Introduction to Modern Analysis

Apr 2021

He's a nice guy, but a very slow teacher and doesn't cover much material quickly and crams the rest in at the end of the semester. Not good for learning when you have 1.5 weeks to learn measure theory in Modern Analysis 2. His first exam had a mean of 75%, 2nd exam a mean of 50%, and the final's mean was 90% for some reason. I failed the first midterm, did well on the last two exams, and still got a B so he's not out to destroy your GPA. If I hadn't failed that 1st exam I'd probably have gotten a B+ or an A-, if that's the information you want. I didn't find his lectures very enlightening or helpful since he spends most of the time drawing pictures and handwaving the math, so you need to learn the material on your own. His office hours weren't very helpful for learning. When I'd ask why he took off 8/10 points in a proof he'd tell me "your logic was there and it's the right idea but your answer was messy." When you ask some professors how to improve, they'll provide techniques or ways of thinking for attacking problems, or tell you when you were doing something wrong. Hui Yu would never elaborate beyond "see when Theorems break as you get rid of assumptions" or refer to his solution when I asked, so I had to find ways to improve my proof-writing on my own. It's still shoddy, honestly. One time I wrote a "less than or equal to" inequality instead of a strict inequality and he took off 19/20 points for the problem. I asked him why after seeing the solution which was almost identical to my solution, and he said "the rest was correct but the inequality was wrong." I imagine if you're a perfect student or a good student he's a generous grader, but if you're just an average student like me prepare to have a hard time academically even if you love the material. He's much nicer than some of the other professors here.

Dec 2018

Dr. Yu is a very solid professor. Analysis is inevitably a hard subject and I think that Dr. Yu makes it as painless as possible. He's a decent lecturer.His lectures are generally regurgitation of the textbook's proofs (baby Rudin) with some pretty helpful commentary and details sprinkled in. Even though he stays very close to the textbook, I would still recommend going to class. Dr.Yu provides the motivation and bigger picture that is notoriously absent in Rudin. One downside (or upside) is that Dr. Yu moves very slowly and he doesn't cover a good chunk of the material in both Analysis I and II. He's also an incredibly nice guy and really helpful during office hours. In terms of exams and grades, he's extremely generous. The tests are very fair. It should be possible to get above a 90+ on each of them without being a math genius if you understand the material. That being said, it is very hard to understand the material, so the averages are probably going to be in the 60-75 range. There are always a few extremely easy questions that are practically reproving a theorem in the book, and then a few trickier ones that require some thought but aren't crazy hard. The curve is very nice and he factors in how much effort you show in the class. For Analysis I, if you go to class, put in the effort, go to office hours, and do around average on the tests you'll get an A minus. In analysis II the class shrunk to around 12 people and I suspect he gave everybody an A/A-. Now a few random notes about Dr.Yu that you should note. First is that the homeworks are completely unrelated to the tests. Dr. Yu pretty much assigns problems from Rudin at random so the homeworks might take anywhere from 10 minutes to 15 hours. Of course you can always look up the solutions online. Second is that Dr. Yu grades tests exceptionally fast. The Analysis II final ended at 4:00, he had them up by 4:58. The downside of this is that he grades so fast that he doesn't read your proofs carefully and your grade is practically a random variable. that being said he's very generous with partial credit. So if you write something half coherent he'll give you half credit on a problem. So no matter how clueless you are on a proof write something and throw some epsilons and deltas in there for good measure. Overall, Dr. Yu is a great professor to have for Analysis. You'll learn alot and your grade won't suffer.

Jan 2013

I agree with both of the previous reviews. Fabio Nironi has no idea how to teach a class. After observing other analysis classes, there is a clear difference between the teaching those professors gave and that of Nironi. They helped, he recited. De Silva was amazing, Nironi was terrible. Take it with her or whoever else. The last review talked about how Nironi made the midterms 20% and 30% and the final 50%, and I agree it's unjust. Yeah maybe people cheated, but that's the real world and Nironi, after all that schooling, should learn to deal with it. That's why the homework is only 20%, if people didn't cheat the homework would be 100% rather than putting 80% of the grade on random sit-ins prone to error. That's not to say cheating is a good thing. I'm sure the class would have loved to have done the problem sets, but you see other professors assign 4-5, maybe 6 problems a week from Rudin. And those problem sets take forever. Nironi assigns 10, and they're mostly the most obnoxious problems Rudin provides. The class almost felt like this guys revenge on people who didn't appreciate math like he did. I mean the proof of Rolle's theorem, probably one of the easiest questions on the exam, still takes a fair amount of ingenuity to pull out of thin air. He has no idea how to write a test because the problems he assigns for homework can take 3 hours to hammer out in your head, and the questions on the exam were harder than those on the homework. It's not like he went out of his way to help anyone. He didn't host review sessions, answer questions well, post any notes after like the second week, collect the problem sets (we had to drop them in his drop box on our own time), or even tell us when he assigned homework or when it was due. His method was simple, Assign the first three questions from the last chapter and the last 7 (very annoying ones) from the previous chapter. He did squat for this class. Office hours were pointless or absent, and you'd never see your problem sets even if you attended because he tended to lose them in his train wreck of an office. The only point where I disagree with the previous reviews is where they claim he cares or is even a good person. He's lazy (he desperately ran through half the material when he realized he was far behind) but he wants to be brutal out of some misguided idea this will make us all love and be better at math. His worthless presentations and claims of A+'s are just excuses for his apathy. I'm dead serious, stay away. No matter how well you'd do with him, you'd do better with someone else teaching.

