Linear algebraic groups and invariant theory

求解多项式方程是数学中最古老的问题之一。解决这个问题的自然方法是寻找一系列替换（也称为Tschirnhaus变换），将这个方程简化为可以容易求解的形式。古巴比伦人早在公元前1600年就知道如何对二次方程进行这种计算。文艺复兴时期，类似的方法导致了三次和四次方程的解。在19世纪初的世纪阿贝尔和伽罗瓦表明，一般多项式方程的程度高于4不能解决的自由基。然而，人们可以问，用契恩豪斯变换可以把这个方程简化到什么程度。十年前，对这个问题的思考使我和我的合作者们产生了本质维数的概念，这个概念在多项式理论内部和远远超出多项式理论的范围都被证明是富有成果的。这个建议的目标之一是继续这项研究在两个令人兴奋的新方向，一个在传统的设置代数群，theother在新的设置代数栈。

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.