Linear algebraic groups

每个人都知道，有些形状比其他形状更对称。例如，圆形比正方形更对称，而正方形是所有矩形中最对称的。19世纪伟大的德国数学家费利克斯·克莱因倡导一种几何方法，即通过几何形状的对称性来研究几何形状。一个给定图形的对称性可以通过一个接一个地应用来相乘；在现代语言中，它们形成了一个代数结构，称为“群”。 克莱因倡导的方法现在经常用于纯数学和应用数学的许多领域。对称群在理解和描述几何形状的性质以及代数和微分方程、纠错码等方面很重要。15年前，Joe Buhler和我为具有规定对称性的几何图形X指定了一个数值不变量。这个不变量是介于0和X的维度之间的一个整数。它是一个图形Y的最小维度，这样人们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”。因为对称群在数学的许多领域都很普遍，所以这个概念在许多不同的情况下都被发现是有用的。特别是，它被证明与古典代数中的许多问题有关。在过去的四年里，由于从代数几何中引入了强大的新方法，这一主题的研究步伐加快了。去年，国际数学家大会的代数部分有一场关于本质维度的讲座，另一场关于典型维度的讲座，这也是我在作品中引入的一个相关概念。我的建议的重点是促进这些领域和相关领域的知识现状。

Everyone knows that some shapes are more symmetric than others. A circle, for example, is more symmetric than a square, and a square is the most symmetric of all rectangles. The great 19th century German mathematician Felix Klein advocated an approach to geometry where geometric shapes are studied via their symmetries. 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". The approach advocated by Klein is now routinely used in many areas of pure and applied mathematics. Symmetry groups are important in understanding and describing properties of geometric shapes as well as algebraic and differential equations, error-correcting codes, etc. Fifteen 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. Because symmetry groups are prevalent in many areas of mathematics, this notion has been found to be useful in many different contexts. In particular, it turned out to be related to many questions in classical algebra. The pace of research on this topic quickened in the past four years because of the introduction of powerful new methods from algebraic geometry. Last year, the algebra section of the International Congress of Mathematicians featured a lecture on essential dimension and another one on canonical dimension, a related concept which was also introduced in my work. My proposal is focused on advancing the current state of knowledge in these and related areas.