Lambda the Ultimate

Sorry, yes, what I meant by the "very strong emphasis on calculus" was that there are 6 required courses (24 credits) in Calculus. I haven't been able to get ahold of what books they use yet, but I'll do so by contacting one or more of the teachers directly.

Thank you for informing me about ABET, I had not heard of it before today.

Waiting until Calc 4 to do multivariate calculus is weak. It's literally as slow-paced as possible. They cover things like Green's Theorem in the fifth course, Advanced Multivariate Calculus. If there is not a heavy emphasis on proof in the coursework, then this is clearly a curriculum slanted towards engineers and not pure mathematics (and likely not theoretical physics, either). Unfortunately, the course descriptions are infuriatingly vague and do not even discuss performance requirements of students e.g. ability to do proofs.

On the other hand, the undergrad curriculum broken up into small chunks makes it easy for graduate students needing missing credits to take some undergrad courses before starting real coursework.

BTW, ABET by no means "great teachers". It just means certain quality standards are met.

Yes, I agree. Even at the previous community college that I was at (a few years ago) these subjects were at least introduced earlier. I don't recall the name of the textbook right now.
Also, yes, it is really annoying to look at these course descriptions. I noticed that, in c.s. at least, a few of the prerequisites were wrong or missing.
By the way, I will be attending a community college before I go on to Portland State for unrelated reasons. However, I see that they're rather aligned with the uni: http://spot.pcc.edu/academ/math/descriptions.htm.
The one thing to consider here is that it may be possible to take some courses in parallel (as it was in some cases at my previous CC), even if they are prerequisites. As usual, the course descriptions give no information regarding this.
Anyway, at least I a better idea of what to look into now.
Thanks again.

I looked at Oregon State's curriculum which seems a bit more organized. As I suspected, some of the calculus courses are intended to be done in parallel. That is, the third and fourth semesters consist of two parallel courses in calculus each. Since everything else here (credits, course numbering, etc) seems to be the same and they are in the same state, I suspect that this is the same for PS. I'll double check though.

Hmm, I haven't considered this in quite a while since all the MS/MA programs that I encounter seem to require a BS or BA first. I'll have to look into that more.

Unfortunately I haven't worked extensively with calculus since high school and a bit of physics in college; although I did have a exceptionally good high school calculus program and I still have a pretty good command of much of the material.

Calculus class would also be a good time to study things such as floating point arithmetic, numerical analysis, and automatic differentiation outside of class. I feel more than a bit cheated that I had not even heard of automatic differentiation until a few years ago, at a time I don't use calculus often and certain parts of my understanding are a touch rusty. :-)

Thanks for the paper using calculus in type theory. I do appreciate that as motivation. One thing I did think about is that there's no way I'd want to skip out on calculus completely as what kind of mathematician doesn't really know calculus? That would have me excluded from plenty of interesting conversations, I'm sure. On the other hand, the more I look at some of these curricula, the more they look like a "calculus major" instead of "mathematics major."

Anyhow, thank you for your input, sigfpe.

- Tony

Funny.. I'd suggest skipping calculus and doing real analysis instead (or maybe complex analysis for a generally wacky time.) (Vector) calculus is often geared towards those not planning to advance further. There are a few theorems that are spectacularly presented in a typical vector calc class, but I'm not sure it's worth it at the expense of other topics. (If you want to be practical, linear algebra and stats take you far, and are chock full of cool yet practical results at the introductory level!)

Being taught math in a way that sets you up for connections to CS... is unlikely, or at least relative to doing it on your own. There are a few exceptions -- information theory, whatever logic/philosophy, classes, number theory, anything the applied statistics/numerics side -- but that's been my experience. Connecting to PL is way rarer (e.g., certain types of logic, category theory seminar @ stanford last year) which is why self-study, perhaps based on such materials, makes more sense to me (though you likely miss out on most of the message this way).

Which other topics do you expect from an analysis course that you wouldn't learn in a calculus course?

