# [MATH V4062] Intro to Modern Analysis II

- Departments: Mathematics
- Professors: Daniela De Silva, Evgeni Dimitrov, Patrick Gallagher, Bin Guo, Clement Hongler, Walter Neumann, Ovidiu Savin, and Xiangwen Zhang

General appeal to CU math professors:

Please, please, please, please, *please* stop teaching analysis as simply Rudin, expanded. Please.

I believe that there are good ways to use Rudin, but they are not to plunge us into Rudin and hold our heads down. There is very little of the beauty of math in this process; it retains its purity in the sense that it is true and very little is assumed, but not its elegance, as it has no meaning; it is just words (prove [this statement]) that expand into more words (ok, so [this statement] means [this expansion of this statement] which means [stuff with epsilons and deltas] then we can apply a theorem which means very little to us although we vaguely remember proving it so we suppose it's true and which seemingly came out of nowhere to get more different epsilons and deltas, and then TA-DA! QED) as a logical or memorization game.

If you _do_ assign Rudin (please don't!!!), please explain what these things mean--- as pictures, or intuitions, or explanations of the desires mathematicians have for their definitions--- in class instead of simply following the damn book. Also, please give us a reference text, as I have not been able to find one that isn't just completely different.

The homeworks were fine. (A little long, but that's to be expected in this sort of math class.)

I wish the tests were not repeats of the practice tests, as this means it is not at all valuable to try to actually understand the concepts and methods of analysis and instead is important to train ourselves on the practice exams and more or less memorize the answers given by the professor. As a result, people for whom rote memorization or test-training is incredibly hard (yes, this includes myself) do poorly. What are we testing, here? Speed? Memorization? I may be biased, but am I not right?

Sometimes I close my eyes and dream of a world where analysis is taught as a subject to love, understand, and admire--- perhaps in a project-based class (as apparently they have in UChicago). Where my effort in the course is driven by honest desire to learn and absorb the material instead of guilt and fear of the exams. Where I exit lectures with a fuller understanding of math instead of a vague sense dread (oh God, what will it be like next semester) and horror at my own incomprehension and inability to follow the class.

Due to no fault of the professor, I honestly believe that if it were not for my other math class this semester, I would drop out of the math major entirely because of Modern Analysis I.

The professor: Professor Dimitrov was quite a good instructor. He was very calm and friendly, which I really appreciate, and his lectures made sense (he was always well prepared and answered any questions that came up fully). He had us ask questions on Piazza as well, and was always really prompt in responding. I think Professor Dimitrov tried to make Rudin more palatable, and indeed, some proofs which I could not follow in Rudin (who is, to put it lightly, terse) made sense in lecture. But I wish he just ditched Rudin and taught us. If it weren't for Professor Dimitrov, though, I don't think I'd've made it through the course.

#### Workload:

Quite a bit. We had homeworks every week, which took a while but varied (5-10 hours, maybe). We also had two midterms, which were incredibly stressful for me, at least. And of course a final. Grade weights respectively 25%, 2 x 20%, and 35%.

I thought Professor Bin Guo did a great job. It's a really hard class that I would not advise taking for fun, and you need to attend lecture, do the homework, get help, and study to stay afloat, especially second semester. You also really need to ask questions in lecture. Otherwise, he will keep rolling through the proof, leaving you half an hour behind.

The tests were really hard, which allowed him to grade fairly with a gorgeous curve. I warn that what he calls "easy" is easy FOR HIM, not for me anyway.

I disagree with the comment written below that he can barely talk and smile at the same time and find that more likely an inappropriate reaction to his accent, which was a little hard for me to follow at first, but I figured it out, especially because he writes everything on the board. He does follow the textbook pretty strictly for the first semester but leaves it completely for Lebesgue Theory in Analysis II, which makes lecture absolutely necessary.

