Lackluster Grade in Real Analysis; Next Course to Take?

[ah25]

I got a lackluster (but not atrocious) grade in Real Analysis I, at an un-prestigious/low-ranked school.

 

This could be a problem for my graduate school aspirations, since it is the most advanced math course I have so far. It could look like I am hitting my "math ceiling" to adcoms, and it is arguably the most important course for Statistics PhD admissions. 

 

Otherwise, my record is decent, with a GPA of about 3.8 between my mathematics and statistics courses, including the Calc Sequence, Linear Algebra, Math. Stats, and Probability. I have already graduated. 

 

I can use the summer to do post-bacc work at a well-regarded university that is highly ranked in statistics. 

 

I know, ideally, it would be to take Real Analysis II and do well. However, it does not appear to be offered during the summer (usually a spring course), and I would like to be ready to apply by the fall. 

 

Here are some of my options for the summer at this university:

 

Real Analysis I - nominally the same course as I had before, but sounds more rigorous and expansive than my previous course 

Upper Division Linear Algebra - ditto; nominally the same course as I had before, but sounds more rigorous and expansive

Abstract Algebra I - has lower division LA as a prereq

Numerical Analysis - has lower division LA and Calc III as pre-reqs. Sounds application-focused, involving calculations and approximations. However, it's listed on Harvard Biostatistics and Berkeley Statistics sites as "encouraged to have" and potentially "useful" to have, respectively 

Complex Analysis - has Real Analysis I as a pre-requisite. Possibly could show that I can do well in something "beyond" Real Analysis? Mentioned by Berkeley Statistics as potentially "useful", and also by UChicago Statistics

 

 

Any thoughts? I feel a bit torn... if I had to choose, I would pick Complex Analysis right now. 

 

Also, would I be shooting myself in the foot by doing any of the above 5 instead of just waiting a year and taking Real Analysis II?

 

 

[MathCat]

It's not clear to me if you know this, so I'll just say it: numerical analysis is a completely different beast than real analysis. It might be good to have, especially if those programs suggest it, but it's not going to show you can do the stuff you did in real analysis.

 

That being said: what did your real analysis I course cover? There's a fair chance complex analysis will be quite a bit different too, but that depends on what you saw in that course.

[ah25]

It's not clear to me if you know this, so I'll just say it: numerical analysis is a completely different beast than real analysis. It might be good to have, especially if those programs suggest it, but it's not going to show you can do the stuff you did in real analysis.

 

That being said: what did your real analysis I course cover? There's a fair chance complex analysis will be quite a bit different too, but that depends on what you saw in that course.

 

Thanks for bringing that up! Yes, that was my assessment, as well--that's what I was trying to get at by "sounds application-focused, involving calculations and approximations." The two "analyses" have essentially nothing in common, from what I understand. 

 

My Real Analysis I course covered the first four chapters or so of Rudin's Principles of Mathematical Analysis: Real and Complex Number Systems, Basic Topology, Sequences and Series, Continuity. 

 

The Complex Analysis textbook will be Brown and Churchill or Gamelin, from what I have seen from past syllabi--covering topics such as functions of a complex variable. Cauchy's integral theorem, power series, Laurent series, the residue theorem, singularities, and more. 

[MathCat]

Well, I'd say more real analysis would be beneficial, but perhaps not serve your purposes if you don't get a good grade.

 

Numerical analysis is completely different. I've never actually done any, but (I believe) it is about doing computations on computers. There are lots of issues of efficiency and round off errors, algorithms, etc. I don't really know anything more than that, but it is not at all similar to real analysis. You might like it, I can't say.

 

Complex analysis is a bit nicer than real analysis. As the names suggest, real analysis is about real valued functions, space, sequences, series, etc., while complex analysis is the same thing, but now over the complex numbers. However, the results are actually quite different - a lot of statements that you'd like to make in analysis fail in the real case, but are true in the complex case. For example, a complex function that is differentiable once is infinitely differentiable, a stark contrast to the real case. They both use similar proof techniques, though I'd say complex analysis arguments are usually a bit more topological, since you can't rely on order properties. Do you know how rigorous the complex course would be? Often a first course is more computational and pretty straight forward, while a second course is completely rigorous (and harder).

