Linear algebraic groups and invariant theory

解多项式方程是数学中最古老的问题之一。解决这一问题的自然方法是寻找一系列替换(也称为齐恩豪斯变换)，将该方程简化为一种易于求解的形式。古巴比伦人早在公元前1600年就知道如何计算二次方程。在文艺复兴时期，类似的方法导致了三次和四次方程的求解。19世纪初，Abel和Galois证明了一般的高于4次的多项式方程不能用根来求解。然而，人们仍然可以问，通过齐恩豪斯变换可以在多大程度上简化这个方程。十年前，对这个问题的思考使我和我的合作者产生了本质维度的概念，事实证明，这个概念在多项式理论之内都是卓有成效的，而且远远超出了多项式理论。这项提议的目标之一是在两个令人兴奋的新方向上继续这项研究，一个是在传统的代数群环境中，另一个是在新的代数堆栈环境中。

Solving polynomial equations is one of the oldest problems in mathematics. The natural approach to this problem is to look for a sequence of substitutions (otherwise known as Tschirnhaus transformations) that simplifies this equation to a form that can be easily solved. The antient Babylonians knew how to do this for quadratic equations as early as 1600 BC. A similar approach led to the solution of cubic and quartic equations during the Renaissance. In the early 19th century Abel and Galois showed that a general polynomial equation of degree higher than four cannot be solved in radicals. One can nevertheless ask how far one can simplify this equation by Tschirnhaus transformations. Ten years ago, thinking about this question has led me and my collaborators to the notion of essential dimension, a concept that proved to be fruitful both within and far beyond the theory of polynomials. One of the goals of this proposal is to continue this research in two exciting new directions, one in the traditional setting of algebraic groups, theother in the new setting of algebraic stacks.