So I'm sure the myriad of bad reviews are going to come in about this class but I wholeheartedly disagree with the majority of them PROS: 1) you get to learn from a world-class mathematician (as in, the mathematician of his generation) 2) he's quite enamored with the subject so he tries to get other people excited about it too 3) he's funny and enjoys having class participation so he's very encouraging in class 4) you learn how to compute some really difficult derivatives, integrals, metrics, Christoffel symbols, and some other cool computational mathematics 5) You cover a lot of content but it's all at a reasonable depth (as in, you get a below surface level understanding of most of the content provided you just pay attention and do a problem set once in a while) CONS: 1) Professor Hamilton struggles greatly with technology 2) spends too much time on concepts we all know pretty well and kind of glosses over the more complex topics (ex: we spent a long time discussing tangents but kind of glossed over the definition of a manifold. honestly though, not that much of a con because it's pretty easy to look up what a manifold is and then go on with your day) Could be a pro or a con depending on how you feel about grading: 1) no grades except for the final exam (I didn't really mind it because I'm sure that it made some people completely blow off the class) 2) not super proof-based, if you're looking for a proof-based class, this probably isn't it. Again though, if you see a cool theorem and want proof it's not hard to google it and read up on it TLDR; if you're not self-motivated, you'll probably be SOL in this class
I had really low expectations coming in to this class given the previous reviews, but I thought I'd give it a shot since I'm really interested in differential geometry and it ended up being the only math elective I could fit in my schedule. I can at least say it wasn't as bad as I expected. In fact, I'd say the first half of the class was just like any math course and Hamilton was on the ball. He would come in and lecture, regularly put notes up on courseworks, and assign really long problem sets that I thought I got a lot out of. After the midterm however he didn't assign any more problem sets and I (and many in the class) slowly drifted away from any idea of what was going on. As far as the content: one big problem was that there was no syllabus or outline of the course, and the book he assigns is pretty random; it is very hard to make any connection between the book and his lectures. The order in which he presents topics is actually very logical, but I have not found any book that does it this way. It was like we started from the middle of most books with Reimannian metrics, and worked our way back. Most books start with curvature, which didn't come up until the end of the course. My friends and I used several references throughout the semester, and I think the book closest to Hamilton's treatment is "Elementary Differential Geometry" by Pressley, which really isn't that close. The class really isn't very proof based, I'd say the lecture was more "derivation heavy" than theorem-proof heavy, if that makes any sense. There was a lot of long formula derivation that was like a physics lecture, but it was abstract math. The notation in differential geometry is absolutely insane so taking notes became really tough. What I got out of the class was better intuition of geometry and parameterization, and a better understanding of distances and area. But everything I learned came from the first half, so I really felt like it was half a class. As has been said, Hamilton is a really cool guy, clearly a genius, and more than willing to help anyone who shows up at his office any time. But he's also really aloof; I imagine you could go to his office for an hour every week and have long and pleasant conversations about math, grad school, and Hawaiian politics, and by the end of the semester he'd probably never once ask your name. Another weird thing about him is the lazy/hard-working dynamic he has going on. On the one hand, he couldn't be bothered to correct the midterm before the end of the semester or assign any new problem sets the second half of the semester, yet he would type up really detailed notes with colorful 3D graphs in Mathematica. He would come in and say, "I typed up some notes this weekend about this topic to help you guys." and it would end up being 20 pages. And I've seen the man type on his computer, he types just with his two index fingers. And he always had free time, he'd spend 30 minutes after class answering questions. But for whatever reason he would not grade our midterms (which seriously would have taken him an hour, there were 18 people in the class). In fact we had a TA he could have given the job to, but I think Hamilton is old school so he wanted to grade all of the exams himself. As to the grading: at the beginning of the semester there was a very distinct group of kids who were clearly in this class because of the below comment "No problem sets, no exams, everyone got an A." They took no notes, spent most of the lecture texting, and were utterly shocked when Hamilton started assigning problem sets. Most, if not all, dropped by the time they realized that, yes, there would actually be a midterm. I can say with certainty that everyone did not get an A. So he, or someone, did grade the exams in the end. How he determines the grades are a mystery, but he is certainly very generous, just not on the level he may have been in the past. I don't doubt that some time in the past he may have not given any work, as he really marches to the beat of his own drummer. But if you want to take the class because you're an english lit major who did a search on culpa for "no work easy A", I wouldn't recommend it.
I read the review from the poor individual below who is reconsidering his or her career goals as a mathemetician because of taking Hamilton's class, and I had to offer encouragement (I hope that person is reading!). True, Hamilton is a great mathematician, but his talent has little or no bearing on the class because he doesn't apply himself to the task of teaching undergrads. Maybe he's used to teaching grad students who sort of know what he's talking about already and who are motivated enough to do work on their own, but he lectures at such a high level that we were completely lost, with no textbook to help guide us and no collected homework to test our understanding. There is a reason that every math class you've ever taken had regular homework assignments and at least one exam: you learn by doing, and I for one did no work for this class (besides taking notes) and got a meaningless A. Hamilton is a nice guy and an important mathematician, but when someone gives you a job to do, you should at least try to do it right, and he completely blew off his responsibilities as a teacher for this class. If you got nothing out of DiffGeo, don't blame yourself, and don't be discouraged; I got nothing out of it either, and I still plan to become a mathematician. P.S. For a real class in Differential Geometry, try W4081, Differentiable Manifolds. You may want to take Topology first though.