 

What's your main goal for taking another course?

Edited April 16, 2015 by MathCat

[ah25]

Since my first go at Real Analysis I yielded pedestrian results, my main goal for a summer course would be to just demonstrate to adcoms that I can handle advanced math, particularly proof-based.

 

I presume Complex Analysis would be reasonably rigorous as an upper division course, especially since Complex Analysis's prerequisite is Real Analysis. 

[MathCat]

Well, numerical analysis is probably not that proof heavy, so maybe it's not the best choice. I could be wrong about that - as I said, I've never taken it.

 

If you don't like analysis too much, you could try one of the algebra courses. Those are also usually rigorous and proof-based. Certainly the abstract algebra course should be. It's a different way of thinking - no fussy epsilon-delta proofs, if that's a bonus for you. An introductory course in group theory is the first course math majors take after learning what a proof is at my university, so it's reasonably accessible usually.

 

I think real analysis is the most relevant to statistics, though.

Edited April 16, 2015 by MathCat

[ah25]

Thanks for your suggestion on Abstract Algebra.

 

What caught my eye about Complex Analysis is that Berkeley and Chicago mentioned it by name, and as a successor of Real Analysis I course-sequence-wise, could perhaps help attest to my abilities there. 

[StatsG0d]

I think that of these courses the best indicator of your mathematical ability would be Complex Analysis. I've never taken it, and I've heard it's very difficult. But if you can manage a good grade in that, I'm pretty sure that adcoms will be forgiving on your Real Analysis I grade.

[GeoDUDE!]

Numerical Analysis is learning how to design algorithms to approximate solutions to equations with no real solution (Like Navier-Stokes in 3-D). 

 

I haven't taken an undergraduate real analysis course, but I have taken a graduate level one and if thats any indication, real analysis is a much much more difficult class. I pulled 2  all nighters a week getting the problem sets done for real analysis, where as i was sleeping just fine during numerical analysis.

 

My impression is that if you got less than a satisfactory grade in real analysis but feel you learned a great deal from it then you will be just fine in complex (as far as fine is in these kinds of courses). Just make sure that you go in knowing why you got that grade and what you can do to make a difference in complex. Perahps, even talking to your teachers/advisors about this may help you a great deal.

[MathCat]

My primary interest is in analysis, so I've taken all the real and complex analysis courses my undergrad university offers. Some people find analysis very, very difficult. Others struggle with a first introduction, but then get their foothold and do well. I was in the latter category, but I can't say for you. I don't know anyone who found it easy from the beginning, but I'm sure they exist.

 

If you think that you really understood the techniques by the end and could be comfortable with epsilon-delta style proofs (pretty much all you ever do in analysis), you should take the next real analysis course. I think that's the most relevant to statistics, and will show you can do that stuff. Complex analysis is a bit different from real analysis. It would probably be a bit harder at this point. When I took it, there was a fair bit of background in real analysis assumed, and it doesn't sound like you saw that much. Did you cover differentiation and integration in real analysis I? Taylor's theorem? Those are pretty much where a complex analysis course will go, and then develop some stuff that you don't do in the real case (e.g. Cauchy's theorem, Laurent series, poles, residues).

 

Have you done a complex variables course in undergrad? Many places seem to have a course where you do the computational aspects of complex analysis without proof - e.g. integration with residues, taking nth roots and such. That would make the complex analysis course a bit easier.

[epimeleia_heautou]

Two thoughts:

 

1) Echoing what others had said above, complex analysis is a different beast than real analysis, at least in its more interesting applications. A different style of reasoning is often employed in complex analysis (as opposed to real analysis) and it can be difficult to get used to. I would choose between the real analysis and complex analysis courses based on which one you think you would do better in. The other courses you listed are not as directly relevant to statistics.

 

2) Does the university offer a course in measure theory or probability theory over the summer? Doing well in either of those could be very helpful. But they can also be quite difficult at advanced levels if you have not seen too much proof-based maths.

