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），将这个方程简化为易于求解的形式。早在公元前1600年，古巴比伦人就知道如何求解二次方程。在文艺复兴时期，类似的方法导致了三次方程和四次方程的解。在19世纪早期，阿贝尔和伽罗瓦证明了一个高于四次的一般多项式方程不能用根式解。然而，人们可以问，通过齐恩豪斯变换，这个方程可以简化到什么程度。十年前，对这个问题的思考使我和我的合作者提出了基本维度的概念，这个概念在多项式理论内外都被证明是富有成效的。本提案的目标之一是在两个令人兴奋的新方向上继续这项研究，一个是在代数群的传统设置中，另一个是在代数堆的新设置中。这个提议的另一部分灵感来自1985年诺贝尔化学奖得主赫伯特·豪普特曼（Herbert Hauptman）的一个问题。为了确定一个物理晶体的结构，我们需要知道一定的量，称为“相”。在实践中，这些无法直接测量；然而，人们可以测量另一组称为“可观测”的量。那么问题就变成了：如果“可观测”是已知的，人们能恢复“相位”吗？如果可以，进行计算的最有效方法是什么？乔·布勒和我证明了“相”总是可以从“可观测”中恢复出来。寻找有效的算法来进行计算的问题自然会导致计算代数中有趣的理论问题，与SAGBI基有关。这部分建议的目标是研究SAGBI基的存在性问题，以及在SAGBI基不存在的情况下的替代算法。

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.Another part of this proposal was inspired by a question of Herbert Hauptman, a 1985 Nobel laureate in chemistry. To determine the structure of a physical crystal, one needs to know certain quantities, called ``phases". In practice, these cannot be measured directly; however, one can measure another set of quantities called ``observables".  The question then becomes: if the ``observables" are known, can one recover the ``phases", and if so, what is the most efficient way to carry out the computations? Joe Buhler and I showed that the ``phases" can always be recovered from the ``observables". The problem of finding efficient algorithms to carry out the computations naturally leads to interesting theoretical questions in computational algebra, having to do with SAGBI bases. The goal of this part of the proposal is to investigate the existence problem for SAGBI bases as well as alternative algorithms in those cases where a SAGBI bases does not exist.