Professor: John Etnyre
Office: Skiles 105
Phone: 404.894.6614
e-mail: etnyre "at" math.gatech.edu

Office Hours: 12:30-1:20 Wednesdays (also feel free to set up an appointment via e-mail).

Lectures: Monday, Wednesday, Friday 9:30-10:20 on-line lectures, with a few touchpoints in Skiles 371
The touchpoint are currently scheduled to be reviews for the two test and will occur just before the tests. Attendance is not required, and the in person meetings will be streamed and recorded.

Prerequisites

I will assume that students know vector calculus and linear algebra. It would be helpful, but not essential, that students have had some exposure to proofs as we will be doing many in this class (and you will do some on the homework). Lastly, we will use a little differential equations in the course too, but that will be reviewed when necessary.

Textbook

I will try to make the lectures fairly self-contained so you really need to come to class!. The main reason you will need the books is as another source for some of the material. (If you would prefer not to buy the book, you can probably get by with just the class notes which I will post on-line.)

You might find the following books helpful:

• Differential Geometry of curves and surfaces by DoCarmo
• Differential Geometry of Curves and Surfaces by Kristopher Tapp (available in pdf through our library)

You also might find the following on-line notes helpful:

• Prof. Shifrin's notes Differential Geometry: A First Course in Curves and Surfaces
• Prof. Ghomi's class notes
• Prof. Cantarella's class notes

Grading Policy

The course grade will be based on the following.

Homework: 50%
2 Midterm: 7.5% each
Term Paper: 35%

The cutt-offs for grades my be reduced from what is indicated below, but they will not increase.

Homework Policy

Every two weeks or so I will assign a set of homework problems. You should check the homework web page regularly, since I will not always announce when I post an assignment. You will have about two week to work on the problems (the exact due date will be on the problem set).

Exams

Midterm 1 is tentatively scheduled for Thursday, September 18th
Midterm 2 is tentatively scheduled for Thursday, Novembe13 th

Term Paper

Each student, or group of up to three students, will write a term paper for the class. The paper will cover some topic in topology, differential topology, differetial geoemtry, algebraic topology, or its application to some other area (physics, biology, data analysis, etc.). The idea is for you to explore some area you find interesting related to the class.

The paper will need to have

1. a significant mathematical component (that is proofs, computations, or the like) and
2. have a good exposition (that is, written well enough to other students to learn something from the paper).

The target audience for these papers is other students. In fact part of the grade on the paper will be your providing helpful and constructive feedback to other students. Knowing who your audience is will help you while writing the paper and by seeing what others are doing well and not so well, you will be able to get a better idea how to be a better writer yourself.

Your grade on the paper will be determined by the following

• 5% Consulting with me by the end of August 31 on a topic for your paper
• 15% Draft of your paper turned in by October 30
• 20% Constructive feed back you provide on other groups papers (by November 6)
• 60% Final paper turned in by November 20

When you turn in the draft of the paper, you will e-mail me a copy of the paper. I will then assign other students to read your paper and provide feedback (there will be a feedback form to fill out for the paper you read. The feedback will be constructive and kind. (If you make negative comments that are un-helpful and un-constructive, then I will not give the comments to the student and you will get a 0 for this 20% of the grade.) I will give you this form (or one very like it) to fill out where you will answer a few questions about the paper and provide some short written feedback. I will collect the student feedback and it to you shortly after November 6. You can use this to make the final version of your paper to be turned in November 20.

The paper will need to be 4 to 8 pages (you can talk to me to get approval if you have a good reason for the paper to be shorter or longer) and be turned in as a pdf document. You should try to write the paper in TeX or LaTeX (ask me if you do not know about this), but I will accept any pdf document (so you can use Word or some other program to create the paper if you like). I will post the final versions of the paper on the class Canvas site so other students can read them if they would like to do so.

• Your paper can be on anything you like, but here are some thoughts to get you started.
  • The most obvious choice for a topic is some part of differential geometry not covered in this course:
  • Minimal surfaces
  • Riemannian geoemtry
  • Fiber bundles (aka twisted products)
  • Curvature comparison theorems
  • Hyperbolic geometry
• Topics from differential topology
  • Smooth manifolds
  • Immersions of S^1 into R^2
  • Transversality
  • Degree theory
  • Whitney embedding theory
  • Handlebody theory Morse theory
• Fun theorem from topology that we do not cover in class, examples
  • Jordan curve theorem
  • Classification of surfaces
  • Classification of 1-manifolds
  • Metrization Theorems
  • Fixed point theorems
  • Direct limits and/or Cantor sets
  • Triangulation of surfaces
• Applications of topology (many of the examples below can be found in Introduciton to topology: pure and applied):
  • Phenotype spaces (evolutionary biology)
  • Geographic Information Systems
  • Cosmology (what manifold do we live in?)
  • Studying DNA via knots and this paper
  • Fixed points and economics
  • Topological data analysis
  • Topology in chemistry and this paper
• Browse some books like
  • Three-Dimensional Geometry and Topology, by Thurson
  • Knots and Links, by Rolfsen
  • The Shape of Space, by Weeks
  • The knot book, Adams

How to write and structure your term paper:

• Here are some guidelines for writing good mathematics by Francis Su
• To see the structure of a well written math paper check out the notes in The American Mathematical Monthly or articles in Math Horizons.
• You can also find some nice expository articles on the arxiv. Warning: not everything here is well written! Some examples to consider are this paper on the Jones polynomial and this paper on contact geometry. These are both longer than your paper, but give an idea for the structure of a math paper.

Miscellaneous matters

• Academic Integrity. All students are expected to comply with the Georgia Tech Honor Code.
• Students with Disabilities and/or in need of Special Accommodations. Georgia Tech complies with the regulations of the Americans with Disabilities Act of 1990 and offers accommodations to students with disabilities. If you are in need of classroom or testing accommodations, please make an appointment with the ADAPTS office to discuss the appropriate procedures.
• Intent for Inclusivity. As a member of the Georgia Tech community, I am committed to creating a learning environment in which all of my students feel safe and included.  Because we are individuals with varying needs, I am reliant on your feedback to achieve this goal.  To that end, I invite you to enter into dialogue with me about the things I can stop, start, and continue doing to make my classroom an environment in which every student feels valued and can engage actively in our learning community.