[MathCat]

Two thoughts:

 

1) Echoing what others had said above, complex analysis is a different beast than real analysis, at least in its more interesting applications. A different style of reasoning is often employed in complex analysis (as opposed to real analysis) and it can be difficult to get used to. I would choose between the real analysis and complex analysis courses based on which one you think you would do better in. The other courses you listed are not as directly relevant to statistics.

 

2) Does the university offer a course in measure theory or probability theory over the summer? Doing well in either of those could be very helpful. But they can also be quite difficult at advanced levels if you have not seen too much proof-based maths.

Yeah, I think real analysis is the most relevant, and would help with taking measure theory or probability down the road. But a rigorous course in either of those would probably be too much at this point.

[Mechanician2015]

An undergrad course in Numerical analysis won't focus on demonstrations nor proofs. You basically implement algorithms to solve problems through arithmetic operations(it often involves coding/programming).[Paradoxical. A first course in Numerical analysis is actually more about "Numerical methods" than about the actual analysis of convergence/proof of such methods].

 

It won't serve to show your analytical skills, but whenever you read "simulation", "computational", "numerical", you are dealing with a branch that requires some knowledge on numerical analysis. It might not impress the AdComm, but it can be a major advantage once you are in(depending on what you want to do).

[ah25]

Two thoughts:

 

1) Echoing what others had said above, complex analysis is a different beast than real analysis, at least in its more interesting applications. A different style of reasoning is often employed in complex analysis (as opposed to real analysis) and it can be difficult to get used to. I would choose between the real analysis and complex analysis courses based on which one you think you would do better in. The other courses you listed are not as directly relevant to statistics.

 

2) Does the university offer a course in measure theory or probability theory over the summer? Doing well in either of those could be very helpful. But they can also be quite difficult at advanced levels if you have not seen too much proof-based maths.

 

Thanks, but Measure Theory isn't available! There is a calc-based probability course, but I already have that. 

Edited April 17, 2015 by ah25

[MathCat]

Measure theory and probability (at the level of rigour we're talking about) would require you to have taken all this real analysis stuff already. So it's a good idea to have it.

[ah25]

Measure theory and probability (at the level of rigour we're talking about) would require you to have taken all this real analysis stuff already. So it's a good idea to have it.

 

Yes, Real Analysis II would be optimal over the summer, but that unfortunately is not available/offered.

Edited April 17, 2015 by ah25

[bayessays]

How bad of a grade are we talking here (people here have wildly different versions of what a "bad" grade is!)?  If it's in the C range, I'd say take Real Analysis 1 again if you're sure you can do better this time around.  Maybe even with a B-.

 

Keep things in perspective.  Schools like Berkeley and UChicago are going to be hard for anyone to get into, especially somebody who goes to an unranked school and didn't do great in real analysis.  But if you got a B and have 3.8 otherwise, you're still probably a good candidate AS IS for a lot of fantastic PhD programs.

Doing well in real analysis will definitely look the best.  Numerical analysis would be very useful and look pretty good, but not make up for real analysis.  Linear algebra is always useful, but since you had a lower-division one and did well I don't think it would add much to your application.  Abstract algebra and complex analysis will help show you are good at math, but won't be very useful.

[Ellies]

The first four chapters of principles only get you started in real analysis; at a minimum you need to do the differentiation and integration chapters (5 and 6) too. I would do the course again and make sure you excel in it, then read the next few chapters yourself; it should be easier if you understand the first 4 chapters well.

Also, since this course is at a higher ranked university, chances are you may get a bit further this time around.

Edited April 17, 2015 by Ellies

[MathCat]

The first four chapters of principles only get you started in real analysis; at a minimum you need to do the differentiation and integration chapters (5 and 6) too. I would do the course again and make sure you excel in it, then read the next few chapters yourself; it should be easier if you understand the first 4 chapters well.

Also, since this course is at a higher ranked university, chances are you may get a bit further this time around.

