MATH670 Introduction to Differentiable Manifolds

Spring 2009 : MATH670 Introduction to Differentiable Manifolds

General information
Prerequisites
Textbook
Homework
Grading scheme
Course syllabus

General information

Lectures: Monday, Wednesday, and Friday 1-1:50pm in room ENGRG E206
Instructor: Justin Sawon, Weber 114
Office hours: Monday 2-3pm, Tuesday 2-3pm
Contact details: phone 491 6005, email sawon@math.colostate.edu

Prerequisties

For this course I will assume you have some knowledge of multivariable calculus, linear algebra, and elementary group theory. Some basic point-set topology would also be useful (for example, an understanding of compactness and connectness), though if you have taken some analysis classes you should have learned about the topology of Euclidean space, and that should suffice.

Textbook

Though I don't plan to stick exclusively to any textbook, the following two books will be available from the bookstore. They provide an excellent introduction to differential topology and geometry, are not too expensive, are short enough that you might read them in their entirety, and are concise enough that you will learn a lot if you do so. However, these are not required purchases, and in fact there are several copies in the library.

John Milnor "Topology from the Differentiable Viewpoint", revised edition, Princeton University Press (November 24, 1997).

John Milnor "Morse Theory", Annals of Mathematic Studies 51, Princeton University Press (May 1, 1963).

Homework

There will be weekly homework. The homework assignments will be available each Wednesday from this webpage (next to the syllabus), and should be handed in the following Wednesday at the beginning of the lecture.

Grading scheme

Course grades will be determined based on homework (75%) and a final project/presentation (25%).

Some suggestions for projects:

topological manifolds which don't admit smooth structures
exotic smooth structures on seven-dimensional spheres
classification of topological four-manifolds
Donaldson theory in four dimensions
manifolds of intermediate differentiability C^1, C^2, etc.
complex manifolds and almost complex manifolds
Gromov-Hausdorff distance between topological spaces
a C^1 counter-example to Sard's theorem
non-embeddings results

(More details on project topics.) If any of these topics seem interesting and you'd like further details, please don't hesitate to ask.

Titles and schedule for presentations.

Course syllabus

The syllabus below will be updated as the semester progresses.

Week	Material covered	Homework
Jan 19-23 No class Monday (MLK Day)	Overview of differential topology Brief review of topology topological spaces, continuity, homeomorphisms Hausdorffness, paracompactness, second countability	No homework yet.
Jan 26-30	Topological and smooth manifolds coordinate charts, atlases, smooth structures maps, submanifolds Tangent spaces function germs, derivations equivalence classes of paths	Exercise sheet 1. Due Wednesday 4th February.
Feb 2-6	Vector bundles topological vector bundles bundle homomorphisms and the Rank Theorem pull-backs, bundle maps and linear maps smooth vector bundles, the tangent bundle	Exercise sheet 2. Due Wednesday 18th February.
Feb 9-13 No classes Mon/Wed	example: the tangent bundle of S^2	No homework this week.
Feb 16-20 Extra class Tue 4pm	orientations of vector bundles partitions of unity Riemannian metrics	Exercise sheet 3. Due Wednesday 4th March.
Feb 23-27 No classes this week	Start thinking about a topic for your project.	No homework this week.
Mar 2-6 Extra class Tue 4pm	Local and tangential properties Inverse Function Theorem and the Rank Theorem submersions, immersions regular points/values, critical points/values Sard's Theorem, Brown's Theorem	Exercise sheet 4. Due Friday 13th March.
Mar 9-13 No class Wed	proof of Sard's Theorem Brouwer Fixed Point Theorem	Exercise sheet 5. Due Friday 27th March.
Mar 16-20 No classes	Spring Break	Started your project yet?
Mar 23-27 No class Mon	Embeddings in Euclidean space Whitney Immersion Theorem Whitney Embedding Theorem	Exercise sheet 6. Due Wednesday 1st April.
Mar 30-Apr 3 Extra class Tue 4pm	Dynamical systems vector fields and their flows Integrability Theorem for Vector Fields Homotopy and isotopy mod 2 degree of a map	Exercise sheet 7. Correction to #4: "Find an injective immersion of R in R^2..." Due Wednesday 8th April.
Apr 6-10 Extra class Thurs 11am	Theorem of Thom (isotopies embedded in diffeotopies) Brouwer degree Hairy Ball Theorem Hopf fibration S^1->S^3->S^2	Exercise sheet 8. Due Wednesday 15th April.
Apr 13-17 Extra class Tues 4pm No class Friday	Vector fields and the Euler number Isolated and non-degenerate zeroes Indices of zeroes of vector fields Poincare-Hopf Theorem	Exercise sheet 9. Due Friday 24th April.
Apr 20-24 Extra class Tues 4pm	Morse theory Non-degenerate critical points Morse Lemma Reeb Theorem Cellular decompositions of manifolds Cellular homology and Betti numbers	Exercise sheet 10. Please think about these problems; we will discuss them in class.
Apr 27-May 1	Poincare-Hopf Theorem revisited Presentations: Wed: Nick Fri: Bethany, Ryan	Titles and schedules for presentations.
May 4-8	Presentations: Mon: Dusty, Steve Wed: Elin, Tim Fri: Luke, Shawn

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This page last modified by Justin Sawon
Saturday, 25-Apr-2009 20:14:17 MDT
Email corrections and comments to sawon@math.colostate.edu