Math 545/546
Topology and Geometry of Manifolds
Judith Arms
Winter/Spring 2005, Monday/Wednesday/Friday 1:30-2:20
Manifolds are arbitrary-dimensional generalizations of curves and surfaces
-- spaces that locally look like Euclidean space but globally may not, just
as the sphere locally looks like the plane. They are the basic subject
matter of differential geometry, but also play a role in many other
branches of pure and applied mathematics. For more discussion of what
manifolds are and what they are good for, see the first chapter of
Introduction to Topological Manifolds, by John M. Lee (the text for
Math 544).
Text for Math 545/6: Introduction to Smooth Manifolds, by John M. Lee.
Topics: differentiable manifolds, vector fields, flows, the
Frobenius theorem, Lie groups, homogeneous spaces, tensor fields,
differential forms, Stokes's theorem, deRham cohomology.
Prerequisites:
- Math 544 (or equivalent, by permission of the instructor).
- Linear algebra and calculus: the material reviewed in the appendix
of the text.
- Differential equations: basic facts about existence and uniqueness
of solutions to ODEs; elementary techniques for solving first-order
equations and systems.
Grades: will be based on homework and a final exam. The latter may include take-home and/or in-class work.