Mathematical Sciences: Algebraic Geometry and Category Theory

数学科学：代数几何和范畴论

• 批准号: 9303216
• 负责人: Mikhail Kapranov
• 金额: $ 7.29万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-06-01 至 1996-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9303216&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Algebraic; Geometry; Category

This award is concerned with research in algebraic geometry and category theory. The principal investigator will work on using Chow varieties to obtain desingularizations of quotient varieties by algebraic groups and to construct higher-dimensional generalizations of Grothendieck-Knuidsen moduli spaces of stable punctured curves. He will also develop the formalism of braided monoidal 2-categories and apply it to the study of Zamolodchikov's tetrahedra equations. In addition, he will study Koszul duality for quadratic operads as motivated by graph cohomology; polyhedral syzygies among Steinberg relations in order to get a more formal algebraic approach to higher algebraic K-theory; and the setting for generalizing the Langlands correspondence to 2-dimensional schemes by using a certain generalization of a concept of operad. This is reasearch in the field of algebraic geometry, one of the oldestparts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origins, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in theoretical computer science and robotics.

该奖项涉及代数几何和范畴论的研究。 首席研究员将致力于使用 Chow 簇来通过代数群获得商簇的去奇异化，并构建稳定穿刺曲线的 Grothendieck-Knuidsen 模空间的高维推广。 他还将发展辫状幺半2-范畴的形式主义，并将其应用于扎莫洛奇科夫四面体方程的研究。 此外，他还将受图上同调的启发，研究二次运算的 Koszul 对偶性； Steinberg 关系中的多面体联结，以获得更正式的代数方法来实现更高代数 K 理论；以及通过使用操作数概念的某种概括将朗兰兹对应概括为二维格式的设置。 这是代数几何领域的研究，代数几何是现代数学最古老的部分之一，但在过去四分之一个世纪中取得了革命性的繁荣。 在其起源中，它处理可以通过最简单的方程（即多项式）在平面上定义的图形。 如今，该领域不仅使用代数方法，还使用分析和拓扑方法，相反，它正在这些领域以及理论计算机科学和机器人技术中得到应用。