I don't understand the question -- they're completely different courses. In a sense, analysis and abstract algebra are the first real foundational math courses, with linear algebra acting like an early teaser. In analysis, you relearn half of what you were taught before (e.g., what is a real number, one or two forms of continuity, covers, mipoint theorems, etc.), and this sets you up for topics like topology and foundations of probability (e.g., metric spaces and martingales, and various analytic tricks). Vector calculus is more like a continuation of the high school math curriculum and thus sits on a foundation that is not rigorous so you miss out on the generality (or not) of the results. Not to say it isn't useful -- physics and engineering live on this stuff.

Thanks. I asked because in my university there is no course called calculus, only analysis. We learn what is a real number, convergence, midpoint theorems, differentiation, power series, integration etc. in a course called Analysis I. Vector differentiation/integration & the theorems that come with it (green,gauss,etc) in a course called Analysis II. So this is actually analysis and calculus mixed together?

Analysis I is usually rigorous calculus. Analysis is more general than calculus, though, including (at least) differential equations and functional analysis (Hilbert & Banach spaces, etc).

Calculus courses usually teach you how to use Calculus, Analysis courses typically teach you how to prove that Calculus works.

Analysis also generalizes concepts in Calculus. For example, Calculus studies the convergence of sequence of real numbers. Analysis covers the convergence of sequence of real functions, both pointwise and uniform convergence.

My university experience was that nearly all of the math and physics majors I knew had "placed out of" calculus when they entered as freshmen. And this was 40 years ago....before the wide use of AP classes in much of the U.S. Maybe my friends were unusual...

I found the OP's comment that one of the colleges he is considering emphasizes calculus...this just doesn't square with any of the undergraduate math syllabuses (sylabbi?) I have seen, at least for major schools.

My freshman year I took a couple of linear algebra classes. Very useful. One was oriented toward manipulating matrices, the other was more abstract.

An analysis class I took started with the construction of the reals via "Dedekind cuts" and introduced more rigorously a lot of stuff I found very useful many years later when my interest in math and CS was rekindled. Really, it was an introduction to point set, or general, topology. Notions of open sets, fixed points of mappings, compactness, and the whole "abstractness" point of view which makes much of theoretical computer science and category theory much more digestible. (It also makes the Haskell and FP talk of catamorphisms and suchlike a lot more "familiar."

A clue is that there's a catchphrase in a lot of books: "mathematical maturity." I just don't think studying a lot of calculus is any real substitute for studying abstract algebra, real analysis, and some general topology.

So for someone thinking about a research career in theoretical computer science, my advice would be "go light" on physics-oriented calculus and go heavier on at least some of the upper division math major classes. (Most of the abstract algebra, real analysis, and topology classes happen in junior and senior years.)

But as sigfpe noted, calculus does have a lot of interesting problems. For a motivated student, a lot of windows into other areas. But my experience with college-level calculus is that neither Apostol's nor Spivak's idiosyncratic texts are used; rather, the focus is on solving integration problems and the like. I think anyone who _stopped_ after calculus would be mystified by, say, Mac Lane's "Categories for the Working Mathematician."

Of course, a lot of cool math can be learned on one's own. In fact, for the more abstract stuff, I prefer to work at my own pace....so if I have to spend 4 weeks figuring out why "the product of compact spaces is compact" is so darned important, I can spend this time. And maybe explore it with some Haskell code....

--Tim May

...calculus is"

...which is why some courses in 'topology' actually end up being a course in real analysis, like the one I sat in on my last semester in college.

I think being exposed to a wide spectrum of math is beneficial. For example, the Mean Value Theorem taught in calculus courses does not hold for geometries not dependent upon the Euclidean Parallel Postulate. Thus, any result that depends on MVT cannot be used in such a geometry. This sort of understanding, of forming minimal models of tasks, such as finding a parallel line, is what math should teach you. You might not realize your MVT-derived results are dependent on a 'constant' such as EPP, but if you are exposed to analytical approaches to non-Euclidean geometries (e.g., hyperbolic geometries), you would learn otherwise.

