M390C: Geometry in Group Theory, Spring 2020
Daniel Allcock
unique number 53289: MWF 11-12 in RLM 9.166;
my office hours: Wed 1-2 and by appointment in RLM 9.112
My email: (my last name)@math.utexas.edu.
my office phone number is 1-1120, but it's not very useful.

Homework will be due approximately every two weeks. Policies about who does what problems are at the beginning of hw#1. The general philosophy of undergrads having to do more problems comes from the expectation that grads have theses to write. I am contemplating having student presentations on selected papers later in the semester. Exactly what I do will depend on who stays in the class. But you should also expect to be responsible for something along these lines.

A template for building a dodecahedron (You need two of these to build one dodecahedron.)
Tutte's 8-cage (a graph for hw2)
Casandra Monroe has put online some notes of my lectures. Thanks Casandra!
Notes on inversion in circles and accompanying pictures. tex source too, in case you need it.
hw1 and its tex source (due Fri Jan 31) (Correction: in 8, both r and s have norm 2.)
hw2 and its tex source (due Fri Feb 7) (Correction: in 6, the group G should be taken to be simple. That makes it so the normalizer of a subgroup of the special form cannot be all of G.)
hw3 and its tex source (due Fri Mar 6; NOT Feb 28)

Course Description:
This class will be more about specific groups than group theory, so it could be taken concurrently with 380D. The idea is more to introduce lots of groups that everyone should know, rather than to propel people to the research frontier. For me group theory is more about geometry than algebra, so expect a heavy geometric emphasis. We will meet some Lie groups, but they are not the emphasis of the course, since we have a whole course devoted to them. Prerequisite: M380C, or in some cases M373K+L. Undergraduates who have taken 373K but not L should take M373L rather than this course.

Topics to be covered:

Bieberbach groups (discrete groups acting on Euclidean space).
Coxeter groups (groups generated by reflections).
Hyperbolic geometry and its friends.
Finite groups beyond the Sylow theorems (eg, some transfer and fusion theory; characterizations of some of the simplest simple groups).
Isometry groups of Euclidean lattices (eg, the E8 and Leech lattices).
Braid groups and Artin groups, including in algebraic geometry.
Geometry of surface groups and free groups.
(depending on interest) SL2Z and modular forms.