Kui Ren is a fantastic professor. He's super nice, a great teacher, down-to-earth (albeit a bit shy), and has an all-around good introductory complex variables course. I personally feel he's one of the best professors at the school, especially if you are not a quick learner. He is readily available for office hours and explains every single step so thoroughly but efficiently that he could teach a toddler complex variables. The two classes, MATH3007 and APMA4204, are equivalent and the math department views them that way as well. When I took the class, Kui Ren used Gamelin Complex Variables and Fokas Complex Variables. It's not Honors Complex (Ahlfors), if that's what you're looking for. The grading is really nice. He's a tad harsh on the 1st midterm, but he generously re-weights tests and homework and doesn't try to hurt you. The workload is one problem set per week and Kui Ren covers both theory and applications equally without compromising either. He uploads lecture notes for each class online and posts worked-out solutions to all homework and midterms, plus sample midterms. The one downside is he talks really slowly, says "um" a lot and swears during class when he makes a mistake or says something in a way he deems to convoluted. For some people it can seem like he doesn't know what he's doing, and I saw people judged him from that impression.
DO NOT TAKE THIS CLASS WITH LETOURNEAU. it seems like a really cool subject, and maybe letourneau would have been good at teaching something else, or with a few more years under his belt; but for now, do not take functions of a complex variable with letourneau. i got an A in the class, so i say this without any spite: this was the worst-taught class i have ever taken. before i give my opinion on why the class did not suit me, i can offer 2 concrete reasons why it is a class that you should avoid at all costs: 1. the class does not cover all of the material that you need to learn in the subject, and the topics that were left out were the most important ones in the course (fourier, laplace, and schwarz-christoffel). the bottom line is, you will either need to go learn these topics on your own or take another class in complex analysis if you actually want to use any of the most important applications of the subject. 2. the professor will not match the amount of work you put in (unless, of course, you put in little to no work). letourneau was completely unprepared in lecture. he had very few office hours and frequently cancelled or rescheduled them. there were significant errors on every posted homework assignment, and they often went uncorrected until just 2 or 3 days before the assignment was due. there were frequent errors on the posted solutions. homeworks were not returned until over 2 weeks after they were handed in, and there were frequent grading errors. there were errors on the exams as well, and there were often crucial errors in the lecture notes. on a more subjective note, i found it unacceptable the way he handled the lectures. for one, the lectures didn't prepare you for the exams. in fact, in lectures, he deliberately misled you about the exams when asked specific questions about their content. the exams were hard, but they would have been much more manageable with the exact same questions if we had been prepared in class for those problem types. even if the exams had been closely related to what was in the lecture notes, i still would have been very upset with the way the lectures were conducted. i'm sure there are plenty of professors who don't care enough to prepare thoroughly, but i would hope that they at least follow the book, or use the notes of a previous instructor. i don't know if it was foolishness, arrogance, or complete incompetence, but letourneau seemed to think he could grab a few theorems from book sections and then structure a 2.5 hour lecture around them, on the fly, from his own subject knowledge. he would often start an example, spend a few minutes struggling through it, and then say: "oh wait, this actually won't work," and then cross it all out and think of a new example. another laughable element was his "proof sketch" method, which consisted of a bunch of disjointed mathematical phrases, with inconsistent notation, scattered across 3 or 4 blackboards and connected by a tangle of long, squiggly arrows intended to reorganize his out-of-order thought train. it was as if we were the guinea pigs for a mock, live-action improv textbook that will never be written. in retrospect, i highly regret attending the lectures. also, the ta (jeff) was even less prepared than the professor. i went to 2 of his office hours over the course of the semester, one at the beginning and one near the end. both times, every question that was asked, directly from the homework, struck jeff as the first time he had ever seen, heard, or considered it, and the students sat around while he tried (and often failed), for an inordinate amount of time, to figure out how to approach the problem. the only difference between the 2 sessions i attended was that the first one, which was the first or second meeting of the semester, was well-attended, whereas the second one was almost empty. the only other people there by the end of the semester were a few students who clearly went every week just to exchange answers and work through difficult problems together (without the ta's help). this class was an abomination, and i'm horribly disappointed that columbia allows this kind of thing to happen. while it stood apart as the worst class i've ever taken, it wasn't alone in the realm of terribly conducted columbia math courses, and this pattern clearly reflects a lack of oversight on the department's part. at least when it comes to math, there seem to be no resources whatsoever devoted to ensuring teacher quality, and especially ta quality. it is understandable for an elite university to place greater importance on the research contributions of faculty than on their teaching contributions, but not all professors have to teach! isn't that what adjuncts and tenures are for? why do they give assistant professors and professorship fellows any classes at all? they've got plenty of veterans who've been doing this for ages, and plenty of money to fill in the gaps by bringing in adjuncts who can be hired specifically for their teaching ability. i'm not saying columbia should get rid of letourneau, because i'm sure whatever scholarly contributions he makes are worth his salary and office space, but that doesn't mean the university needs to plague students with his horrible teaching and neglectful class management. poor form, columbia, poor form.