#### Workload:

TONS of studying for the tests

Only two homework problems/week are mandatory and graded lightly, but the rest are necessary for the tests, and some can take a while if you don't figure out the trick

Didn't need to read the textbook because he went through it in lecture

I had Professor Guo for Analysis II in Fall 2016 - I'm writing this now since I realized he didn't have any reviews yet on culpa. I believe it was the first semester he taught at Columbia (he got his Ph.D. just in 2015) and he did a pretty decent job considering that. I've heard that after his first semester of teaching he made tweaks to his exam difficulty and I've heard only good things about his classes since.

Lectures:

- Professor Guo is very clear and helpful during lectures. During class he would pretty much just go through the content in Rudin (with some added intuition here or there) and if you ever had a question about anything he would give a good answer.

Exams:

- The exams were very very difficult. I was in a class of 15 or so people who were all math/CS majors and the average on the midterm was a 26/60. The format was 6 problems, 1 from the homework and 5 original problems which required a lot of ingenuity to solve -- they weren't just simple applications of the material. Of course, this means that people who knew the material but didn't have the intuition for the tricks needed to solve the exam problems ended up doing poorly, which is understandably never a good feeling. The final was 10 problems which were all original.

#### Workload:

Homeworks:

- His homeworks were pretty hard, but short -- just 4 problems. About half of the class would regularly go to the TA's office hours for help, and a lot of the time it would be us doing the problems together with the TA, since the TA didn't know how to do them immediately. Probably spent 4-5 hours on homework per week.

I had a great time in analysis with Zhang- for an 8:40 upper-level math class, this is really saying something! Although the material had the potential to be hard, Xiangwen was really clear and made following his proofs as easy as possible. Especially in analysis II, where the topics are used regularly in other fields, he did a great job of motivating the material.

Perhaps because the quality of the textbook (Rudin) deteriorates in the second half, Zhang did things his own way a lot in the second semester, giving different and often better proofs and instruction than Rudin. Accordingly, he also wrote a lot of the problem sets in analysis II instead of assigning book problems. For this reason, I liked the second semester better than the first since Zhang's mathematical personality started shining through.

The main problem with the class was pacing. In the first semester, we were supposed to get through chapters 1-7 in Rudin, but got only through 6. A chapter behind going into the second semester, there was a real time crunch towards the end and Zhang had to rush through the last two chapters. We spent but a single day on differential forms, which was the topic I was most excited about. It was a shame that the treatment of measure/Lebesgue theory was so light because a lot of the work we did for the Riemann integral (function spaces, Holder inequality, etc.) was obviously leading up to the idea of the L^p space.

The exams were very fair. The first few questions were True/False with justification, which were normally pretty straightforward. Zhang also always put a couple of questions that were either identical or very similar to HW/practice exam problems, so you are definitely rewarded for doing the work on your own. The averages were quite low, generating a large curve, so one could do quite well on the exams just by doing these easiest problems well, though no questions were impossibly difficult.

#### Workload:

Very Manageable. The weekly problem sets have 4-5 mandatory questions but generally only 1 or 2 require substantial effort. He also provides optional practice problems on every problem set that are great for studying from (Hint Hint).

Overall Xiangwen is one of the best professors I've had at Columbia so far. This was his first year teaching the course (and any course I believe) and this seemed to affect his pacing of the course material at times (we spent way too much time on chapter 1 in Analysis I and had to rush at the end), but otherwise he did a great job.

Xiangwen's lectures are clear and easy to follow and he tries to both explain the motivation/significance for the theorems he proves as well as make sure students understand the argument itself. He is also nice and easy to talk to during office hours and usually responds within ~12 hours to emails.

For homework he assigned 3-4 problems to be graded each week and typically 2-4 more as practice. I personally rarely did the practice problems when they were assigned, but instead used them as study problems before exams. The exams themselves were moderate. He bases them a lot on the practice exams he posts and the homework problems so if you study those carefully and really understand them you should do well.

#### Workload:

3-4 graded problems a week. Difficulty varies, but typically 2-5 hours of work. 2 midterms for Analysis I, 1 for Analysis 2.

To those who have not yet gotten the memo: De Silva is one of the nicest and best math teachers you will have. Sometimes you have to see it to believe it. This semester was marred by the fact that De Silva was going through a brutal (her words) pregnancy, so she was always tired and sometimes would miss classes and/or office hours. Her dedication to the class despite this was truly extraordinary.

