# Math 733: Geometric Topology

## Jason Parsley

### Announcements

• Problem Session meeting time: Fri., 2-3 pm, room 125
• Solutions to the midterm exam are posted under Notes.
• Final projects: presentations are during our exam period: Mon., May 3, 2-5pm (pizza!)

### Course information

Meeting times: TuTh 12-1:30. Tu in room 122; Th in room 125.
Problem Session time: Fri., 2-3 pm, room 125
Midterm Exam: Th., April 8
My office hours (334 Manchester):

• Tu. 10-11am
• TuWTh 3-4pm
• by appointment/drop-in

### Website Info

We'll post most everything for the course on this site, including assignments, projects, Maple code, and various other stuff. For grades, online office hours, and large presentations, you'll have to go to the Sakai course management site. (Sakai is an open-source alternative to Blackboard that WFU is moving towards; alas, there's still bugs with it, so we'll use my website.

### Course description

In this course we start with a review of vector fields, including their curl, divergence, and gradient. We move to classifying vector fields that occur on subsets of R^3. For instance, the curl of any gradient is zero; are there non-gradients whose curl is zero? Russell Crowe, portraying John Nash in A Beautiful Mind poses this question and claims it will take

for some of you, many months to solve; for others among you, it will take you the term of your natural lives.

We will solve it by March. As Crowe indicates, the result depends on which subset of R^3 we consider.

We will then define differential forms and explore their dual relation with vector fields. We use differential forms to define what cohomology means in a three-dimensional setting before defining the linking number of two knots. We'll talk a good amount about knots and how they relate to certain vector fields. Time-permitting, we let cohomology guide us into a brief discussion of wonderful objects in topology known as characteristic classes. The nice thing is that it's all done in 3 dimensions, so we'll have lots & lots of pictures to work with.

Prerequisites for this class are only multivariable calculus, linear algebra, and algebra (but not 731). You don't have to even remember multivariable calculus, because we'll review it. Talented, motivated undergraduates should do well in this course.