Apr 7
Math/Stats Colloquium: Caroline Abbott (Brandeis)
Tue, April 7, 2026
• 4:00pm
- 5:00pm (1h) • Boliou 104
Title: When Surfaces Stretch and Twist: Understanding Mapping Classes
Abstract: Surfaces, like the sphere, the torus (a donut), or more complicated shapes with multiple “holes,” are fundamental objects in topology. One way to study a surface is to look at its symmetries: all the ways you can stretch or deform the surface continuously without tearing or gluing. The collection of these symmetries forms the mapping class group. A remarkable result, called the Nielsen–Thurston classification, says that for surfaces of finite type, every such symmetry falls into one of three types: periodic (repeating after some number of steps), reducible (preserving some simpler structure on the surface), or pseudo-Anosov (exhibiting a kind of “chaotic” stretching behavior). In this talk, we will introduce these ideas through concrete examples and build intuition for how these different behaviors arise.
We will then turn to the much less understood world of infinite-type surfaces, whose topology is dramatically more complicated. Many classical tools break down in this setting, and the Nielsen–Thurston classification no longer holds in its original form. We will give a taste of what goes wrong and highlight new phenomena that arise. The goal is to give a glimpse of how a well-understood theory evolves and becomes richer when one moves beyond the finite-type setting.
Event Contact: Sue Jandro