I agree. If the list you gave me is comprehensive, I think there is probably a lot more to see in the other real analysis I course. That seems like the best bet to me. Unfortunately they have the same name, so it might look like you are repeating the course. Some of my applications had me include a description of what courses I've taken + texts used, so that would show that it's not a total repeat course.

[ah25]

Thanks for the additional suggestions on just taking/re-taking Real Analysis I! 

 

The grade I got before on this course was a B+, by the way. I realize to some it may seem like I'm making a mountain out of a molehill, but I really feel like I have very little margin for error, especially on this particular course. 

Edited April 17, 2015 by ah25

[bayessays]

I think retaking a course you got a B+ in would be overkill.  I still think numerical analysis would be the most useful and some programs specifically look for it.  But I also think if you got an A in Abstract Algebra or Complex Analysis, that would do the most to alleviate any concerns about the B+ in analysis (which I don't think you need to be too concerned about).

What type of PhD programs are you aiming for?  You mentioned UChicago.  You would need to take the math GRE, in which case abstract algebra would probably fill the most holes to help with that test.

[ah25]

Yeah, I will be aiming for the top programs--don't know which specifically yet. However, I don't think I will go for any school that would require the Math GRE--even with Abstract Algebra under my belt, I don't think I would do too hot--very tough to do well against the competition on this test!

Edited April 17, 2015 by ah25

[epimeleia_heautou]

For what it's worth, I had some success this past application season despite having worse grades in a lot of upper level maths courses (even in measure theoretic probability, which is something you don't want to do poorly in when applying to stats PhD programs). What made things work out was that I had strong letters from professors who knew me well and who could vouch that my grades were not indicative of my true mathematical ability. (It also helped that I was able to pursue activities in between graduating from undergrad and applying that helped to illustrate this.) Your mileage may vary (this will depend in part on the reputation of your undegrad institution, among other things), but if you can work closely with a professor/academic mentor who can write a good, personal letter discussing your mathematical aptitude and ability to do original work, then I wouldn't worry about a B+ in real analysis. If that means taking time between graduating and applying, that shouldn't hurt your chances.

[wine in coffee cups]

Here's my contrarian vote for the advanced linear algebra course, assuming that's the theoretical and proof-based version of the more mechanical lower-level course. Special matrix decompositions (SVD, QR), eigenvalues and eigenvectors, orthogonality, changing bases, projections, block matrix properties, etc. are sooooo important and useful in statistics. I did well in the lower-level linear algebra class but didn't feel like I really understood a lot of this until I took the proof-based upper-level version.

Edited April 19, 2015 by wine in coffee cups

[velua]

I'd say Complex Analysis is your best bet as long as it's a "real" (oops) complex variables class, not some easy engineering class where you sit around applying de Moivre's Theorem. Abstract Algebra could also be good.

Sounds like your analysis class didn't cover too much if you didn't get into derivatives, integrals, or sequences of functions. I don't like Rudin, especially not for integrals.

I took both real and complex classes at both the undergrad and grad level. I didn't like real at either level to be honest. Complex was alright at the undergrad level, all about computation, the class was mostly engineers, not math majors. The grad complex analysis classes I took were great though. There was a lot of nice analysis arguments in the material, similar to real analysis, but the graders weren't anal about things like showing something is < epsilon instead of 3epsilon, etc. The material got hard when we got into several complex variables though.

As for real, I want to mention how great Bartle and Sherbert's book is. I used it in my undergrad real analysis class, and I hated the book back then. But then when I was using Rudin in grad school, I pulled out my old Bartle book and it made things so much nicer for me, especially during the integration chapter.

I think you can do fine in complex even if you didn't get into Taylor's Theorem in real analysis yet. I mean you learn all that stuff about derivatives as limits, Riemann sums and Taylor's Theorem in freshman calculus, even if at a higher level, so you can still understand what's going on.

I wouldn't worry about a B+ though.

I also agree with the last poster that a good linear algebra course is important if what you did in your course was just row-reducing and inverting matrices and so on. This is especially true if you still need to take the Math GRE in the fall.