Essential dimension and related topics

19世纪伟大的德国数学家费利克斯·克莱因率先提出了利用对称性来理解几何形状的想法。通过一个接一个地应用，给定图形的对称性可以相乘。在现代语言中，它们形成了一个代数结构，称为“群”。群论已被证明是几何学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及其在代数簇和其他代数和几何中感兴趣的对象上的作用有关。20年前，乔·布勒和我为具有规定对称性的几何图形X赋予了一个数值不变量。这个不变量是介于0和X的维度之间的一个整数。它是图形Y的最小维度，这样就可以在不丢失任何对称性的情况下将X“压缩”到Y。我们称这个数为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很普遍，本质维度和相关的规范维度的概念在许多其他情况下也被证明是有用的。自那以后，许多数学家利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有两场关于这一主题的讲座(一场关于本质维度，另一场关于典型维度)，2012年和2013年，AMS和CMS都因在这一领域的工作而颁发了研究奖。我建议继续在这一领域工作，探索新的方向，包括在模表示理论中的新兴应用。

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.