Plenary Speakers

An algebraic topologist’s view on the disciplinary boundaries of pure or applied math

Abstract: Algebraic topology is one of the youngest fields of pure mathematics, having only come into existence about 120 years ago.  Topology, or the study of mathematical shapes, is generally considered to be a sub-discipline of “pure” mathematics - meaning purely abstract, purely esoteric, purely foundational and not necessarily applied.  However, in the last few decades a number of areas of application have arisen in a variety of subject areas, including big data, materials science, social networks and others.  In this talk, I will give the briefest of introductions to the idea of “applied topology”.  Finally, I will explain how my experience with this change in my own discipline motivated me to co-found (with Jim Colliander) the Math to Power Industry program, and why I believe this program is important for researchers, students and industry professionals alike.

From A-D-E to Fermat to ...?

Abstract: Vertex operator algebras (VOAs) are a mathematical approach to quantum field theories which won their creator a Fields medal. But they are highly complicated structures -- if you haven't already learned what they are, it is surely best if you never do. Fortunately, you don't need to know anything about them to follow my talk, though they'll be the shadows lurking in the background. In a sense I'll sketch, VOAs behave as an upside-down version of groups. This helps, because groups are much easier to understand. Now, one of the greatest results in Algebra in the 20th century was identifying the finite groups which, like lego pieces, snap together to form all other groups. With a handful of exceptions, these lego pieces are all of a common form (called `Lie-type'). The upside-down metaphor then predicts that the VOAs of Lie-type (with a handful of exceptions) are maximal. This is now a theorem. But something unexpected and quite interesting happens when you look at those exceptions, and that is the punchline of my talk. No knowledge of VOAs and groups will be assumed. These exceptions lead us to A-D-E, which is a simple pattern permeating math, then to curves associated to Fermat's Last Theorem, and after that ...???

Points and distances - what do we really know about them?

Abstract: The ancient Pythagorean theorem gives a formula for computing the Euclidean distance between two points. It is simply astounding that a concept so simple and classical has continued to fascinate mathematicians over the ages, and remains a tantalizing source of open problems to this day.

Given a set E, its distance set consists of numbers representing distances between points of E. If E is large, how large is its distance set? How does the structure of a set influence the structure of distances in the set?

Such questions play an important role in many areas of mathematics and beyond. The talk will survey a few research problems associated with Euclidean distances between points and discuss recent breakthroughs in some of them.

The presentation is intended to be an introduction to a vibrant research area; no advanced mathematical background will be assumed.