# Math 670

**Instructor:** Clayton Shonkwiler

**Time:** Monday, Wednesday, Friday, 3:00–4:00

**Location:** Online (see Canvas for coordinates)

**Office Hours:** Tuesday 3:00–4:00 and Wednesday 4:00–5:00 (see Canvas for coordinates)

**Email Address:** clay@shonkwiler.org

**Syllabus**

## Overview

The course will be an introduction to differentiable manifolds with an eye towards Lie groups and homogeneous spaces, as well as toward symplectic geometry.

We will start with the basics of differentiable manifolds (tangent spaces, vector fields, Lie brackets, etc.) and come to grips with differential forms and tensors. With that background under our belts we will be able to dive into the study of Lie groups. Lie groups are groups which are also manifolds – such as matrix groups like \(GL(n,\mathbb{R})\) and \(SU(n)\) – and they are central to much of modern mathematics.

Building on this knowledge of Lie groups and their associated Lie algebras, we will finish by introducing symplectic geometry in the setting of coadjoint orbits and their natural Kirillov–Kostant–Souriau symplectic form. While it arises quite naturally in this setting, symplectic geometry has its roots in Hamiltonian mechanics and is relevant to a very large number of pure and applied areas of mathematics.

Some familiarity with point-set topology, multivariable calculus, and linear algebra will be very helpful, but otherwise the course should be reasonably self-contained.

There is no official text for the course, but the following books may be useful resources:

*Foundations of Differentiable Manifolds and Lie Groups*, by Frank W. Warner*Calculus on Manifolds*and*A Comprehensive Introduction to Differential Geometry*, by Michael Spivak*Differential Geometry and Lie Groups I & II*, by Jean Gallier and Jocelyn Quaintance*Differential Topology*, by Victor Guillemin and Alan Pollack*Lectures on Symplectic Geometry*, by Ana Cannas da Silva*Introduction to Symplectic Topology*, by Dusa McDuff and Dietmar Salamon