Instructor : Dr. David Groisser
- Office Hours:
Tentatively, Tuesday 7th period (1:55–2:45), Wednesday 5th period (11:45–12:35), and Friday 9th period (4:05 – 4:55). My Friday 9th period office hour may start a few minutes late, since I'll be coming from an 8th period class in another building.
My office is Little Hall 308. Please arrive early in the period or let me know to expect you later; otherwise I may not stay in my office for the whole period.
- Textbook: None. I teach from my own notes and, for certain material, draw on several of the references listed here.
- Tentative, approximate weekly schedule of lectures. Click here.
- Syllabus (course content): MTG 6256–6257 is a year-long graduate sequence that introduces the tools of differential geometry and differential topology (see the sequence summary). MTG 6256, the first semester of the sequence, will be devoted primarily to the basics of manifold theory and of calculus on manifolds. Topics will include a brief review of advanced calculus from a geometric viewpoint; definition and examples of manifolds (including Lie groups and submanifolds of Euclidean space); maps of manifolds; critical points and the Regular Value Theorem; Sard's Theorem (possibly deferred to spring); vector fields, flows, and Lie derivatives (some of this may be deferred to spring); exterior algebra and differential forms; integration on manifolds; the Stokes Theorem and de Rham cohomology; vector bundles and tensor bundles (possibly deferred to spring); Riemannian metrics; introduction to Riemannian geometry; and connections and curvature. Riemannian geometry and the theory of connections are large topics; most of this material will probably be deferred to the second semester, MTG 6257. Please see prerequisites for MTG 6256 below.
- Course-grade components: There will be no exams. Your final grade will be determined entirely by homework, assuming your attendance is good. If your attendance is poor a grade penalty may be imposed.
- Homework: I expect to assign and collect from five to eight problem-sets over the course of the semester. The problem-sets will include some problems that are mandatory and some that are optional. I will grade some subset of the mandatory problems. How many problems I grade will depend on how many students handed in the assignment, how successful they were solving the problems, and how well-written their solutions are.
Doing well on the graded subset of the mandatory homework problems will be enough to earn an A in the class, and I don't think that you'll find the mandatory problems excessive (if you've met the prerequisites for the course). However, to get the most out of the course, you should do as many of the optional problems as you can. The more time you put in, the more you will learn. My intent is to give students who want to learn a great deal the opportunity to do so, without requiring any of you to do a lot more work than you'd have to do in other 6000-level courses in this department.
See More about homework below.
- How to get the most out of lectures: To keep up with the lectures and to learn the material really well, my best recommendation i that you go through your notes from each class before the next class, filling in any gaps, trying figuring out anything you didn't understand at the time, and determining what you still don't understand and should ask me about. Time permitting, the best thing you can do is rewrite your notes. Do not expect to understand everything I say in class at the time I say it. I will sometimes make comments that are intentionally cryptic, will sometimes deliberately omit some steps in proofs, etc., to force you to think more about something. (Sadly, I will also no doubt make some comments that are unintentionally cryptic, and some mistakes that I can't claim are intentional ...) The deepest understanding will come only when you think about the material on your own. This will take you a good deal more time than the hours we spend together in class.
- Prerequisites for MTG 6256: Essentially, these can be summarized as "Everything you would learn as an undergraduate (pure-)math major". Please see the sequence summary for details. Note: undergraduates are not allowed to take this class without my explicit permission. No high school students will be given permission.
- More about homework: Even when homework is well-written, reading and grading it is very time-consuming and physically difficult for your instructor. Please do not make this process more burdensome than it intrinsically needs to be. So:
- The homework you hand in must be neat, and must either be typed (in which case LaTeX is preferred) or written in pen (not pencil!). Please do not turn in homework that is messy or that has anything that's been erased and written over (or written over without erasing), making it harder to read. Anything that is difficult for me to read will be returned to you ungraded.
- Use WIDE margins (left, right, top, AND bottom), and leave plenty of space for me to write comments. Do not expect anything to be graded that runs down to the bottom of the page or comes close to any other border of the page.
- Use plain, white, unlined, printer paper with no holes. A package of 500 sheets does not cost much, and should be more than enough for the whole semester (probably the whole year). Do not use any other type of paper (e.g. notebook paper or looseleaf paper), and do not appropriate a pack of paper from one of the department's printer rooms.
- Staple the sheets together in the upper left-hand corner. Any other means of attachment makes more work for me. Learn how to use a stapler effectively. (This takes only seconds to learn, but appears to be unknown to most persons under 30 (40?) years old. The trick is to lean on the stapler with two hands and your weight, not squeeze it with your grip). For your pages to turn easily, orient the staple diagonally (southwest/northeast); this works much better than a staple that's parallel to the top or side of the page. The staple should be close enough to the corner that when I turn pages, nothing that you've written is obscured. Check this after you've done your stapling; I won't grade anything that I can't easily see in its entirety when I turn pages.
- Write in complete, unambiguous, grammatically correct, and correctly punctuated sentences and paragraphs, just as you would find in (most) pure math journals and textbooks. Do not hand in anything you've written hastily or haven't proof-read.
- Partial proofs. If a problem is of the form "Prove this" and you've been unable to produce a complete proof, but want to show me how far you got, tell me at the very start of the problem that your proof is not complete (before you start writing any part of your attempted proof). Do not just start writing a proof, and at some point say "This is as far as I got." Otherwise, when I start reading I will assume that you think you've written a complete and correct proof, and spend too long thinking about, and writing comments on, false statements and approaches or steps that were doomed to go nowhere.
To me the following points are self-evident, and I apologize to anyone who agrees and is offended by my stating them. But I've learned through experience that I need to say them explicitly, even in 6000-level classes:
- I assign homework problems because I want you to figure them out, not go xson a treasure-hunt through the literature (offline or online). If I limit myself to assigning problems that I think are unlikely to have solutions or hints somewhere in some book or online resource, you will not be getting the best education I can give you. When I know that something is a worthwhile problem for you to work on, and even struggle with, I don't want to have to worry about whether a solution (in part or in whole) exists in some textbook or online resource.
That does not mean you are forbidden ever to look at textbooks or online resources. But solutions to hand-in homework problems should be worked out on your own. If you find yourself looking at a textbook or online resource while you are writing up a solution, that solution is not your own.
- You should first try all the problems yourself (alone). After attempting the problems, you may brainstorm with other students in the class for general ideas, but you may not completely work out hand-in problems together. You are also not permitted to split the workload with other students, with each student in a group writing up some solutions that all group-members hand in, or that all group-members work from in writing up what they're going to hand in.
Goals of course: For the student to master the course-content.