Like the other reviewer, I took the both semesters of this class online, given the COVID-19 pandemic. For context, this was my first time taking a proof-based math class, and I had zero prior exposure to both linear algebra and multivariable calculus. I'd had nothing like that in high school (just the necessary scores to get into this class) and thought this would be a way for me to pressure-test whether or not I wanted to, or even could, be a math major. Overall, I think Warner was a solid professor, though (at least in my case) it was definitely worth it to take this class. For someone with my background (i.e. not very much), I could roughly follow what was going on in most lectures but needed to dedicate a substantial amount of time to develop more intuition. I dedicated a lot of time each weekend, especially in Honors A and the first few weeks of Honors B when I was still getting used to proof-writing and lin alg, in asking questions in class/office hours/via email to Warner (who makes himself SUPER accessible - literally I've sent him like 6 emails at one point and he always replied), reaching out to and discussing proofs with my more experienced and knowledgeable classmates, and just thinking very hard to make sure I understood the theorems from lecture and the HW readings and really made my best effort to get a perfect score on every pset. Like one past reviewer said, the "math geniuses" of the class were always super kind and willing to help, running review sessions to discuss psets, so that was a super valuable additional help and source for reassurance alongside Warner's own patience with answering your questions. I'll also just say that he structures lectures quite well, usually offering a motivation for what we're doing or describing the proof before getting into the mechanics of it, which is a great way to get context for what he is doing and makes it easier to follow along. The only part of the 1 year course where I struggled to follow lectures was during the last three weeks of Honors B, which was when we did multi (the semester had already been compressed bc of irregularities in the 2020-21 school year so he was rushing through a LOT of material and often not doing the proofs for a lot of it in class), and I think even after having finished I'll want to revisit that to deepen my intuition. Also hopefully for anyone taking this after this year, that won't be as much of an issue, providing that the CU admins don't create such an abbreviated spring sem. In terms of exam prep (note that this might change in the post-COVID world): He gives us practice exam problems before each midterm and finals (all exams open-book b/c of COVID, we had a timed, regular midterm during Honors A, and two short timed MCQ midterms in Honors B, and a long take-home final for both semesters). I did the midterm ones and found they were at-level for the Honors A midterm; for the Honors B midterms, not *exactly* because the exams were in a multiple choice format while the questions were just in a regular format (the first time he was trying this out), but still similar difficulty level. I also went through and synthesized all my lecture notes and pset results (which btw are SUPER useful - we prove a lot of important results in HW that you may want to cite in exams) for each midterm, as a way to review and also have a good reference since all midterms were timed, even if they were open-book. For the finals, bc they were take-homes, I'd just skim the final review psets and spent most of the week he gave us just working through the actual final exam. The Honors B final was harder (and had a lower average) than the Honors A one, but in both cases it definitely helped to plan WELL in advance during the week he gave us to finish it so one wasn't trying to get through a bunch of hard math problems at the last minute. And again, he was kind enough to answer questions about what results we could directly cite/what we had to prove, what certain definitions meant, typos, etc. The final format both times was 10 T/F questions, 5 "problems" (just regular, pset length problems), and a choice of 1 of 2 "long problems" which extend something we did in class but are a new subject material. Grades: In the end, somewhat to my surprise (in a good way, I guess?), I got an A for both semesters of the sequence (with 94% weighted average overall in honors A and a 90% weighted average overall in honors B). The grade is 40% homework (he drops your lowest score of all the weekly psets and then averages the rest), 40% final, and 20% midterm. I've left with slightly less sanity and a good deal of sleep deprivation, but with a genuine interest in continuing to study math instilled by Prof Warner's interesting lectures and overall consideration for all of us this year, especially given the pandemic circumstances. Which is all to say, as someone who is no "math genius" coming to Columbia with fairly average math major credentials, this was hard, but doable with the support of Prof Warner and my classmates and a lot of dedication, stubbornness to "stick with it," and hard work, and most certainly worth it for me.
Keep in mind as I write this review that I took this class online, so the quality of the course may change depending on this. That being said... many of the reviews beforehand kind of hyped-up Prof Warner and, while I think he seems like a cool guy, sometimes his classes were a little hard to follow. The class was kind of just him introducing new topics and then writing the proofs for them. I've never taken a proof-based math class, so I don't have anything to base this off of, so I imagine that there isn't much else that you can do other than this. At the same time, I feel like there could have been a little more advice given on certain types of proofs that we may encounter (such as epsilon-delta proofs). I also feel like he could've drawn way way more pictures (which he did start to do towards the end). I will say that I've learned a lot from this class, but I think a lot of it came for the work that I put into the homeworks. However, I know that on my side, I didn't really interact with him much. I wish, for example, that I had gone to his office hours way more often, but I ruined my sleep schedule beyond repair this semester and couldn't wake up before 12. Now that I've laid out the "cons," Prof Warner is DEFINITELY NOT the worst professor you'll ever have. Yes, his lectures can be sometimes hard to follow, especially at the start, but I do feel like he responds well to criticism (which is the reason why I think he started to draw more photos). I can also imagine that he was teaching this class at a very rough time, so this is why I emphasize that I can only come from the perspective of someone who's taken his class during the quarantine.
