# [MATH V4061] Intro to Modern Analysis I

- Departments: Mathematics
- Professors: Daniela De Silva, Patrick Gallagher, Bin Guo, Walter Neumann, Fabio Nironi, Henri Roesch, Ovidiu Savin, and Xiangwen Zhang

Roesch is a pretty polarizing guy. Like he definitely did a great job of adapting to the online setting so I'll definitely give him credit where credit is due. He was a pretty good educator. I mean, nothing spectacular but he gets the content through to you. Unfortunately, all but like one of his graders this semester were kind of rude so that didn't help. Additionally, he's an incredibly harsh grader. Bordering on unfairly harsh grading. I enjoyed analysis and the workload was not bad at all. Just wish he wasn't such a harsh grader.

TLDR; average all around, harsh grading

#### Workload:

~1 pset a week (sometimes there wasn't an assigned PSET)

2 midterms (roughly equal in difficulty, though most people found the second midterm to be harder)

1 final exam (a little bit harder than the midterms and the mean score was about a 75)

I was very impressed by how Bin Guo manages to smile and talk at the same time. However...

His lectures bear little difference from an audiobook of Rudin. He nearly always gives the same exposition and examples, and when Rudin reads "Remark: [...]" Bin Guo will say, "Let me make a remark..." I also noticed that many people took pictures of the board instead of taking notes, which meant that either he was going too fast or he was not terribly interesting. Re: the latter, I will grant that verifying that calculus works is not the most exciting subject matter in the world.

He is also not very helpful outside of class. I once asked him to explain why my proof on the midterm was incorrect, because all he wrote was "?", and all he had to say was, "This part of your proof is OK. The rest is bad and not good

#### Workload:

Homework (15%, one dropped) consisted of 2 or 3 problems, half of them out of Rudin. Apparently cheating was rife.

Exams were a bit too long. The midterms were reasonable, the final less so (10 questions, each with multiple parts; the mean was 38%). Know your counterexamples for the true/false questions. He provides practice problems, which are somewhat similar.

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.

Excellent professor. Excellent and thorough teaching. Prof Zhang deserves many plaudits for managing to make an inherently difficult subject much more accessible without diluting any of its content.

The course material follows Rudin (standard intro to modern analysis textbook). It helps tremendously to attend each and every lecture. It is much easier to learn from Prof Zhang's lectures (and also his office hours) than by spending hours deciphering the enigma that is Rudin. You will be grateful for that.

#### Workload:

Quite reasonable. 3 problems a week.

Exams are reasonable. The practice exams are very helpful for preparing.

Modern Analysis I is hard as balls, and you're going to struggle regardless of who's teaching it. Study hard and learn to forget about your grades during the time you're taking it.

That being said, Xiangwen Zhang is fresh out of his PhD program at McGill, and someone thought it would be a fantastic idea to have a newbie teach one of the harder upper-division courses in the Math department.

I don't say this to put down Zhang, as he is one of the nicest, most approachable profs I've seen at Columbia so far. He WILL take the time to teach you something again during office hours if you let him know you're struggling. However, his freshness showed and I think it seriously hurt a few people who decided to take the Analysis sequence before the Algebra sequence (read, people who've taken Calc I-IV, Linear Algebra, and ODE and figured Analysis was the next logical step in the major).

A major problem with this course lies with the nature of the material and the students themselves: HW assignments tend to be 3 questions long, all including fairly involved proofs. WITHOUT FAIL, people COPIED these proofs off the internet and tried to wing it during exams, which WILL NOT WORK.

Exams however, followed practice exams closely. A good way to prepare for exams is to figure out the practice exam thoroughly, and proceed to manipulate the premises of each question to see how answers differ as a result. Do this, and you should have an A- (like I do so far, and trust me, I am not all that good compared to other people in the major).

Note, Zhang will bump the class average to a B+/A-, so heed his advice and study to LEARN, not racking up perfect problem sets, or you will surely SUFFER.

