dixiecgick: trying to solve "real analysis" problem once for all

[dixiechick]

Hello, you guys:

Here comes dixiechick again. Last time, I suggest that we should standardize th math preparation class and it has not been accepted by most people here. However, recently, "thescienceboy" and several other guys had the posts related to real analysis class again, covering from "what are the general class contents for real analysis" to "what class should i choose". Now, i try my best to identify several different levels for this class (at least in this forum):

Real Analysis II:
math symbol and basic proof in analysis; sets; functions; axioms for R, applications of the completeness axiom, countability;

Real anlysis II:
limits (for both sequence and function), continuity, differentiation, integration, fundamental theorem, power series, uniform convergence.

Real Analysis III:
The Riemann-Stieltjes integral;Lebesgue measure and integration;Fourier series and Fourier integrals, incorporated with some metric-transformation

First two are usually the math major-core courses (for sophmore) and the third one is usually an upper-level electives (for Junior or Senior). I checked several university websites here inside the U.S.

How about this suggestion?

[C152dude]

Fix the typos.

Also, this is not going to work. There is too much variability in courses for you to define what is and what is not standard.

[Canuckonomist]

I think its safe to say that this standardization won't work.

My third-year real Analysis class (the second in the stream) covered completely different things. Metric Spaces (and all things related), Completeness, Compactness, Stone-Weierstrass, Implicit/inverse function theorem, just to name a few.

As for preparation, you take all the analysis classes you can (and feel comfortable with taking). The idea that there is a "level of preparation" in analysis is fairly dubious. If there is room left in your degree, take more analysis if you can.

We all appreciate what you are trying to do, Dixiechick, but it just isn't feasible. First we'd have to decide what the ideal "prep analysis" classes would look like, and then you'd need to come up with a "real analysis" equivalent table akin to the "equivalent degree" table most admissions office will put out for degrees from different countries.

Ultimately, we'll need to take things on a case by case basis. But nothing is ever certain -that's why this process is so arbitrary, no one knows exactly what it takes to get in, and we're all making best guesses.

[paradox3696]

dixiechick, any one of the courses listed is GOOD enough of your math competency to go for Econ Phd. I took Real Analysis in undergraduate which was required for Math Major. It was NOT even a choice in the graduate math program of the same school. In graduate school, they offered Advanced calculus I and II, Real Variables (I and II), and Advanced topics in Topology.

Again, any proof-based class shall prepare you well enough for the econ phd.

[dixiechick]

It seems that I have to made several announcement first:

1) Personally speaking, I have finished all these 3 real analysis classes at math department.

2) The reason why I want to make a detailed explanation is that there are tpo many questions in this forum that related to "real analysis". However, since most of us agree that any of these 3 classes will prepare you for your potential ECON ph.d study, then next time, anyone else who does not know which class to take need not to ask question here again. So by doing this, we can save more time and space in this forum for other questions.

3) Meanwhile, note the following: real analysis class is a necessary condition for your econ ph.d application, not a sufficicent condition.

Is that clear enough?

[asquare]

dixiechick, I think the problem is that while the material may have been divided as you describe when you took the classes, that's not true in general. There's less agreement around the course descriptions or the principle that "any of these 3 classes will prepare you for your potential ECON ph.d study" than you imply. And while it would be nice if there were standard courses, we seem to do fine on TM by explaining what topics are covered or what book is used when talking about real analysis.

Finally, real analysis is neither necessary nor sufficient for being admitted to econ PhD programs. There are many examples of people who do not have real analysis being admitted to top-10 programs, which invalidates the claim of "necessary." There are even more examples of people with real analysis who are not admitted; we both agree that real analysis is not sufficient.

[tangsiuje]

Originally Posted by dixiechick

3) Meanwhile, note the following: real analysis class is a necessary condition for your econ ph.d application, not a sufficicent condition.

Actually, even that is misleading. Perhaps you may want to say that it is highly recommended, aspecially if you want to apply to a top school in the US.

I think it's extremely rare here in England, for instance, that students aspiring to do a Phd in Economics take courses in Analysis, or actually any course in the maths department. I'm probably the first one ever doing so at my university - I required special permission from a number of people and I cannot do it within my degree programme (taking it as a course overload). The exception would be those on maths/economics joint degree programs, but they are quite few and almost always go into investment banking and rarely into academia. The maths you need to know in economics is usually taught as an integral part of theory classes (I'm not saying this is a good thing - in my experience, most economists are really crap at teaching maths.)

You may also want to ask yourself what proportion of the incoming class has taken Real Analysis in an incoming econ PhD class. Although the number might be close to 100% in the US top 10, it is not 100%. How about the top 20? How about schools ranked in the 50s? I'm expecting you're considerably below 100% already.

Here, we do sequences, series and completeness in the first semester, and continuity and differentiability in the second (both first-year courses). I guess integration is taught in the third (second-year course), but I will have graduated before I get to that stage so I really couldn't care less. They teach set theory in some other class, and they don't teach proof methods (you're just expected to pick it up along the way). Don't know how relevant that is for your investigation, but probably quite interesting from a comparative perspective.

[thescienceboy]

One full prof, who is considered one of the best researchers/lecturers in my department, said that the role of real analysis and other advanced math courses is overemphasized. According to him, solid knowledge of Calculus 1-3, Linear Algebra, Probability theory, Math.Stats, Econometrics and maybe a couple of other stats courses is surely enough to get into top Canadian schools and do well. His advice is to stress out less about not taking RA, Topology etc. as one can pick up necessary stuff by himself during MA or the first year of PhD or just audit these courses. Of course he was talking about Canadian big4 and American schools of the same caliber, but I think that's the range where majority of TM readers (don't confuse with posters:-) ) hope they will end up. As for the top US schools, I got the impression that it's the letters of reference from famous fellas and the name of your undergraduate institution that brings you in.

[Canuckonomist]

Originally Posted by dixiechick

It seems that I have to made several announcement first:

2) The reason why I want to make a detailed explanation is that there are tpo many questions in this forum that related to "real analysis". However, since most of us agree that any of these 3 classes will prepare you for your potential ECON ph.d study, then next time, anyone else who does not know which class to take need not to ask question here again. So by doing this, we can save more time and space in this forum for other questions.

I love when my posts are ignored. You could easily take the class I mentioned, too, which is Real Analysis, but it's a course that is completely different from the ones you've listed.

Any how. I will also state a case of someone from our MA program who just got into Princeton full-funded without taking so much more than 2 semesters of calculus and two-semesters linear algebra. I believe he didn't have any more math preparation than this, but the guy is brilliant, so they couldn't turn him down. Top of his class out of 100 graduates last year at a top 3 school in canada. He'll probably graduate top of the MA class, too.

I will also follow by saying that he got 100/100 in few difficult classes (3 or 4). And its NOT EASY to get 100 in classes here.

So sometimes all you need to do is just be brilliant to get accepted to places like Princeton.

[econofrosh!]

maybe it's not a question of what our school's standard Real Analysis (aka Advanced Calculus, Real Variables etc. etc.) course should cover but rather which topics under whatever label is sufficient/necessary/advantageous for grad school preparation.

in other words, which topics that are often associated with the course name "Real Analysis" are impt?

(some that have been mentioned here in TM: continuity, convergence)