This is an introductory course in differential topology. The aim of this course is to introduce the basic tools to study the topology and geometry of manifolds (and some other spaces too). (Smooth) Manifolds are "locally Euclidean" spaces on which we can "do calculus" and "do geometry". These spaces are at the center of a great deal of much of the most exciting current research in mathematics and are essential to many applications of mathematics into science and engineering. Throughout the semesterwe will discuss the theory of manifolds and a way to generalize differential, integral and vector calculus. By the end of the course, students should be understand and be able to work with manifolds, tangent and cotangent bundles, vector bundles, differential forms, vector fields and many other things listed in the official course description.

The prerequisites for the course are vector calculus andsome basic knowledge of point set topology. Most topics beyond the most basic material will will be reviewed, at least briefly, when we need it. (Also I am more than willing to meet in office hours and discuss background material here and there should the need arise.) But to get started I have written up an introduction to the course with some of the most important ideas we will need from point set topology. In particular, a very important concept that many people have not seen much of before is quotient spaces. This is a very convenient way to rigorously describe how to build up complicated spaces from simple one.

• Introductory notes, recollections from point set topology and quotient spaces.

Class Notes: Plese find the lecture notes here.

Lectures: Monday-Wednesday-Friday from 11:00 to 11:50 on-line lectures, with a few touchpoints in Coll of Computing 53.
The touchpoint are currently scheduled to be reviews for the test and final. The first will occur just before the test and the second will occur just before the end of the course. Attendance is not required, and the in person meetings will be streamed and recorded.
Professor: John Etnyre
Office: Skiles 105
Office Hours: 12:00-1:00 Fridays (also feel free to set up an appointment via e-mail).
Phone: 404.894.6614
e-mail: etnyre "at" math .gatech.edu

Grading Policy

The course grade will be based on the following.

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

The homework assignments will be posted below and will be due in class on the day indicated on the assignment.  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.

The midterm exam will be in class and I will announce it on this web page and in class at least 1 week before the exam. The tentative dates for the exam is October 21. If you need to miss the midterm exam you must talk to me about this in advance if possible. If you miss the midterm exam for an excused reason you will be given the option to take a makeup exam or skip the exam and have the homework and final exam count more towards your final grade.

The final exam is tentatively scheduled for December 2th from 11:20am-2:10pm.

Textbook

There are two textbook for this class:

• Introduction to Smooth Manifolds by Lee.
• Differential Topology by Guillemin and Pollack

The primary text is Lee, but Guillemin and Pollack is also a good reference and at times has a different perspective on the material. Neither text is required but I will sometimes assign homework out of Lee. You can download an electronic version of Lee from the library.

There are many good textbooks for differential topology. I mention several below. Some of them use a different perspective that we take in this class, but that can be useful to see!

• Topology and Geometry by Bredon
• An Introduction to Differential Manifolds by Barden and Thomas
• An Introduction to Manifolds by Tu
• A Comprehensive Introduction to Differential Geometry, Vol. 1 by Spivak

Term Paper

Each student, or group of two 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 5 to 10 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.

• The most obvious choice for a topic is some part of differential topology not covered in this course
  • Jet spaces
  • Immersions of S^k into R^n
  • h-principle
  • Degree theory
  • Singularity theory
  • Handlebody theory Morse theory
• Topics from differential geometry:
  • Riemannian metrics
  • Connections
  • Symplectic geometry
  • Contact geomery
  • Hyperbolic geometry
• 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
• Browse this page of notes.

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.

Homework

Homework Assignment 1: Due by 5:00 pm August 28

Homework Assignment 2: Due by 5:00 pm September 11

Homework Assignment 3: Due by 5:00 pm September 25

Homework Assignment 4: Due by 5:00 pm October 9

Homework Assignment 5: Due by 5:00 pm November 2 November 6

Homework Assignment 6: Due Never