professor


Apr 2021 
This class is super challenging. With that said, I think Karatzas is a good instructor and I think I learned a lot. I had taken real analysis I but not II. The pace is fast so you may feel like you don't fully understand some topics, but I also think that's just a product of the difficulty of the course. One nice thing about the course is that the beginning is much more typical proofy prob theory (e.g. DCT and BorelCantelli) but the later part switches to Markov chains which is a nice change of pace.
Jun 2018 
Karatzas is a captivating lecturer and an excellent professor. He gave us challenging problems and we discussed them in class. The subject material we covered was clear, interesting, and practical. Though the only prerequisite for this course is Analysis 1, in order to truly understand the subject matter one should take both Analysis 1 and 2. For example we discussed Lebesgue theory in depth. Unfortunately I took this course while taking Analysis 1. This meant that I was way out of my depth. Yet Karatzas made the course accessible to me. His notes, the textbook, and the outside readings he provided were excellent sources that helped me catch up on the necessary background. He grounded every question of probability in a real world example, so that students could understand concepts even if they didn't understand the abstract definition. I spent much of he semester worrying that I would fail the class. Karatzas as a policy did not release the mean and standard deviation on exams, but I'm sure my scores brought the class average way down. If you care about getting homework grades returned, Karatzas is not the professor for you. That said, he gave me a B. My guess is that he gave everyone at least a B, and that grades were mostly made up.
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 
Unless you're a genius, don't take a lower level class with this guy: you will suffer and NOT do well(unless you think a B or B+ is "superb"). He is the only professor I have ever emailed that told me not to email him. I think that should speak for itself.
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 kindergardenrules 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.