I took this class online and it was very manageable. Professor Warner's method of teaching is very clear and I have no complaints. The exams were open note which I greatly appreciated and it made the class a lot easier to do well in. He also drops one homework assignment. I recommend this class both if you want to learn calc iii thoroughly but also for a GPA boost! Also, I'm taking this online so if you take this in person it might be different (especially about the GPA boost part) however Professor Warner is a really great teacher so I still recommend him for an in-person class.
It's not an exaggeration to say Evan's class changed my life. He talks about linear algebra and multivariable calculus in a rigorous yet intuitive way. He lectures for the entire 75 minutes without stopping except for questions, providing definitions and proving theorems. The class goes fast. The material is hard. But with dedication, it's manageable even for someone like me who is mediocre at math. The class is filled with geniuses, but most of the time they're kind and willing to help their classmates. There is a strong sense of camaraderie. Get a few friends in the class to work together on homework. Taking this class with Warner will make you appreciate math in a new way if you haven't really been exposed to pure math and proofs before. This is not for an easy A. This is for a hard-earned introduction to higher mathematics through highly rigorous calculus and linear algebra.
Evan is exactly what you picture when you think of a young genius math professor. He is funny seemingly by accident, but his lectures are of extremely high quality. He goes quite fast sometimes but is always willing to stop and clear things up. He speaks with extreme mathematical precision and rigor, so don't expect many concrete analogies or examples or alternate ways of understanding something -- he wants you to read and understand mathematical proofs in all their purity! The course is very difficult. There's no going around that. You cannot get by with Chegg or constantly going to TA hours (though it helps) -- you really need to put in the time and effort to understand the material and complete the homework by your own hand. It is worth mentioning that my experience in this class featured a high degree of camaraderie among the students; we helped each other out on homework and while studying, which made an otherwise impenetrable class full of difficult material instead quite bearable. You must make some friends in this class to succeed unless you're a genius already. Luckily, a significant portion of your classmates will be geniuses, and they're usually humble enough to help out.
Solid professor. Here are some of my remarks about him. PROS: - Evan has an excellent teaching personality. His lectures are engaging and often funny, and he replies to email with questions/clarifications in the most sincere way possible. Generally he is a friendly person who is truly devoted to teaching and will make your learning experience enjoyable. - His homework are not just straightforward applications of the results we’ve learned in class, but are constructed to deepen the understanding of the materials. There was a decent amount of inquiry on relevant fields (e.g. a portion of homework in the second semester was devoted to homological algebra that wasn’t directly covered in the class), and many problems required a lot of thinking. I admit that some people might like it more easygoing, but I truly find myself more enriched with the material by the homework. - He constructed the lectures so that he will show a core result on the subject (e.g. the nth homotopy group of S^n is Z), and progress through the material as if we are preparing the “tools” to prove it. It made me anticipate what will be coming next as the class progressed, and to have a better view on the subject. - He speaks pretty well and his handwriting on boards is one of the best I’ve seen. CONS: - His class is a bit fast-paced, which made some students to lose track easily in the lectures. This is understandable though, as algebraic topology is not an easy course and students have a relatively higher degree of responsibility to go through the materials on their own. But I would have preferred to omit some convoluted proofs of some theorems and leave them to students to read on their own, as we only went through the subject of cohomology in the last 30 minutes of the last class. - Occasional mistakes in lectures and in homework. (e.g. He told us the slightly wrong definition of a CW subcomplex / It was stated in class that the inclusion A → X is a cofibration if and only if (X, A) is a good pair (in Hatcher’s definition), but it turned out only the if direction holds.) Eventually students will point them out and Evan will inform the entire class by email if there was a mistake, but try to be vigilant as much as possible while going through all the class materials (as you should be when you are studying mathematics!). Also often he got stuck in the proofs during class, which wasted some time. Overall, if you are really willing to learn (a lot) and/or looking up for challenges, it’s definitely a worthwhile class to take. Personally, I will take anything he offers anytime.