[MATH W4061] Introduction to Modern Analysis

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.

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.

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.

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.

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!

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).

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.

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.

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?

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.