exploit the fact that, for a singular space defined over the complexnumbers, the real and the imaginary directions contribute to the topologyin dual ways. This theory is a part of algebraic topology: a branch ofmathematics concerned with the properties of higher dimensional spacesthat are preserved under continuous deformations. Most applications ofalgebraic topology are based on the following key idea: to consider allobjects of a certain kind as points in a single ``configuration space''(sometimes called a ``parameter space'' or a ``phase space''). For example,all possible states of a mechanical system, or all possible situations in aneconomic game (e.g., an auction) may be viewed as points of an appropriateconfiguration space. The topological properties of this space may then beused to determine qualitative properties of the associated objects, such aswhether the mechanical system will evolve towards a periodic motion, orwhether the players in a game have optimal strategies.***