Essential dimension and related topics

基本维度和相关主题

• 批准号: RGPIN-2017-03829
• 负责人: Reichstein, Zinovy
• 金额: $ 3.13万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=688627
• 关键词: Essential; dimension; related; topics

The great 19th century German mathematician Felix Klein pioneered the idea of using symmetries to understand geometric shapes. Symmetries of a given figure can be multiplied, by applying one after the other. In modern language, they form an algebraic structure, called a "group". Group theory has proved to be an important tool in geometry as well as other areas of pure and applied mathematics. Much of my work has been related to algebraic groups and their actions on algebraic varieties and other objects of interest in algebra and geometry. Twenty years ago Joe Buhler and I assigned a numerical invariant to a geometric figure X with prescribed symmetries. This invariant is an integer between 0 and the dimension of X. It is the minimal dimension of a figure Y, such that one can "compress" X to Y without losing any of the symmetries. We called this number the essential dimension of X and noticed that it is related to many questions in classical algebra. Because symmetry groups are prevalent in many areas of mathematics, essential dimension, and the related notion of canonical dimension, turned out to be useful in many other contexts as well. These notions have since been explored by many mathematicians, using a variety of techniques. In 2010 the algebra section of the International Congress of Mathematicians featured two lectures on this subject (one on essential dimension and another one on canonical dimension), and in 2012 and 2013 both the AMS and the CMS awarded research prizes for work in this area. I propose to continue working in this area, exploring new directions, including the emerging applications in modular representation theory.**

19 世纪伟大的德国数学家菲利克斯·克莱因 (Felix Klein) 率先提出了使用对称性来理解几何形状的想法。通过逐个应用，可以将给定图形的对称性相乘。用现代语言来说，它们形成一种代数结构，称为“群”。群论已被证明是几何以及纯数学和应用数学其他领域的重要工具。我的大部分工作都与代数群及其对代数簇以及代数和几何中其他感兴趣的对象的作用有关。二十年前，Joe Buhler 和我为具有规定对称性的几何图形 X 分配了一个数值不变量。该不变量是 0 到 X 维度之间的整数。它是图形 Y 的最小维度，因此可以将 X“压缩”为 Y，而不会丢失任何对称性。我们将这个数字称为X的本质维度，并注意到它与经典代数中的许多问题有关。由于对称群在数学的许多领域中都很普遍，因此基本维数以及相关的规范维数概念在许多其他情况下也很有用。此后，许多数学家使用各种技术探索了这些概念。 2010 年，国际数学家大会的代数部分举办了两场关于该主题的讲座（一场关于本质维度，另一场关于规范维度），并且在 2012 年和 2013 年，AMS 和 CMS 都为该领域的工作颁发了研究奖。我建议继续在这一领域工作，探索新的方向，包括模块化表示理论中的新兴应用。**