The class covers chapters 8-11 in Baby Rudin, including Fourier Analysis, Multivariable functions, Differential forms and Measure/Lebesgue theory.

Strengths: The strengths of the class include Prof. De Silva's own incredibly approachable and kind personality; her dedication to the class and her students; the tests, which are very manageable (see below); and the curve, which is very generous (I got around 5 points below average on each test, got a little above 50% on the problem sets and still got a B+).

But the first point is really key. To give just a few examples: I had three midterms on one day so I asked De Silva if I could do hers on another day, and she agreed and even gave me the same test as the others; she almost always agreed to extensions for homework.

Weaknesses: The one primary weakness may have to do more with the pregnancy than anything else, but the lectures themselves consisted pretty much exclusively of statements of theorems and their proofs, with little in terms of intuitive understanding and examples, though on a couple of occasions she did add a few of these. This is largely to be expected from a pure math course, but if you- like me- are still shaky regarding some of the concepts of Analysis I, you may be very lost in Analysis II. The material here is much, much harder and far less intuitive (even at the end of the semester, when I asked people "Soâ€¦what exactly are differential forms?" they didn't have much to answer me). We also did not have recitations, which made the homework very difficult, though office hours were always helpful and De Silva doesn't object to you using online solutions as a last resort if needed (though these existed for only ~50% of the problems). However, all of the above is mitigated by the very generous curve, so I wouldn't worry about your grade too much unless you want straight As (or if you're very bad at memorization; see below).

#### Workload:

Two midterms, one final, around semi-weekly problem sets (6 total this semester).

Exams: Each exam consists of 60% memorization, i.e. 20% definitions (given in advance), 20% proofs of specific theorems (also given in advance), 20% homework problems (solutions given in advance). On one exam, an easy problem replaced the HW question. All in all, if you feel you can memorize things well, you should be able to start at 55% for each exam (assuming a ~5 point margin of error for miswriting a definition or something). If you can do around half of the doable question and steal a few points off the last, you'll do fine. If you can actually solve the doable question, you'll do great. Note: whatever you do, write everything you can possibly think up for the last two questions, including random verbal explanations and such. She's very willing to give partial credit.

Homework: do in groups, go to office hours, look up solutions as a last resort. You should be fine, since averages will be very low.

Seriously, do anything you can to take this professor. The previous reviews for her analysis class have said this, and I feel obliged to repeat it. This applies to any class she is teaching.

She does a great job teaching the material. The second half of Rudin is even more dense than the first, and she does a great job picking out what is important. There were a few times this semester when she covered material differently than it was covered in Rudin, in which case she made lecture notes available online. De Silva seems like sheâ€™s constantly looking out for questions during lecture, and sheâ€™s also very helpful during office hours. Sheâ€™s also a wonderful person who really seemed to want us all to understand the material.

The exams are very fair and there are no surprises. She tells everyone exactly what to expect and posts a review sheet with all the important terms and theorems.

#### Workload:

20% Homework Assignments - There were 6 of these, each consisting of 3-4 problems from Rudin. We definitely were given ample time for each.

20% Midterms (2)

40% Final Exam

Professor De Silva is by far the BEST math professor I have had during my time at Columbia. She is an excellent lecturer and very good at both understanding and answering students' questions (both of which are often a weak point of math professors in my experience). The subject matter is definitely challenging, and having a good professor can be the difference between being totally lost and understanding exactly what is going on in the course, especially considering the textbook for the course (Rudin) is a bit intimidating, especially at first. Overall, Professor De Silva made a very challenging two semesters quite enjoyable. If you have an opportunity to take a class with her, you definitely should.

#### Workload:

Generally weekly problem sets (only a few problems, but they take a LONG time; really helpful for understanding the subject matter and practicing proofs), two midterms, cumulative final

I had a terrific time in Prof. Hongler's Analysis II class. He is very genial and approachable, and he tries to make sure that the students understand what is going on, even if that means spending some extra time reviewing material from previous lectures.

