# Graduate Student / Postdoc Seminar

#### Invariant Polynomials and Quotients of Spheres

**Speaker:**
Ben Blum-Smith

**Location:**
Warren Weaver Hall 1302

**Date:**
Friday, March 10, 2017, 1 p.m.

**Synopsis:**

Classical invariant theory is about trying to describe the collection ("ring") of multivariate polynomials over Q or C that are unaffected by ("invariant under") some specific group of coordinate transformations. The theory is mature, and there are many powerful general theorems that describe this ring's good properties.

Surprisingly, a number of these theorems fail if the polynomials are restricted to integer coefficients.

In this talk, we investigate one such failure. In the classical case, all the invariant polynomials can be written uniquely in terms of a preselected few. Over the integers, this works for some transformation groups but not others. Which groups?

In another twist, this question turns out to be closely related to a question in pure topology: if a group of transformations acts on a sphere, is the quotient a nice topological space like a sphere or a ball? or is it something more exotic?