Fall 2018

Math 868 -- Geometry and Topology I

Math Tools

MWF 9:10--10 am                      Student talks and homework sessions:
C304 Wells Hall                          Tuesday 2-3 and Thursday 10:20-11:10


Prof. Thomas Parker
D216 Wells Hall
Office hours: Mon 2-3, Tues. 1-2, Thurs. 2-3
                      and by appointment
Alan (Zhe) Zhang (TA)
C-234 Wells Hall

Office hours: to be announced

Goals: This course introduces the intuition and techniques used to study manifolds. Manifolds are the natural setting for calculus in its most appealing and flexible form, and are the primary objects of study in much of modern geometry and topology.

The course will start with a rapid introduction (which should be a review for most students) to linear algebra, metric spaces and multi-variable calculus. Then come the main topics of the course: differentiable manifolds and tangent spaces, vector bundles, transversality, calculus on manifolds, differential forms, tensor bundles, the Frobenius Theorem, the deRham Theorem and cohomology groups. If time permits, we will cover the beginnings of Riemannian geometry.      Course outline   

Background: The official prerequisites are a 400-level course on Abstract Algebra and one on Real Analysis. In reality, the main prerequisite is a solid knowledge of multi-variable calculus and linear algebra and some knowledge of basic point set topology (open sets, compactness).

Text: Introduction to Smooth Manifolds by John M. Lee.

Other helpful reference books:

Homework:   HW1    HW 2    HW3    HW4     HW5    HW6     HW7      HW8     HW9

Comments on HW 2    HW3 Sol'ns    HW4 Sol'ns    HW7 Sol'ns   HW9 Sol'ns

Math 868 Student Seminar Times and location to be announced.  Seminar Topics and Schedule

During this time students will give informal talks on small topics related to the class material. You have a choice of either giving a talk, or doing a written alternative project (there may not be time for everyone in the class to give a talk).

Exams:   Midterm Review    Midterm Solutions   Final Solutions