The Bianchi-Pinkall torus, image from the virtual math museum
Course topic:
A manifold is a space where you can do calculus. You can imagine a surface (with a well-defined tangent space at every point), but more interesting examples include Lorentzian space-time (our uinverse, according to relativity), configuration spaces of mechanical systems, and many of the frameworks for physics and motion.
This course will introduce you to manifolds and the structures that they are endowed with, such as tangent vectors, boundaries, orientations, and differential forms. The notion of a differential form encompasses such ideas as surface and volume forms, the work exerted by a force, the flow of a fluid, and the curvature of a surface, space or hyperspace. We will re-examine the integral theorems of vector calculus (Green, Gauss and Stokes) in the light of differential forms and apply them to problems in partial differential equations, topology, fluid mechanics, electromagnetism, and areas of pure mathematics such as topology.
Prerequisites:
Multivariable calculus and linear algebra.
Online textbook:
The content will largely follow the excellent textbook/notes that Prof Sjamaar has put together specifically for this course, available at this link
Other reading?
For a refresher on vector calculus, and/or just a great exposition of how to think about integrals and why you should even care (!), I highly recommend the book Div, Grad, Curl and All That by H. M. Schey (any edition is fine). Used copies are easy to find and inexpensive. It is one of the most pleasant-to-read math books I know, the kind of thing you keep on your bookshelf forever and lend to your friends. If you have the time to read parts of it sometime early in the semester, I encourage you to do so!
Provisional format:
We meet Monday-Wednesday-Friday 11:30-12:20. On most weeks, the Monday and Wednesday classes will be lecture style over zoom, and the Friday session will be something more interactive, with a combination of group activities/problem solving, and office-hour style discussion. The precise format of this will vary week by week, and will depend on how many students are enrolled in the course. Attendance is required unless you have exceptional circumstances, in which case you should talk to me (the professor) and we'll work something out.
Both the instructor and TA will hold regular office hours, at times to be announced.
Required supplies:
Students in this course need to have some method of writing mathematics on something that can be seen over zoom on the Friday sections of class. Some options are:
1. Access to a whiteboard/chalkboard during Friday class time.
2. Window writer erasable markers and a window that's easy to see (check for glare/backlight!)
3. A tablet/ipad + pencil/styuls.
4. A big pad of chart paper like this: https://www.amazon.com/Pacon-Chart-Tablet-x32-Sheets/dp/B000KIE2BK/ and a fat marker.
5. (trickier but possible) a document camera or a home-made version using your phone as a document camera, such as that described here: https://www.rochester.edu/college/cetl/assets/pdf/diy-document-camera.pdf
Note: holding a normal size sheet of paper up to your computer webcam really doesn't work well enough! We have done extensive tests on this!
Evaluation/assignments:
There will be weekly problem sets for homework like a typical non-virtual math course. Homework is due Mondays by 11am, to be submitted on gradescope. Rather than have a heavily-weighted final exam, we'll have multiple short quizzes, a written final project + short presentation to be done individually or in pairs (more on that below), and two timed at-home tests. Assignments and test will be submitted using gradescope.
Grading scheme:
Quizzes: 10% (lowest score dropped)
Attendance/group activity participation: 5% (yes, it's ok to miss a couple classes for reasonable reasons!)
Two tests: 15% each
Homework: 25% (lowest score dropped)
Final project: (proposal + written project + in-class presentation. Rubric to be distributed) 30%
Working together: Please *do* work on your homework with other students, but write up your own solutions individually. Best practices for doing this are to discuss in groups (share a screen, share a whiteboard, have a conversation, etc) and then put that away when you start writing your own solutions. Tests and quizzes must be done on your own, without help or discussion with others.
Academic integrity: Each student in this course is expected to abide by the Cornell University Code of Academic Integrity. Any work submitted by a student in this course for academic credit will be the student’s own work. You are prohibited from buying, selling, or accessing course materials through internet sites such as Chegg, CourseHero, and Slader -- Original course materials are intellectual property that belong to the author and are not a student’s property to sell, repost, or distribute!
A word on the final project:
As mentioned above, the material in this course has many applications in physics, engineering, and some areas of pure math, so the final project will be an exploration of one of these connections in some area that interests you. I will provide a list of suggestions for starting points, but you will be responsible for selecting and refining the topic and scope. You can also propose your own idea - talk to me about it! Can it involve programming (mathematica, etc)? Yes! Can it involve a specific application, a historical example, a connection between a concept we learned and another area, learning the proof of a theorem we talked about but didn't prove, etc, etc? Yes! We'll discuss this more at some point in October when it is time to start thinking about it. Presentations (using keynote/powerpoint/other visuals) in front of your classmates will occur the last week or two weeks of classes, with a written component due at the end of the course.
Accommodations: If you have an SDS accommodation, or require an accommodation at some point in the semester due to extended illness or other circumstances, let me know as soon as possible so we can work something out!
More course info, and all homework, discussion, zoom links and video will be on the course Canvas site.
The Bianchi-Pinkall torus, image from the virtual math museum