Mathematical Sciences: Operads, Representation Theory and Algebraic Geometry

数学科学：运算、表示论和代数几何

• 批准号: 9623044
• 负责人: Mikhail Kapranov
• 金额: $ 7.35万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-06-15 至 2000-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9623044&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Operads; Representation; Theory

Kapranov proposes to extend the prior study of moduli spaces of stable curves by means of operads and their duality to more sophisticated moduli spaces of stable maps and Fulton-McPherson compactified configuration spaces. He proposes to relate two branches of representation theory which previously were not connected: theory of automorphic representations and theory of affine quantum groups, by looking at Eisenstein series and their functional equations from a new point of view. He also proposes to study Hecke operators on vector bundles on algebraic surfaces over finite fields, in order to establish an analog of the geometric Langlands correspondence, a fundamental principle in the theory of vector bundles on curves. Working with V. Ginzburg, Kapranov proposes to study natural analogs of Hecke algebras for matrix groups over 2-dimensional local fields like the field of power series in two variables over a finite field. In addition, Kapranov proposes to generalize the theory of Drinfeld modules to algebraic varieties of arbitrary dimension, again with the aim of finding the correct analog of the Langlands correspondence. He also proposes to look for a generalization of the theory of character sheaves to the case of $p$-adic fields, which would involve stacks. This research is in the fields of algebraic geometry and representation theory. Algebraic geometry is one of the oldest branches of mathematics which initially studied plane geometric images defined by simple equations but in last few decades has developed very extensively, finding applications not only inside mathematics but in such diverse fields as particle physics (geometry of microscopic degrees of freedom), car design (use of algebraic surfaces to develop new aesthetically appealing shapes), robotics, theoretical computer science and others. In particular, algebraic geometry over finite fields (to which a large part of this proposal is devoted) has found applications in constructing error-correcting codes, optimal net works and in several other important problems related to communications and information transmission.

Kapranov建议通过运算及其对偶将稳定曲线的模空间的先前研究扩展到更复杂的稳定映射的模空间和Fulton-McPherson紧化配置空间。他建议有关两个分支的代表性理论以前没有连接：理论的自守表示和理论的仿射量子群，通过寻找爱森斯坦系列及其功能方程从一个新的角度来看。 他还建议研究Hecke运营商的向量丛代数曲面在有限领域，以建立一个模拟的几何朗兰兹对应，一个基本原则理论的向量丛曲线。工作与V.金兹伯格，Kapranov建议研究自然类似的赫克代数矩阵群在2维局部领域一样，领域的幂级数在两个变量在一个有限的领域。此外，Kapranov建议将Drinfeld模理论推广到任意维数的代数簇，目的也是为了找到正确的Langlands对应。他还建议寻找一个概括的理论的性质层 到$p$-adic字段的情况，这将涉及堆栈。 这项研究是在代数几何和表示论领域。代数几何是数学中最古老的分支之一，它最初研究由简单方程定义的平面几何图像，但在最近几十年中得到了非常广泛的发展，不仅在数学中而且在粒子物理等不同领域都有应用（微观自由度的几何学），汽车设计（使用代数曲面来开发新的美学上吸引人的形状），机器人技术，理论计算机科学和其他。特别是有限域上的代数几何（本建议的很大一部分用于此）已在构造纠错码、最优网络以及与通信和信息传输有关的其他几个重要问题中得到应用。