Dec 2012

I agree with the previous post. To sum it up, I think Nironi's teaching style punishes, rather than encourages, students who are learning the subject for the first time; for example, from personal experience, he doesn't give much partial credit on exams, if it all. Finally, at the last second (way after finals), he decides to completely disregard the homework portion of the grade (20% of the final grade), saying that most students copied from online sources (which I assume is b/c while most students did relatively well on the homeworks, only a handful did well on the exams). I don't agree with this in three aspects: 1) homework problems are much easier than the ones on the exams b/c they're open book and there's no time constraint 2) even if students did find solutions online, it's not that much different than looking up solutions in the student manuals, which is available for many math classes 3) while some may have cheated, some may have actually gotten help from other students or tutors. To drop 20% of the final grade and make the final exam worth 50% after the exam had already been taken, I think it's unfair. Nironi is a likable professor but his teaching style discourages students who are learning the subject for the first time. The material as difficult as it is, on top of that, with no homework grade and no partial credits on exams and an exam format that favors students who have a strong background in the subject (exams are structured in a way that there are more questions than you have time to answer, which naturally favors students that are familiar with the subject from the start), his teaching method is extremely unforgiving for newcomers. I'd recommend taking this class with some other professor.

Dec 2012

Unfortunately previous reviews for Nironi also hold in proof-based classes. I'd like to say he's very dedicated to the subject and looks to share that inspiration, but I feel there are some problems with that. First of all he quit posting homework solutions after the fourth problem set, and when people requested more solutions he emailed the class and told us to Google them. In fact, he didn't even hand back our homework, you had to be lucky enough to find him in his office. Second his office hours were a real pain, immediately after the class at a time where almost anyone would have another class to attend, and he wasn't particularly personable when he was available. of all he pretty much just lectures straight from the textbook, it even sounds as if he's reading to you. Some people find it a virtue when a professor follows the textbook closely, but that wasn't really the case. Rudin's exposition is so much better than most people didn't really take any notes or pay much attention at all to his lectures, which are particularly hard to follow anyways. Third, he mostly, with some notable exceptions, just lectures straight from the textbook, it even sounds as if he's reading to you. Some people find it a virtue when a professor follows the textbook closely, but that wasn't really the case. Rudin's exposition is so much better than most people didn't really take any notes or pay much attention at all to his lectures, which are particularly hard to follow anyways. As for those exceptions, he adds in a ton of topology, far past what Rudin found necessary. Some of the definitions in Rudin are esoteric but that's because they make the material more fluid. Finally, while on the note of topology, quite frankly if you haven't taken topology and algebra don't take analysis with Nironi. He really presumes you know a lot of mathematics beforehand even if he claims this is where everyone should start in college. I was in topology for about half the semester and I could clearly see that the handful of us had a clear advantage over those who didn't know what all his additional topology was going on about. This was particularly obnoxious on the first midterm and to some extent the second midterm where half the material was literally topology, which is fine only as far as the second chapter is concerned, not when he expects you to pull the product/box/discrete topology out of some spark of innovation and work with it. Other notes, his exams were pretty much a free for all, especially the first one wherein he just put a ton of questions without even knowing how many points there were total and told us to run with it. I guess that's fine if you want to do a ton of stuff on an exam, but he wound up curving the average, a 48 or so out of 200 something to a C. There was a clear distribution based on previous background. Second midterm was alright, and the final was a pain. I guess he also realized at the second to- or last week of the semester that he was pretty far behind, so he pretty much crammed differentiation and integration into a week or two and made a problem set due Monday (we were a Tuesday Thursday class) without any announcement (I was particularly unhappy about this since I turned it in unknowingly on Tuesday and have yet to get a grade for it) and then he jammed both the integration and differentiation into a problem set due on study week. He tries to be generous with presentations and extra credit. I did a presentation, its 5 points and given his curves Im not quite sure how valuable that was, if those points are even for tests. The extra credit assignment I handed in twice and he never noticed I guess, I'd have given it to him at office hours but, you know... Let's put it this way: If you have a lot of math background, you'll kill in this class. If not, stay out. Either way, you'll probably enjoy it a lot better with Savin, De Silva, Masdeu, whoever else is teaching, and you'll have a reason to come to class. I really think he's a nice guy and he probably means well, he just needs to figure out what he's doing. Maybe disregard this review when it turns brown/green with the "5 years ago" statement, he needs to grow has a teacher.

