Reducing structural groups by slicing objects

Humans naturally try to understand complex structures by dividing them into distinct classes, and listing the possible classes. For example, Aristotle devised a classification of living things. In mathematics, often algebraic structures have infinitely many classes, but sometimes it is possible to parametrize these classes by a finite number of variables. The minimum number of parameters needed is called the "essential dimension" of the structure. Computation of this number for various algebraic structures is considered a deep problem, and has recently attracted the attention of many mathematicians. It is hoped that these efforts will give insight into the original algebraic objects and their classification.

This project aims to improve the known bounds on the essential dimension for actions of certain matrix groups. In some cases the gap between the bounds is quite large, for example we only know that 9 <= ess.dim.(E_8) <= 231, where E_8 is the 248-dimensional exceptional group. This gap could be viewed as a measure of our collective ignorance about the classification of the given algebraic structure. There is increasing interest in the exceptional groups from the theoretical physics community, and there is evidence that E_8 symmetry occurs in nature, according to recent experimental physics [Garibaldi and Borthwick]. By developing techniques to decrease the size of gap, in this project I intend to provide new insights into our understanding of the underlying structures.

To achieve this aim I intend to apply a recently developed reduction method to the group. Another consequence of such a reduction is the potential simplification of a very old question in invariant theory known as the rationality problem. That problem seeks to understand the orbits of an action, by not only counting the number of variables needed to parametrize them, but also to measure how independent the variables are of each other.

There has been a growing trend in many areas of mathematics to extend results that were originally discovered over the real or complex numbers to more general fields. For me a driving motivation is the belief that many phenomena of algebraic groups which depend on the base field are really just different perspectives on the same underlying object. Evidence of this is the classification of (split) simple algebraic groups being independent of the ground field. This remarkable fact binds together a wide range of mathematical disciplines, from Lie theory, symmetries of differential equations (which was Lie's original motivation), the study of singularities in algebraic geometry, to finite group theory, and number theory. Contrast this with the classification of simple Lie algebras, which is much more complicated in prime characteristic than in characteristic zero. The new reduction techniques I intend to use are attractive because they were developed in a characteristic-free way, and are therefore no longer restricted to the real or complex numbers.

Most of the impact of the proposed research will be within the pure mathematics subject areas of algebra and algebraic geometry. In the Pathways to Impact document I discuss three further areas of impact:

1) Training and development of an aspiring mathematician (the RA).

2) Increasing by one (the RA) the number of mathematicians who are fluent in editing Wikipedia and have made significant contributions to the articles.

3) Potential impact on theoretical physics, mainly through an improved understanding of certain representations which are important to quantum information theorists.