In terms of logistics, this class is run in a pretty standard way following Rudin's book. That is, there are weekly homeworks (consisting of about 3-4 problems each, from the book), 2 midterms, and a final exam.

For the exams, he had more questions than there were available points, so you could choose which ones to answer. That being said, he did not make all the "extra" questions impossible, as some professors are wont to do.

#### Workload:

Weekly homework (11 total), 2 midterms, Final

Prof. Hongler is great. He presents material clearly, usually follows the book and makes notes when he departs from it. This helps keeping the material organized. He's occasionally humorous, making jokes about his study in undergrad, and in earlier study. Lectures were a joy to attend, even though they were at 9AM. To do it again, I would have reviewed more consistently, so I could better keep up with what was going on. Indeed, Analysis II is challenging and is a lot of work, but it's straightforward if you're diligent.

Prof. Hongler seemed to care about student performance, and always made himself available for answering questions. Great professor; great class.

#### Workload:

Indeed, this class seems to be much more work than the average Columbia college course, perhaps only exceeded by stuff in engineering. Weekly problem sets; each took about 10-15 hours to complete; if including learning the stuff on which the set is assigned, even more time. Challenging course load, but no problem was obnoxious and without purpose.

First of all, I took this class four years ago, so it's possible some things may have changed, but I doubt it. This review comments on the material in this class, its difficulty and what Savin is like as a teacher.

As others have said, this class is much harder and requires a much higher commitment than calculus or linear algebra. To really learn the material and get a good grade you're going to have to know Rudin backwards and forwards. You will also have to do all the problems Savin assigns, even the ones he doesn't collect, and then some. Be prepared to spend more time studying and doing problems than in any other class you've taken. The students who succeed are the ones who love math, I mean really love it, and expect to use this material for the rest of their lives. If you're looking to challenge yourself or take an interesting math elective, don't take this one--at least not with Savin. To give you an idea, the kid sitting in front of me during the 4061 final handed in a blank exam. That said, as long as you have the necessary preparation and motivation you don't have to be some magic genius to do well here. In my case I had already taken Honors Math which was key since it covers a lot of the methods used in this class. The optimization course may also cover some analysis techniques, but I never took it.

Although Savin is indeed a super nice guy, he is by no means an easy grader. He told me that he wanted 20% of the grades to be an A- or higher. I don't know if he was actually that harsh, but that was what he said. Despite that he was the best lecturer I've encountered--and I've had quite a few classes in math and the humanities--simply for the depth at which he understands the material and his willingness to show that understanding to students. I don't know if he is Russian, but he has a very dry, ironic, very Russian sense of humor. Think Vladimir Putin but nicer. Lectures were very clear and businesslike without extraneous non-mathematical information. This puts off some people but I kind of liked it.

Also, 4062 is harder than 4061 partly because the material is just harder and partly because the quality of Rudin's exposition deteriorates significantly in the second half--particularly in the chapters on power series, Fourier series, the gamma function, and differential geometry. I found Munkres to be much easier for diff geometry.

It's also worth noting that there seem to always be a few grad students taking this as a gut course--I think the entire first year class in the Statistics department was taking it when I did--and that this effects the grade distribution.

#### Workload:

Lots. Two midterms + one final. Do as many problems in Rudin as you can.

The man. The myth. The Romanian.

On the first day of 4061, Savin told us of an important aspect of analysis - in every problem, there is never any more or any less information needed to solve it than what is given. I felt this was a theme of his teaching throughout the course; he presents no more and no less than exactly what we need to know about each concept and topic. A variety of people take analysis for various different reasons. As someone who is took it for a solid foundation in pure mathematics and for plain old intellectual growth, I could not have asked for a better instructor than Savin.