Dec 2009

Pinkham is absolutely horrible! He is by far one of the worst professors I've had at Columbia. The subject matter was the potential to be very fascination, but Pinkham absolutely kills everything. He throws notes on the board, barely explains anything and is not capable of answering even the most simple questions. Also, he is incredibly capable of putting anyone and everyone to sleep. He is so unhelpful during office hours, it is not even worth going. He assigns problem sets that take hours to complete and does not teach you the necessary material. The TA (Irina) was amazingly helpful, and even admitted that Pinkham teaches horribly. Also, the textbook (Pedregal) might be the worst book out there. If you take this class, be prepared to work your butt off and learn absolutely nothing. Avoid this class if you can...

Jan 2007

In my opinion, this man is a horrible professor. He can easily put people to sleep. He just talks about examples from the book and doesn't completely go through them but expects us to know how to and puts the hardest questions on the test and fails a lot of students.

Jan 2006

WARNING: he does not give standard Columbia grades. Pathological exams. Decent, but overrated, lectures. He introduced new material for the final at the "optional problem solving session." On the friday before a monday exam, we were forbidden from emailing him. His TAs (who are also his PhD students) regularly skipped their office hours. He refused to even discuss the content type or range of material on the exams, and then complained that students kept asking him about the exam. Even by upper level math course standards the class was full of very smart people, but he apparently set the curve somewhere between a C and B-. I know someone who did all the work but failed.

Dec 2005

Karatzas is a brilliant man -- one of the only few math teachers who knows his stuff well enough to teach it well. The downside is that this is a very difficult class for people who are not used to rigorous theorem proving. Analysis was traditionally taught by Gallagher, who I've heard was so boring that nobody attended class. I guarantee that Karatzas is just the opposite. His teaching style is extremely engaging, and though keeping with mathematical rigor, he knows how to use intuition to get the message across. As for the competition in this class, be prepared to meet some of the smartest people on campus, since this is a required class for math majors. But that shouldn't deter you from taking an absolutely enlightening class. Just forget about pumping up your GPA -- take this class for your enjoyment of learning.

Dec 2005

Dr. Karatzas is very smart, but not a man you want to see at 9 am. Don't try to eat food in class (or you'll get yelled at for sure!) Don't interrupt the lecture... he might make you look dumb. Don't be late or you'll be forced to withstand a very evil glare. Don't expect to get out of class at 10:25 - the lectures regularly run past 10:30!! Luckily you can still do this course without attending lecture. He posts all of the notes online. The homework is hard and the grading is extremely harsh all around, so be sure to write everything up very precisely. The average on the midterm was 10/30. Actually a score of 10 was considered "superb" by this prof!! Some superfreak math types got in the high 20's though, so I don't know what the curve will be like (oh well). My advice for this course is to learn everything from the textbook/online notes, skip the kindergarden-rules lecture, and make friends with the TA.

Nov 2005

OK some basic facts about this course: 1. This course is compulsory for all Math, Applied math majors, which means all the math freaks, undergraduate Math TAs etc. will take this course. 2. Analysis is a prerequisite for Economics and probably some other SEAS PhD students, so expect a small number of graduate students in this course too. So what's so special about Professor Karatzas, or how is he different from Professor Gallagher? Well first of all, Professor Karatzas actually assigns homework, and homework is graded harshly. Second, while Professor Gallager mostly gives T/F questions and asks for short proofs which require good memory in midterms, Professor Karatzas actually gives impossible problems in the midterm. Average of midterm is 10/30, possibly the lowest class average I've seen in my entire life. The third thing is that Professor Karatzas mostly teaches graduate students. Analysis may be the most "elementary" course he has ever taught in the past 5 years, so he expects all students to be really capable and enthusiastic. He starts writing on the board at 9 a.m., and usually does not stop lecturing until 10:30. He is also very strict in class, and does not tolerate any kind of stupid questions/ whining in more lenient grading/ less homework etc. Also, he actually scolds students in class. He's apparently appalled with the results of the midterm, and more than once he's been asking us to revise, pay attention in class, and do the problems. That being said, Professor Karatzas is EXCELLENT in terms of his ability to explain super abstract concepts. He basically follows Professor Gallagher's handwritten notes, and adds a few more tips on his own, so by taking Professor Karatzas' class you're actually learning from 2 great Math professors! I may have a problem with understanding other Math professors e.g. Sean Paul in other below 4000 level courses, but for Professor Karatzas, you're almost certain that you will learn something out of every lecture as long as you pay attention. Conclusion? If you're an above average student in other lower level Math courses (i.e. A range in every single below 4000 course), and really want to learn more and challenge yourself, Professor Karatzas is the best bet.After taking this course you'll probably laugh at the kids who get A in the Calc series and claim that they are good at Math. If you're an applied math/ any other major who just wants to fulfil the major requirement and finds a banking job? Sorry Professor Karatzas will kill you and your transcript.

