Mathematicians Solved a Notorious Old Problem, Shaking Up Abstract Algebra
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Representation theory transforms abstract algebra groups into things like simpler matrices.
The field’s founder left a list of 43 problems for others to study, iterate on, and prove.
One mathematician has helped to prove both one of these and another problem just this year.
Mathematician Pham Huu Tiep and his colleagues have made a proof of one longstanding open problem and one additional foundational question in a subfield of abstract algebra known as representation theory. Tiep, a professor at Rutgers University, paired that breakthrough with a second in the same subfield, bringing his recent paper total to three on these two breakthroughs alone.
Before we get to the good stuff, here’s some quick and dirty context. Algebra is a huge umbrella term within mathematics. It’s the first thing we learn after the basics of arithmetic, introducing variables as something we can work with and solve for. Abstract algebra, as an extension, twists those basic variables into shapes and sets that all have their own set of operations.
Taking that idea one specific step, representation theory is a way to shift our understanding of the most abstract of those algebraic objects into things that are easier to manipulate, and draw conclusions from them without quite as much work or need for posing like The Thinker. In a sense—though, not a literal one—it’s analogous to how we can reduce the equation of a curve to an equation for its straight-line derivative using calculus. Mathematicians can redraw a ring or group as a less complex linear algebra structure, and then they can play with it using the full, large toolkit of linear algebra.
Tiep et al’s breakthroughs in the field appear in Princeton University’s journal Annals of Mathematics (twice!) and Springer’s journal Inventiones Mathematicae. His coauthors include Robert M. Guralnick, Michael Larsen, Gunter Malle, Gabriel Navarro, and A. A. Schaeffer Fry.
The first of these papers describes a proof of mathematician Richard Brauer’s longstanding Height Zero Conjecture. In this context, ‘height’ is a quantity of something’s magnitude or complexity—for some polynomials, this might be the highest degree, or exponent. And in representation theory, ‘height’ applies to something called abelian groups. In an abelian group, all the elements in the group can be rearranged in any order. Each element has a ‘height’ that describes whether or not it can be divided cleanly by another element. If so, the result of that division (by the lowest qualifying value in the group) is the height.
Brauer laid out a list of problems that help to define representation theory. Number 23 is the Height Zero Conjecture, in which Brauer claims that abelian groups must have a particular quality related to the heights of their elements. Tiep helped to prove part of the conjecture in 2013, and has worked “intensively” on the rest since then, he said in a Rutgers statement. Brauer, Tiep said, was a “rare intellect” as though “from another world,” and Tiep was surprised when he did eventually manage to solve the immensely complicated problem.
The second breakthrough may not be as glamorous as conquering a Brauer problem, but it puts in place another building block of representation theory. Tiep, Guralnick, and Larsen studied a particular aspect of finite classical groups, producing two papers on the subject.
In the first, Larsen and Tiep helped to prove existing beliefs about how an abstract algebra group can be instead represented as a matrix, and that the representational matrix itself can be described using its diagonal elements, or its trace. Each element in the group can have the trace attached, creating the “character” of the group representation. Larsen and Tiep refined a mode of thinking about these characters by adjusting how they can be bound using not just the character, but multiple other features of the group. This should be able to help us understand more about the trace from the matrix.
In their second character paper, they deepen this understanding of the character and trace bounds by analyzing a ratio of components of the character. All of this work layers incredibly carefully on the steps laid out by previous work—both by these mathematicians and by colleagues for generations prior. Even Brauer, the unusual intellect who influenced everyone on these three papers, didn’t prove his outstanding problems. He knew they were right, but it’s taken decades to prove just some of his long list.
In the statement, Tiep said of luminaries like Brauer, they are “capable of seeing hidden phenomena that others can’t.” It’s true that Brauer established this field and its fundamentals, but Tiep and his colleagues continue to chase those leads and solve mysteries that even Brauer could not during his lifetime. To the rest of us, much of this work may be inadvertently hidden, but everyone’s contributions matter.
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