Essential dimension and related topics

伟大的世纪德国数学家菲利克斯·克莱因开创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，它们形成一个代数结构，称为“群”。群论已被证明是一个重要的工具，在几何以及其他领域的纯数学和应用数学。我的大部分工作都与代数群及其对代数簇的作用以及代数和几何中其他感兴趣的对象有关。20年前，我和乔·布勒（Joe Buhler）给一个具有规定对称性的几何图形X指定了一个数值不变量。这个不变量是一个介于0和X的维数之间的整数。它是一个图形Y的最小维数，这样人们就可以将X“压缩”到Y，而不会失去任何对称性。我们称这个数为X的本质维数，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都很普遍，本质维数和相关的规范维数的概念在许多其他的背景下也很有用。这些概念已经被许多数学家探索，使用各种技术。2010年，国际数学家大会的代数部分有两个关于这个主题的讲座（一个关于本质维数，另一个关于规范维数），2012年和2013年，AMS和CMS都为这一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.