Topological Techniques for Computing with Perverse Sheaves

9971030Grinberg该奖项支持deJong教授的博士后助理Mikhail Grinberg的研究。反常束是代数拓扑的一部分，大约有20年的历史。它们已成功地应用于复杂代数奇点的研究。也许最值得注意的应用是与代数群相关的空间和分层，产生了表征理论的新见解。在接下来的两年里，格林伯格将从事四个研究项目，他们的共同目标是发展这一学科的计算技术。具体来说，重点是计算轴和分层的微局部不变量，以及在对称存在的情况下计算附近循环轴。第一个问题是理解复半单李代数中nilcone的微局部几何。二是研究由E. Vinberg最近引入的半单群的包络半群的几何。第三个项目是探索准圆锥对称在微局部层序研究中的可能应用。最后一个项目，与D. Kazhdan合作，是研究奇异变量中有限余维的形式弧的空间。整个工作的共同主题是学习提取代数群及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。广义地说，反常束理论是一种系统的方法，它利用了这样一个事实，即对于复上定义的奇异空间，实方向和虚方向以对偶方式贡献拓扑结构。这个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。代数拓扑的大多数应用都基于以下关键思想：将某一类对象视为单个“构型空间”（有时称为“参数空间”或“相位空间”）中的点。例如，机械系统的所有可能状态，或经济游戏中的所有可能情境（游戏邦注：如拍卖）都可以被视为适当配置空间的点。然后，这个空间的拓扑属性可以用来确定相关对象的定性属性，例如，机械系统是否会朝着周期性运动发展，或者游戏中的玩家是否具有最佳策略

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.***