Linear Orders

Math 191a: Linear Orders

Fall 2022

Class Info

Class times: Tu/Th 10:30am - 11:55am
Class location: 289 Linde
Office: 204 Kellogg
Office Hours: Tu/Th 2:30 - 3:30pm

Course Description

Prerequisites: Ma 5 or equivalent, or instructor's permission.

Welcome to 191a! This course is an introduction to the structure theory and combinatorics of infinite linear orders. Topics include Cantor's characterization of the order type of the rational numbers, Hausdorff's inductive characterization of the scattered orders, Dushnik-Miller and Sierpinski's exploration of suborders of the real line, Morel's results on the arithmetic of order types, Laver's proof that the sigma-scattered orders are well-quasi-ordered by embeddability, Todorcevic's and construction of Aronszajn and Countryman lines via his method of minimal walks.

Textbook

I will be posting my lecture notes on Canvas regularly, and there is no required textbook for the course. However, nearly all of the material we cover in lecture can be found in one of the following sources:

Cardinal and Ordinal Numbers by Sierpinski (to my knowledge the first textbook on abstract order types)
Linear Orderings by Rosenstein (still the most comprehensive introduction to linear orders)
The Arithmetic of Order Types by Morel
Trees and Linearly Ordered Sets by Todorcevic
Coherent Sequences by Todorcevic

Assessment

I will regularly post problems related to the material in lecture for you to think about, but won't require written solutions. Most of these problems are exercises intended for practice, while a few are elementary questions I don't know the answer to. If you're ever interested in discussing these problems, or working on some more substantial open problems, let me know!

At the end of the quarter, I'll ask you to prepare a talk on a topic of your choice (e.g. a problem, or theorem, or paper) related to linear orders and present it to the class. I'll make myself available to help with these presentations.