Essential dimension and related topics

点击翻译按钮获取中文摘要

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.**