In short Mike Miller is the most committed, kindest, and one of the smartest professors I have had at Columbia. Although our topology class was no joke (especially the algebraic topology section), I ended up learning a lot. The problem sets and curios were something I got to look forward to because they were designed to teach you a lot of the material and ensure you actually knew how to apply these problems to new and difficult situations. My only advice would be to add some easier problems at the start of a problem set just so we can get used to new concepts and definitions since the problem sets always looked very scary at the beginning. However, Professor Miller was always available on the class discord and his office hours were always immensely useful so I never felt stressed about the problem sets. The midterm was well-structured and I ended up learning a lot from the midterm. We were given the option to choose a final project and a final exam. I found the final project to be quite a bit of work but it definitely felt like it was the better option in comparison to the final which would have a lot of algebraic topology and some manifolds. I think the algebraic topology section was a bit too fast-paced for a lot of us. I also think part of the reason why I found it more difficult was because I also had to focus on a final project on a topic not really related to algebraic topology.
The review about 30 hours per week below is no joke! Sometimes even more. But I would say he is really a great professor if you want to learn the material. Sometimes I put in more work and learn less if I have a bad professor but not in this case at least I can say
I am completely new to the topology and I study so hard to catch up with him. I did get great scores... but in order to get great score, I spend 30 hours per week on his stuff. this is ridiculous. And I am pretty sure almost all of my classmate, except one or two who had learn topology and even did some topological research before, were confused and cannot stay with him. I can tell from the last minute question that they ask before the midterm--some of the questions are fundamental (still hard for a neophyte) and they still have trouble with those trivial things(relative to what Dr. Miller give us in the HW and practice exam.) Just let you know. The class begin with 40-ish people after 2 weeks of the shopping period. but now( 10th week?) we only have 23 people in the class. think about it my friend. And by the way, one of my friends, a math major at another Ivy League U, told me that getting to homotopy and stuff in algebraic topology within half a semester is really crazy. it took him 2 and a half semester of topology course to get there. this instructor really need to reschedule his syllabus. But I had to say that he is friendly. And he is very available if you reach out. Just keep in mind. He is so smart. If you are not so sure weather you are ready to take his topology class... then it's very likely that you are not ready. Be careful. As something would cause you several hours to figure out but those results are trivial for him.
Easily the best math professor I've had at Columbia. Assigns a decent amount of work; however, is incredibly devoted to making sure students understand the material and incredibly kind. He wrote probably pages about my proof style etc in response to my midterm and gave very specific reasoning when I was wrong. Additionally, gives guided proofs that he calls "curios" that take you through some interesting implications or alternative definitions of material in class. He never checks or asks if anyone does them so they really are optional, not like "optional," but they're very interesting and rewarding. He also gives helpful visualizations during lectures and talks frequently about common practices in math, textbook choice, etc. He's incredibly available as well. At least this semester, Fall 2020, when most of us aren't on campus, he set up a discord server for the course for him to answer questions outside of Office Hours and he's incredibly responsive and helpful. It has made me feel much more comfortable to ask questions and it means when you want to ask a question outside office hours, you don't need to construct a somewhat stilted email because it's a bit of a more natural format for questions. I also highly recommend going to his office hours because he's really not at all intimidating and is very approachable and often even if your question is somewhat outside course material he'll have interesting reading suggestions, e.g. I went into talk about whether a theorem about unique limits could be more generalized (not really) and he gave me suggestions for readings about some topological properties we'd not talked about in class but were relevant to the problem and then some problems in a textbook that were also helpful. In this specific instance I learned about the "long line" which is now a favorite topological space of mine! I really cannot recommend enough Professor Miller, at least for topology. If you're a math major and you get the opportunity to take his course you really must.
I would not call this class a success. Neumann had trouble proving any non-trivial theorem successfully; he almost always failed to finish the proofs, and had to revise them later or refer us to the book. This is very disappointing, as the whole point of a math lecture is to prove theorems. At times, he attempted to present topics that are not standard to an intro topology class, for example, CW complexes, the classification of 2-surfaces, and ultrafilters. These lectures were interesting, but felt out of sequence; I was not prepared enough to get much out of them. By the time we got to algebraic topology, lecture was spent on excessive examples of graphs, their covers, and their fundamental groups. This was not very conducive to learning the material as he did not effectively connect the examples to the relevant theorems.
This was definitely one of the best classes I’ve taken at Columbia. Paul Siegel put incredible amounts of effort and enthusiasm into teaching this course and this played no small part in making the course such great experience. Professor Siegel is an excellent lecturer who presents the material at a reasonable pace with high levels of enthusiasm in addition to a good amount of involving the class. He’s very organized also does a great job answering questions without getting sidetracked. While the material can be very dense, he does a great job keeping a relatively light tone with humor and interesting asides. Additionally, he typed lecture notes that paralleled the lectures for the entire course, and these were extremely helpful. They totaled to about 50 very dense pages by the end of the semester. As the previous review mentioned, the material was related chapters 1-5 and chapter 8 of Munkres, but he also added several topics and examples of his own choice that were not in the textbook which greatly enriched the course. It was clear that the put so much thought into what he covered into this class, which was a refreshing departure from professors who essentially read from the assigned textbook during lectures. The same creativity extended to the choice of homework problems, but this often meant that the homework assignments could be very difficult. He definitely went out of his way to choose interesting and relevant problems, and many of these were great. At times he was a bit too ambitious, but he acknowledged this after the fact and made adjustments for future homework assignments. He also wrote great solutions for all of the homework assignments. The exams were very straightforward and he made it clear that he didn't want them to be stressful. We were given a good idea of what to expect from each one, and he wrote practice exams to help us prepare. The problems he expected us to solve for the first time were orders of magnitude easier than the homework problems, and there were also problems from the previous homework on the exams. I only have one complaint, and that’s the this was the only upper-level math class he has taught here! I know that many of us would've love to take more classes with him in the future.