The textbook, "Principals of Mathematical Analysis" by Walter Rudin has been the standard analysis textbook for over half a century and has not been revised since 1976. In the first semester I cursed the thing's existence, as did most of the class. Even the TA said it is really not a good book from which to learn analysis for the first time. Aside from the prose, the problems were a new form of hell to all of us. An illustrative example was one particular problem that the TA solved during office hours with a very clever use of concavity and inequalities. When we asked him how he knew to do that he replied, "Well, a few hundred years ago someone sat around for a really really long time and figured this out, then he told someone, and that person told someone else, then years later someone told it to me, and now I'm telling it to you." The point being that there were many problems where once you see the solution your only reaction is, "How on earth was I supposed to know how to do that?" The answer is either someone tells you, or you are the guy or girl who sits around for a really really long time and figures it. However, by the end of the year, some of us really began to appreciate Rudin, and today I think it's one of the best mathematics texts ever written.

I sympathize with those who complain about the amount of memorization required for the exams, and I can say that the run up to each exam was never a pleasant experience - one that I'm glad to be done with. However, Savin never assigns irrelevant things for the sake of assigning them, nor does he put any of the long and obscure homework problems on the exam; in each proof there is something important that he thinks you must understand. Memorizing in this class is not like memorizing a bunch of chemical names from flash cards, it really forces you to condense and rewrite the proof in a way that you understand it, and make sure that you understand every key logical step. From my own experience, there were original exam questions that I was only able to answer because of my careful study of the proofs he forced us to know. So putting the effort into knowing those proofs can be a two for one deal.

One unfortunate part of 4062 is that the first topic, Fourier Series, was so goddamned confusing that a lot of people dropped after the first midterm (hell, some dropped in the middle of the first lecture.) Part of the trouble may have been that it was a continuation of the last topic in 4061, and the winter break severed the continuity. But after that chapter, the course shifted to completely different topics which were much more comprehensible and I felt were the most enjoyable of the entire year. Having talked to people who have taken 4062 with other professors, and having seen the syllabi of others who have taught it, Savin is one of the only professors who thoroughly covers every chapter in Rudin. In particular, most either skip chapter 10 entirely (Integration of Differential Forms, the longest chapter in Rudin,) or give it a very weak treatment. This is unfortunate as I found it to be my favorite part of the course.

Savin's personality is really hard to describe. He is never overenthusiastic and never bored; if I had to describe his personality in one word: non-quirky. His lecture persona is that of the foremost expert of avant-garde film who has been sent in to the Amazon rainforest to explain the plots of various David Lynch films to the indigenous tribes. He is extremely approachable and always willing to help, though he never makes small talk or ever talks about anything other than the task at hand. He's one of the only professors that I've had who has not once said so much as a sentence about his outside life during lecture. You'll never hear anything like, "This reminds me of a joke my thesis advisor told me when I was a graduate student.." If you didn't hear it from other people, you would never know that he is Romanian, that his wife is Daniela De Silva, that he has a daughter, or that he eats and sleeps just like every other human being on this planet. This made him all the more intriguing and added to the cult status he holds with some of us who stuck with him for the whole year.

I would take absolutely any class he teaches without hesitation.

One aside about the course itself: an interesting social phenomenon in Modern Analysis is a fear that permeates through the class of phantom "math geniuses" who somehow know everything, dwell in the back of the lecture hall, and ruin the curve for everyone. One of the main reasons that so few people from 4061 go on to 4062 is that they think that 4062 is going to have all the "geniuses" from 4061. I'm going to call bullshit on this. From my observation, the people who did really well in the class were not savants, but those who worked hard and had a good amount of time to dedicate to the class relative to other courses.

#### Workload:

"Weekly" problem sets in 4061. During an exam week or around a vacation we got a break, so in the end we there were 9 total and he dropped the lowest one. In 4062 they were more like every other week, since the proofs done in class were longer and it took a long time to get through each topic; there were 6 total.

Two midterms each semester with the same format:

20% - A mix of 4 or 5 of the following: definitions, stating theorems, providing examples/counterexamples, or calculations.

20% - One homework question: can be either one turned in or one of the assigned problems "to know."

20% - One proof from a list of 4 to 7 proofs provided a week before the exam.

20% - A somewhat "doable" problem that Savin cooks up.

20% - A somewhat impossible problem that Savin cooks up.

