Essential dimension and related topics

19世纪伟大的德国数学家菲利克斯·克莱因（Felix Klein）率先提出了用对称性来理解几何形状的想法。一个给定图形的对称性可以通过一个接一个地应用而成倍增加。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及其对代数变异和代数和几何中其他感兴趣的对象的作用有关。20年前，Joe Buhler和我给一个几何图形X分配了一个数值不变量，它具有规定的对称性。这个不变量是一个介于0和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”到Y而不会失去任何对称性。我们称这个数为X的基本维数，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流行，因此基本维数和规范维数的相关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有两个关于这个主题的讲座（一个关于基本维数，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。我建议继续在这个领域工作，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.