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

Office Hours: Friday 11:50 - 12:50  (also feel free to set up an appointment via e-mail).

Lectures: Monday-Wednesda-Friday  11:00 - 11:50 on-line lectures, with a few touchpoints in Skiles 269
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. There might be a few more touchpionts. The exact number of touchpoints will be determined by student interest.

Syllabus

The official syllabus for the class is at Math 4432 but a more details outline of the course is:

• Intro to topology (topological spaces, basic topological properties, quotient spaces)
• Intro to manifolds and statement of the classification of 1-manifolds and 2-manifolds
• Intro to groups, with a focus on group presentations
• The fundamental group of a topological space
• Studying groups via topolog  
• The fundamental group and knot theory
• Coverings spaces (and more studying groups via topology) 

Textbook

There is no required text for this course. I will try to make the lectures fairly self-contained and post lecture notes on the courses Canvas site. However, there are many good textbooks on topology and algebraic topology. If you would like a text that will cover many of the things in this course I recommend:

• Topology: a geometric approach by Terry Lawson.

• Basic Topology by M.A. Armstrong.

Grading Policy

The course grade will be based on the following.

Homework: 50%
Midterm: 10% each
Term Paper: 30%

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

Homework Policy

Every week or two 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 between one and two week to work on the problems (the exact due date will be on the problem set).

I encourage you to work on these assignments with other students in the class and to use whatever other resources you might have (like me and others in the department), but each problem must be written up in your own words by you. At some point everyone needs to learn TeX or LaTeX so I encourage you to write up your homework using one of these packages, but this is not a requirement. If you would like help getting started with TeX or LaTeX you are welcome to talk to me about it.

Exams

Midterm 1 is tentatively scheduled for Friday, February 26
Midterm 2 is tentatively scheduled for Friday, April 9

If you need to miss an exam talk to me in advance if possible, or if not then as soon after the exam as possible. We will arrange an alternate time for you to take a make-up exam.

Term Paper

Each studentor, or group of up to three students, will write a term paper for the class. The paper will cover some topic in topology, algebra, 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 February 1 on a topic for your paper
• 15% Draft of your paper turned in by March 26 March 29
• 20% Constructive feed back you provide on 2 other students papers (by April 7 April 9)
• 60% Final paper turned in by April 23

When you turn in the draft of the paper, I will then assign other student to read your paper and provide feedback (there will be a feedback form to fill out for each 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 my own and return it to you shortly after April 7. You can use this to make the final version of your paper to be turned in April 23.

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 algebraic topology not covered in this course:
• Homology theory
• Higher homotopy groups
• General homotopy theory (H and H' spaces, etc)
• Cobordism theory
• Fiber bundles or Serre fibrations
• K theory
• 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 (many of these can be found in any topology book), 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.