Math 102: Foundations of Smooth Manifolds

Fall 2011

Instructor	Craig J. Sutton
Lectures	MWF 12:30-1:35 (201 Kemeny)
X-Hour	Tu 1-1:50 (201 Kemeny)
Office Hours	Tu & F 10:30-Noon (Also By Appointment)
Office	321 Kemeny Hall
E-mail	Craig.J.Sutton AT You Know Where
Phone	603-646-1059
Syllabus/Homework	Click here

Course Description:

Manifolds provide mathematicians and other scientists with a way of grappling with the concept of "space." The space occupied by an object. The space that we inhabit. Or, perhaps, the space of configurations of a mechanical system. While manifolds are center stage in the study of geometry and topology, they also provide an appropriate framework in which to explore aspects of mathematical physics, dynamics, control theory, medical imaging, econometrics and robotics, to name just a few. This course will serve as an introduction to the basics of manifold theory and lay the foundations needed to explore problems in which "space" plays a fundamental role.

Topics will include some of the following

manifolds & tangent bundles

vector fields & vector bundles

smooth maps and the inverse function theorem

cotangent bundle & differential forms

densities, integration on manifolds and the generalized theorem of Stokes

Whitney's imbedding theorem

Lie groups, group representations and group actions

homogeneous spaces

Tensor Fields & Riemannian metrics

Differential forms & orientation

integral curves & the existence and uniqueness theorem for ODEs

Distributions and Frobenius' Theorem

Prerequisites: Linear Algebra (Math 24), point-set topology (Math 54) and multivariable analysis (Math 73). It will also help to be familiar with covering spaces and the fundamental group, but the interested student should be able to pick this up on the fly.

Course Objectives: We will provde you with the tools needed to deal with spaces as they arise in the "wild" and on the qualifying exam in differential topology.

Textbook: An Introduction to Differentiable Manifolds and Riemannian Geometry (Revised 2nd Edition), William M. Boothby, Academic Press 2003. (available at Wheelock Books).

Deliverables & (tentative) Grading Guide: The following will probably comprise the written assignemtns for this term.

(Quasi) Weekly Homework: There will be (quasi) weekly homework assignments for this class which will generally be assigned on Wednesday and collected the follwiong Wednesday. No late homework will be accepted.

Oral Final Exam (Closed Book): More details later

Graduate Students who have passed the qualifying exam in topology can enroll in this course and will be exempt from all assignments.

Your course grade will probably be computed as follows.

Homework
70 %

Final Exam
30%

Students with disabilities: If you have a disability and require disability related accomodations please speak to me and Ward Newmeyer, Director of Student Accessibility Services, as soon as possible so we can find a remedy.

Last Updated 21 September, 2011