So, I thought this class started off as a train wreck. It was his first semester teaching and it showed. However, he was extremely responsive throughout the semester. He usually replied to questions on piazza within an hour. He told us to let him know if there's anything we want to do to improve the course. He started off with it being very proof heavy and with long homeworks. He responded to student concerns and while the theoretical base for the class never really changed, he did less proofs and shorter homeworks as time went along. Unfortunately, while I found the course incredibly tedious at first (and agreed with some of the concerns), I found it highly inappropriate that people just completely crapped on him on piazza. I feel that can't be a great morale booster for a first time professor who is NOT THAT BAD. His lectures are fairly decent. His handwriting takes a bit to get used to, but I now know what to expect (but he still writes super tiny so sit up close). I think it is more worthwhile for you to listen intently and try to understand than take notes. He teaches pretty much from the textbook and tells you whenever what he is writing isn't in the text. I used to take notes and not understand, but felt like it got better when I tried to just listen. He does do his own examples and notes, however, so sometimes in the middle of a proof or example he'll just stand and think for a minute to see if what he is saying is right. Sometimes, he'll decide to scrap the example and move on. I never really saw him phased by a question, he could almost always explain it, but he sometimes asked students to come to discuss it during the break (he gives a very generous 15-30 minute break in the 2.5 hour lecture). He splits the homework and exams up for undergrads and grad students. Homework started off with 17 questions, then 7, then 5, usually 3 for both grad and undergrad, then 2 "practical" (computation) for undergrads and 2 "theoretical" (proof) for grad. Exams work the same way, except 2 are for both grad and undergrad, then 1 of each kind. Bottom line: There will be people who bitch who didn't put time into this class. Yes, you'll need to put time in. A lot of it. I highly recommend reading Complex Variables by Flanigan in addition to the quite good assigned book. It took a while for everything to click for me, but when it did, the course became much easier. I did well on (most) of the homeworks and I think I did very well on the final. Completely bombed the midterm. Due to his grading scheme, though, it won't hurt me. Most importantly, I came out feeling like I understood the material very well, though a lot of it was due to my own work. Would I take him again? Yeah, probably. The only thing holding me back was the exams and the final definitely tested to see if you knew the concepts more than how to compute hard math, while the midterm didn't (so he learned). Also, his TAs need to be on the same level. My homework grades varied wildly based on who graded it. I would work on it with 2 different people, we'd all hand in the same exact homework and all get different grades.
Polvani is a great teacher who made complex variables relatively interesting and bearable during the 2.5 hours of weekly lecture, considering that it's late afternoon, when everybody is mentally and physically drained already. He goes through most of the material very thoroughly, minimizing students' pain, by presenting a healthy mix of derivations and applications (+ occasional tangents). He has a cute, but comprehensible, accent and is generally very receptive to questions/comments/corrections to mistakes. He will actually slow down his pace and reiterate certain things depending on people's requests. Make sure to come to class when he covers CHAPTER 6; otherwise, be prepared to spend a billion hours deciphering the hieroglyphics in the book and still getting confused while doing the long, torturous homework assignment over Thanksgiving break (i went to lecture, but still spent most of the holiday doing the HW and suffering from the messy algebra). Be warned that the exams are not simple at all - you need to really know the stuff cold so that time wouldn't be an issue. STUDY, STUDY, and STUDY hard - practice problems, but keep in mind though that understanding of concepts, rather than the ability to perform robotic-like, mindless calculations, is key (think critically!!!). Also, be heedful of the questions on branch cuts and Mobius transformations -> Lorenzo loves these subjects, but did not provide enough examples, and to make it worse, the text we used, Saff Snider Fundamentals of Complex Analysis, was useless for these topics. Enjoyable experience overall, to say the least. Very polite, humble, but funny man who respects each student. Only complaint is that we did not get our final course grade until January.
cool guy, cool ish class. polvani is really, REALLY exuberant about complex variables. he definitely makes the 2.5 hour long class less painful to sit through, and it usually pays off to go to class. the material is a little daunting at first, but the calculations are always simple and solutions are very straight forward. (and there's a solutions manual always floating around that i'm sure you can find.) this class is required for applied math majors, but i think that other majors could definitely enjoy it as well. you really don't need to know any super advanced math for it anyway.
This past semester, I only went to two classes regularly. One was a lab class so it was mandatory, and the other was this. Professor Polvani makes his weekly 2-hour lectures a breeze; he is so enthusiastic about the subject, it makes you get involved too. He is very thorough in his proofs and examples, pausing often to ask for questions or add quirky remarks. In short, Polvani is awesome. The homework generally took about 4 hours each for me although there was one that took more than 10 hours, so just look out when he covers chapter six. Otherwise, I found doing the homework really helpful since the exams covered the same material. The midterm itself was not too difficult since the material covered to that point is largely basic. The final was harder, but not too much so. The curve is not bad, though there were only two A+'s.
By far one of my favorite professors at Columbia. He is completely approachable and very helpful. Also he might be one of the nicest people I have ever met, always encouraging questions during class even though some were really stupid. He is a little scattered and tends to make errors with his calculations on the board but always encourages students to correct him. He makes a difficult subject easy to understand and fun. The only problem I had with the course was that it was a 3 hour lecture on Monday nights. Since I came straight from another class it made for a long day, but he gave the class a break. The lectures aren't mandatory but you should stay. They are usually helpful for an understanding of the subject before attempting the homework and because he is awesome.
I never thought I'd look forward to my last class of the week -- a TWO AND A HALF HOUR long class on Thursday evenings from 4.10-6.40pm. But Prof. Polvani (he insisted we call him Lorenzo) made complex variable come alive for those 2 hours! Lorenzo is a fantastic professor; it was his first time teaching complex variables but he was absolutely fluent and competent. He answered each and every question the students posed to him -- and by Thursday evenings brains are pretty fried -- and never revealed a sense of impatience or annoyance. He took us through various theories, stopping for a moment after a parcitularly important one to tell us to "admire the beauty of this tango of equalities!" and often his notes superceded any use of the textbook. He has fantastic board management, and gives a generous 15 minute break in the middle of the 2.5 hours. He is always available to answer questions, and treats the students as they should be treated -- with respect, enthusiasm, attention. He's one of the most amazing professors I've had thus far at Columbia; I strongly recommend a class with him!