The final in 4061 was the same format as the midterms but x2. The final in 4062 had fewer definitions and theorems, and more original problems. Both were very fair overall.

Exam Averages:

4061 -- Midterm 1: 57, Midterm 2: 59, Final: 57

4062 -- Midterm 1: 51, Midterm 2: 72, Final: 56

The best advice I can give for preparing for the exams: do not try to guess what he will put on it, you will always be wrong.

Professor Savin is a great lecturer. He has a simple style that makes it easy to understand the material.

I found the class interesting and fun to attend. Questions were always welcome and he was constantly asking questions for students to jump in and answer which made coming to class a little more exciting. The course was at a reasonable pace and followed the book quite well. I highly recommend taking the analysis sequence with Prof. Savin if he teaches it again --or anything else that he teaches. You will learn a lot.

The workload was lighter than expected (~1 hour for each homework and several hours for the test preparation) and I think it wasn't hard to do well on the tests and homework.

#### Workload:

6 problem sets-- short, easy to medium difficulty.

3 tests: some questions were hard but I imagine there was a generous curve

This class was a never-ending source of stress and a major contributor to a moderate depressive episode for me. It's a 2 semester sequence, and the first semester was a lecture of about 60 kids. Second semester, we were down to about 10. I know that a lot of people like Neumann a lot, and he is basically a nice guy who is pretty approachable, but in general I found his lectures to be quite vague. He didn't state definitions or theorems clearly enough, and often returned to a concept later in the course using a slightly different formulation. Expectations and grading were also not clear enough. Second semester's midterms were not scheduled until the week before they were given, and even then only after I asked 3 class meetings in a row when they would be - although he did let the class choose the dates. Neumann does seem to be a major proponent of grade inflation, however: first semester, I failed the second midterm and still pulled through with an A, and second semester my homework grades were dismally low (enough to make me take the pass/fail option) and I ended up uncovering the grade.

#### Workload:

Weekly problem sets (which can be very difficult), 2 midterms, and a final.

To avoid repeating any opinions hereinto expressed, let me just say that this man gives you a glimpse into how graduate-level mathematics should be done. Having done mathematics research over the past summer, I was amazed at how close Gallagher's method of definition, theorem, proof is to research caliber mathematics. While the protocol that he uses can be construed as boring [xeroxing notes and requiring no homework essentially], one should allow time to digest the notes that he does give.

Bombard him with questions, he will give you great answers. While this class might be a shock to some who have been used to the calculus sequences, it is definitely a great class to test one's aptitude and curiousity for higher mathematics.

#### Workload:

Essentially no written work. He assigns hw problems randomly and infrequently. However, I would recommend reading each of his sectioned notes two to three times to aid in the overall understanding of the material. Exams are usually true false requiring complete understanding to receive full points [no partial credit], but the overall difficulty of the questions is not that bad.

I personally love Professor Gallagher and would take any class he offered. The man is the shining example of someone who loves teaching, loves math, and lives for his job. He explains very well if you are willing to listen, and can go through really hard proofs without even looking at notes. The man LOVES proofs. He doesnt assign a textbook, and instead, writes a packet of notes very every class and gives a copy to each student. You don't have to take notes, so you can just listen. He is also just a very interesting guy - writes and reads lots of poetry, etc. Once in a while he gets caught up in one of these other things and isn't very prepared for class (or like the time he didn't give back the graded midterms until the last day of class!)., but hey, knowing the average Columbia student, who are we to complain.

#### Workload:

Random homework problems (don't worry too much, just hand it in), 2 midterms (memorize the definitions and theorems word for word, final is all true/false

## Directory Data

Dept/Subj | Directory Course | Professor | Year | Semester | Time | Section |
---|---|---|---|---|---|---|

MATH / MATH | MATH MATH W4062: Intro to Modern Analysis II | Ovidiu Savin | 2009 | Spring | MW / 9:10-10:25 AM | 1 |

MATB / MATH | MATB MATH W4062: Intro to Modern Analysis II: Intro Modern Analysis II | Walter Neumann | 2008 | Fall | MW / 11:00-12:15 PM | 1 |