Paul is a great guy whose enthusiasm both for teaching and the material is very obvious. He moves through lecture at a pretty reasonable pace and often asks the class questions about what they think the next step in a proof might be, etc. He is always well-prepared and very organized -- over the course of the semester he typed up 50 pages of notes (updated at least weekly) that detailed nearly everything done in class (complete with proofs, examples, and exercises) and these were enormously helpful. That said, we covered a lot of material (I believe approximately Chapters 1-5, 8) in Munkres and I found the homework to be quite challenging, especially some of the Algebraic Topology at the end. Overall, though this class was probably harder and more work than I'd expected after hearing people who'd taken it previously with other professors, it was very well done and Siegel is very helpful and responsive both via email and in office hours (where he will often go through the idea of the proofs of hw problems).
i have taken 5 classes with thaddeus (too bad he went on sabbatical senior year, otherwise i might have taken more). these five semesters of "thaddeology" have helped me build a very rigorous and exciting foundation for studying higher math, and also introduced me to one particular way of thinking and talking about math. thaddeus is a very nice guy (i'm writing this because there seem to be many people who find him a little intimidating, but don't worry), very willing to answer your questions, and lectures very clearly. he encourages class participation, asking questions to the audience, and creating a question-friendly atmosphere. he lectures with notes, but not from a book, and organizes his presentations very well so that they are not that hard to follow. he is very available during office hours: i remember him answering my silly set-theory questions freshman year with the same patience as the (probably equally stupid) questions about serre duality or sheaf cohomology later on. honors 3+4: his lectures were vey engaging, and this definitely turned several students into math majors. thaddeus is a very eloquent speaker, and this actually helped my english as much as lithum did. although the lectures were fast (or so we thought), they were very funny at times: at the end of first semester, he read poetry to us, and at the end of second semester, he gave away fig newtons and leibniz cookies, to celebrate the inventors of calculus. i would say he is surely the best choice if you would like to learn calculus. topology: on we went after honors, to enjoy more thaddeo-mathematical theatre. the original plan was to take differential topology as well in the spring, but for some god-awfully trivial reason that class was cancelled. the one semester of basic topology was a very nice undergraduate class, taught at a faster pace than honors. professor thaddeus seems to work very hard (just walk by the department at 2 am and look at which offices are still lit), which suggests that he wants his students to do the same. i remember spending a lot of time on the homework assignments, maybe too much. fewer problems might have given me more time to digest the material, but then worrying about finishing homeworks is a sign of undergraduate immaturity. although the material isn't that exciting (point-set topology), this class was probably the best organized one. i have a friend who is planning on typing up the lecture notes. thus, it is probably a very good idea to take an (advanced) undergraduate course with thaddeus. commutative algebra: the journey continued, and the few undergraduates left tackled their first graduate-level class and (of course) had to struggle a lot. thaddeus' lectures were still very clear, but also very detail-heavy (maybe because of all the undergraduates there), which made some of us lose focus on the conceptual understanding of the material. he covered a lot of topics, and expected very hard work from us. but commutative algebra is a subject that never ends - maybe realizing this is far more important than knowing the few selected topics that this course taught. (i have seen the final for the same class a year later with jacquet, and it's much, much easier than ours.) algebraic geometry: only very few of the undergraduates from honors continued all the way to this course. algebraic geometry made me work harder than for most of the undergraduate courses combined. following the lectures was not too hard as usual, but the homeworks were a huge workload. during the last month, thaddeus started (finally) to not focus too much on the details, and left out various proofs. personal conclusion about grad courses: i later took other graduate courses, which emphasized the global picture of the material, and didn't focus on the local details ("the proof can be found in any book on[subject name]"). they also didn't include any homeworks, and i learned a lot more in those courses (basically from studying all day in butler - on many days, and more importantly, talking with grad students). my guess is that now, i would prefer these grad courses over thaddeus-style ones. but i also feel that taking two grad classes with thaddeus made me mathematically very mature. this maturity was absolutely necessary to get into a rhythm of properly working through, and truly beginning to grasp unknown mathematical theories and thoughts. if you feel you lack this maturity, taking a grad course with thaddeus wouldn't be a bad idea. he truly is a very amazing teacher, and columbia is lucky to have him. finally, i know that math professors read culpa a lot. thus, i would like to thank him for the very enjoyable five-semester journey!
Manolescu is really an excellent young Math teacher. His lectures were not dry (i.e. only definitions, theorems and proofs like a text book) but included many excellent examples and insight. He really knows his stuff and speaks excellent english. Furthermore, he was always well prepared in class and helpful in office hrs.
Michael Thaddeus is a terrific speaker. He gives some of the most well- presented, organized, simple, and interesting lectures on terrific topics. There is one problem with his classes, which while not minor, is not insurmountable, he doesn't deal as well with questions from students as many professors. He will always answer questions, but he can be intimidating with his responses and the aura he presents. Even so, I would still highly reccommend taking classes with Prof. Thaddeus, as he gives some of the best classes I have taken.