Jul 2004

Gallagher is the most brilliant professor I have had so far. He doesnt use a book - in fact, writes up his whole course in notes that he copies and hands out, which is very nice, since you dont have to take any! His lecture style is very enjoyable - he lectures completely from memory, which adds some spontaneous insights and jokes to his lectures, and his 50 years of teaching experience make sure you have a great time in class. Did I mention that his lectures are absolutely clear and reflect his unending love for teaching? There are, though, occasional mistakes in his lecctures/notes which he usually corrects immediately (instead of covering them up like some pretentious profs). In one word, a Gold Nugget. Oh, yeah, and he included poems in his notes, too!

Jan 2003

Prof. Gallagher is a genius, on top of being a very nice man. He provides notes for the lectures, but people seemed to be taking their own notes anyways (why??!!!!) His exams require understanding of the material rather than just rote memorization, which is what exams should do, in my opinion. However, understanding modern analysis is no simple task. Be prepared to study if you want to do well. No need to go to class if you can understand his notes. Go to office hours if you can - he seems to explain the material better if you ask him specific questions. My only complaint is that the lectures were to dry - I wish he would have done more than simply repeat what he wrote in the lecture notes (although he did it all from memory).

Dec 2002

Professor Gallagher is a terrific professor and a very nice man. His lectures were extremely clear and often quite fascinating. Furthermore, there is no need to take notes, since he hands out photocopied notes for each lecture. (Actually, the notes make attendance at the lectures less than absolutely necessary. However, I went anyway since the lectures were so helpful and interesting.) Gallagher clearly understands and deeply appreciates the beauty of math. While he may appear to be absent-minded and easily distractable, this is not really the case. He can be quite witty and is extremely receptive to questions. When I went to his office hours, he was helpful and patient. In short, I can not think of a single bad thing to say about him. The meaterial for this class was sometimes quite interesting, sometimes not so much. I found the class helpful, in that it provided precise definitions for concepts that I had previously understood only in vague terms.

Jan 2000

Simply put, Professor Gallagher is a mathematical genius. He brings almost no notes to class, but he has yet to be unable to answer any student questions, and he never gets lost while lecturing. His lectures, at least in higher-level math courses, consist solely of definitions, theorems, and proofs. Gallagher and his lectures are laid-back and relatively easy to understand, which is surprising given the complexity of the material (set theory). He throws in some "math humor" occasionally, which anyone in this class should appreciate. He *almost* reaches the level of you actually looking forward to class, but not quite. Now, for the not-so-fun part. His tests, in any math course higher than Linear Algebra, consist of rote memorization of the definitions, theorems, and proofs that the lectures cover. That means you had better be prepared to attend every single class, lest you be royally screwed on the tests. If you go to class, the tests really aren't that bad except for the proofs, which can get very complicated. However, Gallagher usually makes his tests definition-heavy, which is good. He is very picky about wording -- miss a single word in a theorem, and expect some points off. He doesn't fail anyone, at least in this course -- the lowest grade on the first test was called a C-, and he worked up from there. Back to the good stuff, he's always available to meet with students. He schedules late-night review sessions prior to midterms, which can really help if you're struggling. There's not a lot of homework at all in this class, though the problems that are assigned are usually challenging. Overall, I like Gallagher as a professor, and would recommend this course to other students -- at least Math majors looking for an elective.

Jan 2000

Prof. Gallagher is (in my opinion) the greatest professor in the math department and quite possibly the entire school. He is extremely well spoken and knowledgable to the point where he brings no notes to class, doesn't follow a book and hardly ever gets lost. Both midterms are straight foward-definition, theorem, proof. The final is true-false, and is absolutely impossible--bring your anal-butt plug and pray to god your midterm grades were good enough because no amount of studying is going to give you any sense of confidence on the final. However, if you're interested in taking an upper level math class that is actually enjoyable to attend, Prof. Gallagher is a safest bet.

Jan 2000

Couldn't understand a word this man said/wrote in class. Is it an "x", a "n", an "i", a "t", or an "f"? Does it matter?