Math 544/545/546: Topology and Geometry of Manifolds

Dan Pollack

Autumn/Winter/Spring 2002-2003, MWF 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.  In the fall quarter we will concentrate on the topology of manifolds, i.e., properties that are invariant under continuous deformations.  The main goals here are the fundamental group, covering spaces, and the classification of compact surfaces.  The winter and spring quarters will be devoted to the study of smooth manifolds, on which derivatives of functions and maps make sense.

Texts

• [Fall] Introduction to Topological Manifolds, by J. M. Lee.
• [Winter/Spring] Introduction to Smooth Manifolds, by J. M. Lee (to be published in early Autumn 2002).

Prerequisites: In addition to the references below, much of the prerequisite material is outlined in the appendices to the textbooks.

FOR FALL QUARTER:

• Set Theory:  Operations on sets, functions, equivalence and order relations, number systems and cardinality, the axiom of choice. References: Principles of Mathematical Analysis by Rudin, Chapter 1; Naive Set Theory by Halmos.
• Analysis:  Metric spaces; convergence and continuity; open and closed sets; interior, exterior, and boundary; compactness.  Reference: Principles of Mathematical Analysis by Rudin, Chapters 2,3,4.
• Algebra:  Elementary group theory, homomorphisms, isomorphisms, subgroups, normal subgroups, permutation groups, cosets, quotient groups.  Reference: Abstract Algebra: An Introduction by Hungerford, Chapter 7.

FOR WINTER AND SPRING QUARTERS

• Linear algebra:  Vector spaces, subspaces, bases, dimension, matrices, determinants, change of basis formulas, linear maps, kernel and image, inner products, orthonormal bases, linear functionals, dual spaces.  Reference: Linear Algebra by Friedberg, Insel, and Spence.
• Multivariable calculus:  Partial derivatives; the total derivative as a linear map; Taylor's formula in several variables; multiple integrals and the change of variables formula; gradient, divergence, and curl; the theorems of Green, Gauss, and Stokes; uniform convergence.  References: Basic Multivariable Calculus by Marsden, Tromba, and Weinstein; Principles of Mathematical Analysis by Rudin, Chapters 5,6,7.
• Differential equations: Basic facts about existence and uniqueness of solutions to ODEs; elementary techniques for solving first-order equations and systems. Reference: Ordinary Differential Equations by Birkhoff and Rota.

Homework and grading: There will be a homework assignment each week to write up and hand in for a grade.  Your grades will be based 2/3 on homework and 1/3 on a take-home final exam.