Guide to analysis courses

420/507, 421/510, 516-517, 541, 544, 605F

Unofficial and unauthorized guide to the analysis and analysis-related graduate courses at UBC offered in 2005/06. 420/507 and 421/510 are offered every year, others - as indicated. UBC has very few advanced graduate courses in analysis. The cross-listed 507 and 510 cover mostly undergraduate and entry-level graduate material, and 541 is being offered for the first time in decades as far as I know. Some analysis material is included in PDE and probability courses, but I don't recall any graduate courses from 2000/01 to 2004/05 that would specifically focus on advanced topics in analysis.

If you are preparing for the qualifying exam: 507 may be somewhat helpful, in that it reviews Lebesgue integration and various convergence theorems. But students have reported that only a short part of 507 was relevant for this purpose, so if this is your only reason to take 507, you might be better off studying on your own or sitting in on just a few classes. Most of the analysis part of the exam material is not included or reviewed in any UBC graduate courses. (In particular, none of it is included in 510.) You could sit in on an undergrad course (no, you don't have to explain to me why you don't want to do that), or else you just have to study on your own. This is in sharp contrast to abstract algebra and complex analysis, where the exam material is very well covered by 501 and 508 and exam questions are quite similar (sometimes identical) to 501 and 508 homework. Incidentally, 510 can help with the linear algebra part if it includes elements of spectral theory (eigenvalues, eigenvectors, etc.) - which, traditionally, it doesn't.

420/507, Real Analysis I (senior undergraduate) combined with Measure Theory and Integration (graduate): Measure theory for both Lebesgue and more general measures, integration, convergence theorems. Additional topics are often included, e.g. elements of probability (as if we didn't already have several probability courses each year!), Hausdorff measures and dimension. The measure theory part can be a bit tedious - I certainly felt that way when I taught 507 a few years ago - but most mathematicians do need it. Even if you are already familiar with Lebesgue measure, you may still need to learn the more general case.
421/510, Real Analysis II (senior undergraduate) combined with Functional Analysis (graduate): This course doesn't know what it wants to be. Functional analysis is quite distinct from real analysis. Also, "Real Analysis II" might suggest a research topics course in real analysis, and that's certainly not what we have here.

Traditionally, the course starts with several weeks of point-set topology. This is material that topologists don't want to include in their graduate courses, because it would be a waste of their time. Don't get me wrong - point-set topology is a lot of fun, I loved it when I was 16. But most of it is undergraduate material and many graduate students are already familiar with it when they come here. More importantly, it's not analysis. Its inclusion not only leaves less time for the actual analysis content, but also makes the impression that analysis is an uninteresting subject with not enough good material to fill a course. Especially because follow-up topics courses are hardly ever offered. WRONG!

The usual functional analysis content is a bit meager. We cover the fundamentals of Banach spaces, L^p spaces, define Hilbert spaces - but then we stop right there. What, no eigenvalues or eigenvectors? No eigenfunction expansions? No compact or symmetric operators, or spectral theory of any kind? Nope. None whatsoever. You could complete the course without ever knowing what you're missing.

Last year (2004/05), I taught 510 as a functional analysis course, cutting down on the diversions (topology) and including more functional analysis material. Since the calendar description includes topology, I couldn't drop it altogether. In the future, I guess it's up to each instructor what he or she wants to do with the course. The 2005/06 outline looked promising, but students report that at least 1/3 of the course was again wasted on point-set topology and that there is little chance that any serious analysis will actually get done.

Perhaps "Functional Analysis" and "Real Analysis II" should be two separate courses, offered in alternating years if they can't both be offered each year? Some of the 2005/06 proposed topics, such as rearrangement or variational inequalities, would be perfect for Real Analysis II.
516 and 517, Partial Differential Equations I and II: Apparently, the plan is to offer this sequence every other year. 516 is strongly recommended, even if your interests are in analysis rather than PDE. The 2005/06 syllabus includes essential analysis topics that are not always covered in undergraduate courses, such as Holder and Sobolev spaces. 517 looks more PDE-oriented, but still could be of interest for students in analysis, depending on their intended research focus.
541, Harmonic Analysis: Offered for the first time in 2005/06. The first part will cover the basic material: Fourier transform, Plancherel, convolution, inversion, Hausdorff-Young and L^p theory. The second half will be much closer to current research topics. Perhaps a harmonic analysis course should also be offered on a regular basis? It could be useful to most analysis and PDE students.
544, Probability I: Offered each year. I would not have thought of it as an essential course for analysis (or, more generally, non-probability) students. But Mariah, who is working in analytic and combinatorial number theory, reports that this course has been immensely helpful to her in her work. Probabilistic methods are essential in analysis as well as other fields (combinatorics, number theory), but many incoming graduate students don't have that background, so this might be a good course to take.
605F, Topics in Applied Mathematics (Wavelets): Also offered for the first time in 2005/06. Normally, I would not include an applied mathematics course here, as most of them focus on numerical applied PDE. This is not the case here. I would expect that interesting topics from harmonic and functional analysis will be covered, and that the balance between theory and computation will be reasonable.