Albion College Mathematics and Computer Science Colloquium



Title: A Diagrammatic Approach to Invariant Theory and Steenrod Actions
Speaker:Dana Hunter
Mathematics
Kalamazoo College
Kalamazoo, Michigan
Abstract: Algebraic topologists study geometric objects by associating to them algebraic structures which remain unchanged under deformation. One very powerful such algebraic structure is the cohomology ring, along with its Steenrod squaring operations. After a very brief introduction to these topological notions through pictures, the bulk of our talk will focus on examples of algebraic structures that show up as cohomology rings (or images of those rings). Our first example will be symmetric polynomials. Given a polynomial ring in n variables over a field, we can define an action of the symmetric group by permuting variables. The polynomials which remain unchanged by such permutations are called symmetric polynomials and turn out to be a very similar looking polynomial ring! We will give a proof using diagrams, which are highly useful in further cases. We will also define and calculate the Steenrod squares in this setting. We will next look at what happens when we replace permuting variables by sending each variable to a linear combination of other variables, and again look at what elements are left unchanged by all such coordinate changes. This will give us Dickson algebras, which, with their Steenrod squaring structure are an important building block in a new approach to the classic problem of studying families of maps between spheres.
Location: Palenske 227
Date:3/16/2023
Time: 3:30 PM



@abstract{MCS:Colloquium:DanaHunter:2023:3:16,
author  = "{Dana Hunter}",
title   = "{A Diagrammatic Approach to Invariant Theory and Steenrod Actions}",
address = "{Albion College Mathematics and Computer Science Colloquium}",
month   = "{16 March}",
year    = "{2023}"
}