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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 and some 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.
Class Notes: Plese find the lecture notes here.
Lectures:
Monday-Wednesday-Friday from 11:00 to 11:50 in Skiles 308.
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
Classes will be in person. I will record classes so that people can watch them if they are unable to attend class for some reason. I will be experimenting with various technology to achieve good recordings, so the first few might not be optimal, but I hope they will all be reasonable. I am also posting class notes that can be used to keep up with the class if you cannot attend all the classes in person.
It is highly recommended that you try to addend in person as much as possible, the live interaction with the easy ability to ask questions is very helpful in learning the material.
I highly encourage everyone in the classes to get vaccinated and wear a mask. Out of concern for your health and mine, I am choosing to wear a mask in the classroom and ask that you give me and your fellow classmates this same consideration. Masks have been shown to slow the spread of this virus. Your cooperation and understanding on this matter are much appreciated.
Our classroom should have enough seating to allow people to be somewhat distanced from one another and I encourage you to keep as much distance as possible from each other. Also, please use the same seats each day in class. This will help with contact tracking should someone come down with COVID.
Office hours will be held on-line through BlueJeans (links available in Canvas). I do hope to hold some office hours in person, but when and if that happens will depend on how the pandemic progresses. Should we have in person office hours, I would again encourage you to wear a mask while in the office.
The midterm and final exams will both be ON LINE. This should mitigate any problems that might occur due to the pandemic. You will be able to download your exam from Canvas and upload your solutions. The exam will not be actively proctored, but I will be available through BlueJeans if there are any questions during the exams.
All of these policies are subject to change as guidance for the CDC and Georgia Tech change.
The course grade will be based on the following.
Homework: 60%
Midterm:
20%
Final Exam:
20%
The cutt-offs for grades my be reduced from what is indicated below, but they will not increase.
| Average |
Grade |
| in [90,100] |
A |
| in [80,90) |
B |
| in [70,80) |
C |
| in [60,70) |
D |
| less than 60 |
F |
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 22. 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 Wednesday, December 15th 11:20 am to 2:10 pm.
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
- 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
Assignment 1: Due by 5:00 pm September 8
Homework
Assignment 2: Due by 5:00 pm September 20
Homework
Assignment 3: Due by 5:00 pm October 1
Homework
Assignment 4: Due by 5:00 pm October 13
Homework
Assignment 5: Due by 5:00 pm November 5
Homework
Assignment 6: Due by 5:00 pm November 22
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