It's hard to explain what taking a class with Prof. Hamilton is like. On the one hand, you are given the incredible privilege of learning from one of the world's greatest living geometers. On the other hand, all that you will learn, really, is that mathematics is probably the most difficult field of inquiry to pursue, period. Coming into this class, I had planned on becoming a mathematical logician of some kind. Now, I'm not so sure that I can handle a career in mathematics at all. If you thought that "AP Calculus" was a hard class, I strongly suggest that you steer clear of Hamilton's "Differential Geometry" class. Hamilton is one of the most brilliant individuals I have ever met, and if you have an appreciation for analytical skills, in every lecture his will blow you away. Each meeting of the class is a problem-solving tour de force that few if any of the students appear to follow throughout. Indeed, the problems are sometimes so involved and complicated that even Hamilton loses his train of thought, causing him to pause for a few minutes while he attempts to work through the whole problem again in his head. Approximately twenty-four students signed up for the course initially. Today, there are fewer than ten students remaining, and two of them are well into their forties. That is a vivid reflection, I think, of the intensity of the lectures, which are often much too intense for all but the most dedicated students. I have chosen to stick around despite all the shortcomings of the course largely because this intensity excites me. I can't imagine it does the same for most others, and so I should insist that if you are not absolutely certain that this is the class for you, you MUST not take it. Let me get clearer about the problems I have with the course. First, the textbook. I have no idea why the selected textbook was chosen for this class. Hamilton indicated that the book's author is a friend of his, which probably had a lot to do with the decision to use this particular text. There is almost nothing in the book corresponding to what we have been discussing in class. Indeed, it seems that the topics we have been covering are not covered in most differential geometry textbooks. This would not be a problem if the material was not so difficult, but alas, it is. The book seems to be written for students who have already taken an introductory course on differential geometry, something that Hamilton does not expect us to have done. Second, Hamilton is not a very good lecturer. He speaks and works through problems at a rapid pace, which is hard to keep up with. He also never writes WORDS on the board â€“ he only draws figures and writes down sets of equations. The problem with this is that by the end of a lecture, I will have managed to just barely write down everything on the board, but I will not have had time to write down anything he said. As anyone who has taken an upper-level math class knows, figures and equations are unfortunately not self-explanatory, and this means that after class I will have pages and pages of meaningless equations before me, without the slightest sense of what they represent. Third and finally, Hamilton does not appear to take the class very seriously. He shows up to class several minutes late every lecture, and often leaves early because he has other, more important things to do. He has so far assigned only one problem set, but none of us have any notion when or whether he plans to collect it. I do not know whether there will be any midterms or a final, but if there is, there will probably only be a single one at this rate. That worries me, since I could do poorly on it, in which case I would be fucked. All of that said, Prof. Hamilton is, as noted above, a truly remarkable intellect, and he ought to be appreciated by those who can handle his classes. But do take all of these things into consideration when choosing your courses because you just might not find keeping up with him to be an easy task. I know I don't.
Man, there are sure a lot of whining little babies on here complaing about Dr. Hamilton. Bottom line, Hamilton is a genius who loves teaches, gave his time to students, and unlike too many other faculty in Columbia completely devoid of pretense and snobbery truly worthy of punishment. If any student did poor, it was because they were simply lazy and didn't take the steps to amend any problems with understanding material. That said, it is true that Dr. Hamilton did not follow the text or assign regular assignments. But he did follow an outline that students had, and moreoever he gave ample references of books to either purchase or take out from the library and gave what I consider to be the best advise to any student: Study on your own and do as many problems as you can stand and then some! You learn math by doing it. These accusations of him being a disorganized and unclear lecturer are simply outright lies. Some professors truly fall into that category, but not Richard Hamilton.
This man is apparently one of the most brilliant geometers in the world, and he is also a very nice guy. Put him in front of a chalkboard, however, and horror results. He never prepared any notes or xeroxed our problem sets for us, or ever handed anything back! I'm still waiting on a problem I handed in a month ago and a midterm I took at the beginning of October! His lectures are a mess. He just stands in front of the class solving equations without giving any explanation at all about what the terms he is using mean. He usually gets lost near the end of every class, and spends the last five or ten minutes trying to figure out where he made a mistake. He only successfully proves about half of the theorems he presents. And he's been teaching this course for years, not to mention that this is his field of research. The fact that he picked out such an awful waste of money for a textbook when there are so many fantastic differential geometry books out there indicates his enormous lack of interest. For some reason the math department also unleashes him on the freshman in honors math, many of whom tell me that he scared them away from math completely! This man must be stopped before he does further harm to the young minds of the future.