Topological Techniques for Computing with Perverse Sheaves

小行星9971030 该奖项支持deJong教授的博士后研究员Mikhail Grinberg的研究。 反常层是代数拓扑学的一部分，已有二十年的历史。 它们已成功地应用于复代数奇点的研究。 也许最显着的应用已经空间和分层相关的代数群，产生新的见解表示论。 在接下来的两年里，M。格林伯格将致力于四个研究项目，其共同目标是发展该学科的计算技术。具体而言，重点是计算shepherd和分层的微局部不变量，以及在对称性存在的情况下计算附近的循环层。 第一个问题是理解复半单李代数中的nilcone的微局部几何。 第二部分研究了E.文伯格 第三个项目是探索准锥对称性在层结微观局部研究中的可能应用。 最后一个项目是与D. Kazhdan，是研究奇异簇中有限余维形式弧的空间. 贯穿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，反常层理论是一种系统的方法，它利用了这样一个事实，即对于复数上定义的奇异空间，真实的和虚的方向以对偶的方式对拓扑作出贡献。 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，研究高维空间在连续变形下保持不变的性质。 代数拓扑学的大多数应用都基于以下关键思想：将某种类型的所有拓扑看作单个“配置空间”（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，或经济博弈中的所有可能情况（例如，拍卖）可以被视为适当配置空间的点。 这个空间的拓扑性质可以用来确定相关对象的定性性质，例如机械系统是否会朝着周期性运动演化，或者游戏中的参与者是否有最优策略。

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.***