#### Workload:

Weekly, very challenging, 3-question problem sets with 3 optional questions (no credit for those, but they help)

3 exams, all challenging, all follow practice exam closely

This is the best possible way to learn Analysis I, especially for less intense math people, for the following reasons:

1) De Silva is a fantastic lecturer. She always makes sure to make the material accessible to people who digest things more slowly, and moves at a very reasonable pace.

2) She is also unbelievably nice, accessible, the kind of professor you just adore. Go to her office hours (which she'll always be happy to arrange), you'll understand. She's also very laid-back about assignments: at one point she allowed me to re-submit HW after she had explained the problems, the solutions had been posted, and weeks after the due date. She'll always respond quickly, and usually positively, to emails and requests.

3) Put quite simply, I can hardly believe there is a more manageable way of learning Analysis. The exams are easy to do reasonably well on, even if you don't know the material all that well (see below), the HW is more than manageable, and all in all the workload is not particularly heavy. (SEE THE NOTE BELOW ABOUT HW)

To summarize: this is the perfect way to learn Analysis, and there is little doubt in my mind that other classes by De Silva would be similarly excellent. The one tragedy is that I hear she isn't teaching Analysis next year.

TAKE THIS PROFESSOR IF AT ALL POSSIBLE!!!

#### Workload:

1) Weekly Problem Sets (20% of your grade), except for the last month or so of the class that for some reason had only 2 problem sets (meaning there were a good few weeks off). Typically three problems from the book, of which two were manageable and one was hard. However, you aren't necessarily expected to do the problems yourself: there is a recitation which involves solving the problems almost fully, and she'll gladly go over them with you if you ask her.

NOTE: While this is considered only a last resort, De Silva does not rule out using internet solutions for problems from the book (the book is an old one, so there are sites with near-full solutions to all problems). Most of the class copied straight from the web. This hurt them on the tests, but you should know that this is an option, because otherwise it could give other students an edge (my advice: try them yourself, go to recitation, and only if there are still holes left, look online).

2) Two midterms and a final (20% each for the midterms and 40% final). EXTREMELY MANAGEABLE. By which I mean that 60% of every test is memorization (Definitions, proofs from the book and HW problems for which solutions are posted), and another 20% is "doable" proofs (for which you can get decent partial credit even if you know a bare minimum of the material)

Professor Nironi is fantastic, Most of the reviews that have been posted here are very misleading and are probably written by dumb students. I did both modern analysis 1 and 2 under Prof. Nironi's guidance and thanks to him i acquired a solid understanding of Real analysis. Any course on mathematical analysis is bound to be rigorous, if you cant handle mathematical rigor, this course is a strict NO, if you like math and if you are passionate about learning math, then this is the most basic course that you need to master. I didnt have any prior background in topology/analysis or any other fancy stuff that has been mentioned in other reviews. Prof. Nironi explains the concepts very well and his notes is excellent. Infact i prefer his notes to the prescribed text(Rudin) just because its more accessible than Rudin for a novice. He is very helpful(There were instances when he went over the entire proof when i had trouble understanding the proof)and his grading too is fair. I do agree that he gives difficult(but thought provoking)mid term exams sometimes and he does it deliberately to gauge the level of the class. He adjusts the difficulty accordingly in subsequent examinations. Over all it was an enjoyable experience learning analysis 1 and 2 from Prof. Nironi.

P.S. If you cant understand what Prof. Nironi is teaching then you are simply not good enough for studying real analysis , drop the course otherwise you will inevitably screw up and post another misleading review here !

#### Workload:

1 homework a week

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

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.

His teaching style was ok (perhaps he could refer to intuition more to build relatively abstract concepts), but the course emphasized too much memorization. Besides that, there were three problems per homework and all were very difficult. In other words there was no foundation or nothing to build upon. It was sink or swim-- unless you are gifted at memorization, as many people in that class seemed to be. However, while the textbook is cold, dry, concise, and littered with circular logic and fallacy, Savin himself is a good person who would be willing to help if sought after.

I think he could have surveyed notions such as that of continuity at a more philosophical level and attempted to show through emperical tests why they may hold true and aren't arbitrarily defined to create an abstract universe which is nothing like the physical universe.

