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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=663142
• 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费利克斯·克莱因（Felix Klein）率先使用对称性来理解几何形状的想法。给定图的对称性可以通过一个接一个地应用一个。在现代语言中，它们形成了一个代数结构，称为“群体”。事实证明，小组理论是几何以及其他纯数学和应用数学领域的重要工具。我的大部分工作都与代数群体及其对代数品种和其他感兴趣对象的代数群体有关。二十年前，我和乔·布勒（Joe Buhler）和我为带有规定对称性的几何图x分配了一个数字不变性。这个不变的是0和X的尺寸之间的整数。它是图y的最小维度，因此可以将x x到y的x上，而不会丢失任何对称性。我们称该数字为X的基本维度，并注意到它与古典代数中的许多问题有关。由于对称组在数学的许多领域，基本维度以及规范维度的相关概念都普遍存在，因此在许多其他情况下也很有用。此后，许多数学家都使用各种技术探讨了这些概念。 2010年，国际数学家大会的代数部分在该主题上进行了两次讲座（一个关于基本维度，另一个关于规范维度的讲座），在2012年和2013年，AMS和CMS均授予该领域的工作研究奖。我建议继续在这一领域工作，探索新的方向，包括模块化代表理论中的新兴应用程序。**