University of Pennsylvania
Math 600 Geometric Analysis and Topology Fall 2010

Professor : Ryan Blair
Email : ryblair@math.upenn.edu
Office : DRL 4N59
Office Hours : Wednesday 5-6 pm and Friday 10-11 am

Goals: The basic goals of the course are to master this material, to enrich it with many examples, to cultivate talents for invention, discovery and the solution of hard problems, and to help develop high-dimensional geometric vision.

Prerequisites: The prerequisite for the course is a typical advanced calculus course plus familiarity with the basic notions of point set topology: metric and topological spaces, convergence, continuity, compactness and connectedness.

Texts: I intend for the course notes to serve as a text for the course, however it is a good idea to have access to Introduction to Smooth Manifolds, by John M. Lee. Lee’s book will serve as the recommended, but not required, text for the course. However, we will be pulling heavily from several books including Calculus on Manifolds by Michael Spivak, Differential Topology by Victor Guillemin and Alan Pollack and Topology from the differentiable viewpoint by John Milnor.

Grading

Homework: 60% of your final grade.

Homework will be assigned every lecture and will be collected one week later. Homework will be graded on correctness and completeness by Ricardo Mendes. I encourage you to work together on the homework.

Midterms: 10% of your final grade each.
You will have two take home midterms that will represent individual work. You will be given at least two days to complete each midterm. Please feel free to use any class notes and Lee’s book on the midterms, but no other sources.

Final Exam: 20% of your final grade.
There will be an accumulative in class final on December 21^st from 12pm to 2pm.

Topics

Below is the syllabus for Math 600 for the coming academic year. How much we cover of this depends on the background, ability and comfort level of the class. We can expect occasional reordering of the topics, with some to be dropped and others added.

Review of Advanced Calculus: differentiation, inverse and implicit function theorems

Review of Advanced Calculus: integration, Fubini's theorem, measure zero,

partitions of unity, change of variable in multiple integrals

Differentiable manifolds: charts, tangent spaces, tangent bundles, derivatives of maps, examples, partitions of unity, transversality, singular and regular points and values, Sard's theorem

Embedding manifolds in Euclidean space: the Whitney embedding theorem

Vector fields: basic existence and uniqueness theorems for ODEs, flows, tensor fields

Lie derivatives and Lie brackets: definitions and examples, Lie groups and Lie algebras

Riemannian metrics and Riemannian manifolds: basic definitions and examples

Vector calculus in 3-space and in 3-manifolds: gradient, divergence, curl; product rules; tensor notation; Laplacian; electrodynamics and Maxwell's equations; integral calculus of vector fields and Stokes' theorem; curvilinear coordinates

Multilinear algebra: tensor and exterior products, forms, wedge product of forms,

exterior algebra of forms

Differential forms: behavior under maps, exterior derivative, closed and exact forms,

Poincare lemma, definition of De Rham cohomology, foliations, contact forms,

conversion between vector fields and forms, statement of the De Rham

Isomorphism Theorem, orientation of manifolds, Lie derivatives of forms,

Hodge star operator on Riemannian manifolds, exterior co-derivative,

Hodge Laplacian on forms, harmonic forms, all on Riemannian manifolds

Integration on chains and on manifolds: Stokes' Theorem, Green's Theorem, Divergence Theorem

Frobenius integrability theorem: k-plane distribution, integral manifolds, integrable

distributions, foliations; Frobenius theorem via Lie brackets, via differential

ideals of differential forms, via curls of vector fields

De Rham cohomology: Poincare Lemma revisited, relative cohomology,

long exact cohomology sequence, Mayer-Vietoris sequence,

statement of the De Rham Isomorphism and Hodge Decomposition Theorems

Invariant forms on Lie groups: left-invariant forms on Lie groups; bi-invariant forms

on Lie groups; bi-invariant forms are closed; On a compact connected Lie group,

the algebra of bi-invariant forms is isomorphic to De Rham cohomology, existence of bi-invariant Riemannian metrics there, fundamental 3-form.