Also, he could have tried to build understanding more intuitively, as the basic ideas don't come from some arbitrary abstract universe (although that is where they may eventually lead to), they come from the physical universe, so something like, you can take a piece of wood and cut it into two pieces, three pieces, and thereby always find a piece of wood to cut into more pieces-- the set of natural numbers tends to infinity not in any abstract universe only, it also happens in the physical universe. The point is, these ideas have foundation, more or less, in emperical knowledge, and if he could supplement his lectures with more emperical intuition, that would make it probably much more accessible and interesting. Too much memorization killed it for me anyway, I had lost all hope and interest in class.

#### Workload:

2 Midterms, 1 Final, weekly problem sets. 60 percent of the tests are memorization (theorem, hw problem, and definitions), and if you can score that, you will be above average.

Prof. Savin has matured as a professor in a very short time. My roommate took his course in Spring 2008 and had the impression that he did not explain results well in class, and that he sprung surprises on exams. She did note that he was very able to field questions and respond to them effectively.

The best professors are quick learners. I took 4061 with Prof. Savin in Fall 2008 and was struck this morning while sitting in on 4062 (in Spring 2009) that he had been to scanning the room between definitions and proofs to see if students understood the progression of concepts. I've never seen another math professor with that kind of self awareness or ability for self correction. Don't get the proof? Alright â€“ we'll tear it apart and put it together again. Definition not sticking? Let's take a look at another diagram.

For this, I enjoyed Analysis I more than Calculus I.

It's not an easy course by any stretch of the imagination, and the only way to do well is to work like a dog. It's a lot to learn, but in the end you'll know you have earned your understanding and you'll have built something beautiful in your mind. Either that, or you will drop out. The survivors learned to lean on each other, and for what it's worth, the TAs are unbelievably good at what they do.

#### Workload:

One problem set per week, often exceptionally difficult; review your notes with a fine-toothed comb before beginning, get help if you're stumped, and it should not take you more than six hours. Otherwise, good luck.

Two midterms, one very early in the year. Learn your early lessons well, keep organized notes, and attend Savin's OH and review sessions if at all possible. He makes a real effort to prepare you.

One final examination, somewhat easier than the midterms, with two caveats: memorization of complex results will be more important, and the grading is less generous by a small margin. And the quitters will be at home and off the 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.

The previous review which calls this class "miserable" was extremely unfair. Savin's expositions of the material were very good: slow, thorough, clear, with good examples and illustrations of the theory, and certainly no unfathomable logical leaps. Rudin's Principles of Mathematical Analysis, the textbook for the course, is very dense and Savin's good at helping you through it. The thing to take away from my review is this: math majors rarely seem to write reviews on CULPA. If you want to take analysis to look good for business school, don't. If you want to learn some great stuff, please take the class. I know of some econ kids who talked to Savin at the end of the semester and said they'll consider taking Analysis II iff Savin is teaching the second semester.

#### Workload:

Weekly homeworks: three problems from Rudin to turn in per week, and 1-4 "extra" problems "to know" which could be on the exams. 2 midterms, final: all very fair (testing on definitions, theorems, proofs, hw problems), but with the last couple of questions there to measure the density of hair on your chest. Its possible to find the hw solutions online: don't... he called out people who were turning in identical tests and pwnd them by putting the 'turned in' hw problems on the tests.

This was by far the hardest, most miserable class I've ever taken. Savin is brilliant and is quite good at explaining things if you ask, but without being prompted he will gloss over important details, assuming that the class is as brilliant as he is. I am good at math, but not that good at math.

The material is interesting, and after the course you'll be an exceptionally logical person. But don't take it without a backup plan. It's hard as hell, the homework takes forever, and if you're not a natural it's a lot of memorizing infinite proofs and proof methods. It's definitely not for everyone, and it's terrible to get stuck in it if you're not sure you want to be there.

#### Workload:

Weekly problem sets, two midterms and a final. All hard.

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 W4061: Intro to Modern Analysis I | Igor Krichever | 2009 | Spring | MW / 9:10-10:25 AM | 1 |