MATH1060 - Differential Geometry - Fall 2024
Axiom 1. Mathematical potential is distributed equally among different groups, irrespective of geographic, demographic, and economic boundaries.
Axiom 2. Everyone can have joyful, meaningful, and empowering mathematical experiences.
Axiom 3. Mathematics is a powerful, malleable tool that can be shaped and used differently by various communities to serve their needs.
Axiom 4. Every student deserves to be treated with dignity and respect.
This is a course in differential geometry geared toward undergraduates. We will study geometric properties of curves and surfaces (with an emphasis on surfaces). The content requires a clear understanding of multivariable calculus and linear algebra; we will review some of the most important prerequisites before extending them to a new setting. After taking this course, you might consider taking a course on manifolds or Riemannian geometry.
Course Objectives:
-Learn basic properties of curves and surfaces and how to compute them
-Prove some of the classical theorems in differential geometry including Gauss-Bonnet and Theorem Egregium
-Understand local versus global properties as well as intrinsic versus extrinsic quantities
-Improve your ability to read and write proofs
-Improve your ability to present mathematics to your peers
-Enhance understanding of calculus and linear algebra through application
Instructor:
Professor Christine Breiner
Kassar House 312
christine_breiner@brown.edu
TA:
Ethan Bove
ethan_bove@brown.edu
Lecture:
MWF 10-10:50am, Barus and Holley 163
Office Hours:
TBD - totaling 4 hours per week
Textbook:
Differential Geometry of Curves and Surfaces, 2nd edition, Manfredo do Carmo
The grading system is as follows: a 90% guarantees an A, 80% guarantees a B, a 70% guarantees a C.
More info on problem sets, the exam, and the final project can be found under Course Structure below.
Course Structure
Attendance/Participation
While attendance is not part of the course grade, you are expected to attend lecture and participate in any discussion. I will not be posting my lecture notes
Problem Sets
There will be one problem set per week, due (almost) every Wednesday starting on September 11th at 11:59pm on Gradescope. This amounts to about 11 problem sets for the semester. You will be given the course ID in class.
Problem set solutions must be typed in LaTeX. If you’re new to LaTeX, you can either download LaTeX or use Overleaf (you can get a free account with your Brown ID) which allows you to download, edit, and compile LaTeX files online. See here for some commonly used symbols in LaTeX. A basic solutions template is in the first assignment. The most basic thing to know/remember is that math always goes in between dollar signs when you're writing it in the middle of a paragraph. You can center math on the page in equation mode. See the template provided by email for examples.
The schedule of assignments in this course is designed to provide you with regular feedback, allow time to address confusion, and ensure that you do not fall behind on new material. For that reason, no late problem sets will be accepted. That being said, I understand that life often gets in the way of careful planning! I will therefore drop your one lowest problem set score. DO NOT email me to ask for a HW extension. I will not reply. If you feel that you need to drop more than one homework, you need to reach out to an academic dean about your situation.
Exam
The midterm will occur during the week of October 21, and the exact date will be set at least two weeks before the midterm.
Information about the exam structure and content will be available no later than a week before the exam date.
Final Project/Paper
The final project will be a presentation, prepared and delivered in groups of two. The presentations will take place in class with the likely dates being 12/4, 12/6, 12/9, 12/11. More information will be provided during shopping period, including some potential topic ideas. If the enrollment is above 24 by the end of shopping period, we might switch to (individual) final papers since there will be too many presentations to fit in.
Academic integrity
As stated in the Brown University academic code, Brown students are expected to practice high standards of academic and professional honesty and integrity. Among other things, this means that you are expected to adhere to the collaboration policy of this class. If you are unsure if something is academic misconduct, ask me and I will be happy to discuss any questions you have.
COVID-19
Students are required to adhere to the safety requirements set forth by Brown University. For details on these requirements see: https://healthy.brown.edu/. I will likely be wearing a high-quality mask in the classroom. If you have had a COVID exposure or you feel unwell, I ask that each of you wear high-quality (not cloth!) masks in class and office hours. If you are symptomatic or test positive for COVID-19 do not attend class.
Your instructor and TA will each hold 2 hours of office hours per week, at a time and location to be determined.
Office hours are times that I have set aside especially for students to come and discuss math. When you come to office hours, you can arrive at any time that is convenient for your schedule (not just at the beginning). During my office hours you can:
Ask a question about something you are confused about
Go back over a discussion from class
Look for more example problems to work through
Work on your homework alone or in a group
Ask for advice on skills for homework or test-taking
Explain solutions to your classmates
Talk about something entirely different!
I'm looking forward to getting to know you all better in office hours!
Tips for success
Mathematics is not a spectator sport. The best way to gain fluidity with new concepts is to practice them over and over again. Try forming a study group with some classmates and explaining solutions to each other. Challenge each other to explain the reasoning behind every step.
This may seem obvious, but it bears repeating: come to class, come to office hours, allow yourself ample time for the homework.
Read the reading posted in the schedule before class, and make notes about what confuses you. If I don't clear this up in class, ask specifically about your questions.
Try completing the homework first without looking at your notes or textbook or talking with your classmates. The homework assignments are there to help you gauge if you understand the material. It is easy to overestimate how much you understand if you are getting help.
Constructing proofs is difficult, especially as a beginner. Try something! Assume that your first approach won't work (this is completely normal, even among professional mathematicians), and be ready with another approach. Your biggest asset is a willingness to be wrong a lot. Don't try to do a lot in your head -- writing things out can help you stay organized and stimulate new ideas.
Have a growth mentality! I believe that intelligence and ability can be developed and improved upon with effort and support.
I encourage you to:
Engage in challenges
See hard work as a path to success
Treat feedback as an opportunity to improve
Be inspired by the success of others
Say/think:
What am I missing?
Perhaps I need a new strategy.
This is going to take some time.
How are they coming to that conclusion?
That's an interesting idea for improvement.
Read more about the growth mentality and its impact in this article
Accessibility and Accommodations
Brown University is committed to full inclusion of all students. Please inform me early in the term if you may require accommodations or modification of any of course procedures. You may speak with me after class, during office hours, or by appointment.
Here are some additional Campus Resources that may help you this term:
Students Support Services. Deans can be a helpful resource to discuss personal, family or health-related concerns, as well as a potential academic and personal plan. They are available for same-day consult and/or scheduled appointment.
Counseling and Psychological Services. (CAPS) provides a range of mental health services to the Brown community, including individual psychological treatment, mental health support, medication management, crisis stabilization, after hours assessment and urgent care, outreach programming, and groups.