MVT (mean value theorem I presume) isn't a statement of geometry and isn't dependent on the Euclidean parallel postulate.

I had a similar reaction, but then thought maybe he was interpreting the MVT geometrically as "every secant line to a smooth curve in the plane is parallel to the tangent at some intermediate point." This would certainly be false on spherical geometries :).

Edit: Of course, I think I can see my way to the end of a proof that this MVT does hold in hyperbolic 2-space (assuming parallel lines are merely non-intersecting or coincident), so I don't know...

But we can certainly agree that this is not "taught in calculus courses"!

Regarding your first paragraph, I have to agree with that. It does seem tailored for people going into sciences that are being forced to take mathematics.

One thing that I did not previously mention is that, for now anyway, I'm going to have to go to an Oregon university. With that premise out of the way, I think it's worth mentioning that it seems that most of the Oregon universities are the way you describe. I realize there are private schools but I haven't looked into that too much yet. (The financial aspect of that especially concerns me right now.)

Thank you for your second two tips. I'm proud to say that I have, in the past at least, done what you suggest in point 3. I spent a lot of time outside of class during my first semester of calculus trying to figure out what the Taylor series for some function may be and checking with my teachers. As for now, pretty much everything is self-study.

Anyhow, yes, maybe I should be looking into what other Uni's are available (public or otherwise.)

Right, oops. I found a program at OSU called "Mathematical Sciences" which I hadn't noticed before. It seems like it may be exactly what I'm looking for. I'll look into this more.

I don't know much about CS at the Oregon unis. I think some of them have good software engineering programs. There is or was some Haskell research somewhere in Oregon; maybe OGI. Washington is supposed to be very good, but I think they're strong in applications and maybe not so good in theoretical CS.

If you have to stay in Oregon AND you really want to get into research I'd suggest:

0. You should aim to absolutely destroy your coursework thereby getting top grades and thereby getting admission to a top place for your PhD.

1. Go to a uni where there is someone doing stuff that interests you. This way you'll at least have a supervisor for a relevant undergrad/masters project. Do be aware that research interests often change (mine certainly have) and you might end up doing something completely different.

Yes I've heard that UW has extremely good CS programs for both engineering and theoretical work.

Re #0: Absolutely. If I do enter this program at Portland State, that is indeed my plan. And there doesn't seem to be much of an excuse for not doing so since it is so spread out over time.
Re #1 - Changing interests: Yes, I've been thinking about this too. Much of my interest in the theoretical has to do with the practical "issues" we have in software engineering as it is currently practised. So, I could definitely see a change in this if there is a significant change in the industry. On top of that, it may be that doing all of this work that I'll have to do, which is, according to me today, completely unrelated may change my interests.

Regarding Haskell, yeah, I did read that somewhere. I wonder who that was and where it was.

As for what I read about Haskell work being done in Portland, it seems that Galios (http://www.galois.com/) is based there. It seems that they not only do a lot of work in Haskell and for GHC but that they also do some related education as well.

You should consider signing up for the pdxfunc mailing list.

There is or was some Haskell research somewhere in Oregon; maybe OGI. Washington is supposed to be very good, but I think they're strong in applications and maybe not so good in theoretical CS.

OGI shut down their computer science research to focus on health related areas, many of those CS people are now at PSU (e.g., Tim Sheard). PSU isn't such a bad bet, and you get to hang out in Portland rather than live in some small town somewhere. PSU is probably a better bet for CS than UO. But at the undergrad level, just learning and working on cool projects yourself will get you a long way into grad school.

it's just advanced applications of CS, as it relates to health-related areas. It's interdisciplinary and most focused on things like spoken language understanding.

Not all Universities can get the best students; they have to tailor their courses to the students they can get. In the case of PSU it seems they have fairly average students, and their courses reflect that. The best Universities will give you an excellent education whether you take Maths or CS (and you'll get plenty of maths at top CS schools).

PSU hasa couple of excellent programming language CS faculty inherited from OGI. Andrew Black has a good OO rep while Tim Sheard has a good functional programming rep.

Good to know.