Linear algebraic groups and invariant theory

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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.