For fun: favorite UG math course

[bigbadwolf]

Many of the more interesting areas of math aren't in the poll (presumably because prospective quants wouldn't touch them with a barge pole): complex analysis, number theory, differential geometry, algebraic topology, algebraic curves, combinatorics, classical mechanics.

 

[MikePski]

Hm... I'd have to say statistics/econometrics because they were the easiest.

 

[chaoticrambler]

In my humble opinion, it's useful to see it in the finite-dimensional case first, to understand what it says, and the method of proof. Usually a first undergrad course in the subject has the spectral theorem as its high point. The spectral theorem for compact (infinite-dimensional) linear operators needs to use more complex tools from topology for its proof, and this applies to the even more recalcitrant bounded linear operators. The spectral theorems in these cases are generalisations of the finite-dimensional case, which falls out as a corollary. But I reiterate that the f-d case should have been seen first -- it's useful pedagogically. The same applies to dual spaces and annihilators. The Schaum series book on the subject (by Murray Spiegel?) is quite good on the subject and there are a couple of Springer undergrad texts (Curtis? Smith?) that are also sound.

Oh sir I definitely agree. I would be the first to admit my math education hasn't been very well planned, but that is partly due to the fact that I jumped from engineering in undergrad to applied math type stuff in grad school (applied dynamical systems).
I read up material on finite dimensional case on my own to even begin to understand what was happening in the infinite dimensional case. The second course in sequence for functional analysis (which covered spectral theorem) was the hardest course I ever took for that very reason.

 

[IlyaKEightSix]

Many of the more interesting areas of math aren't in the poll (presumably because prospective quants wouldn't touch them with a barge pole): complex analysis, number theory, differential geometry, algebraic topology, algebraic curves, combinatorics, classical mechanics.

Combinatorics may or may not be. I have a friend at Lehigh getting his PhD in info sys engr. with a heavy quant finance application that respects combinatorics.

As for the rest of that stuff you listed...of course quants wouldn't touch it.

Difficult abstract math with no or very convoluted real-world applications is a quick way to get frustrated, stressed out, your GPA dragged through the mud, all while learning relatively little in terms of applied, marketable skills.

I guess that's the difference between engineers and theoretical mathematicians--engineers want to make something work, while theoretical mathematicians simply pursue the elegant and interesting, no matter its lack of actual societal meaningfulness.

 

[bigbadwolf]

I read up material on finite dimensional case on my own to even begin to understand what was happening in the infinite dimensional case. The second course in sequence for functional analysis (which covered spectral theorem) was the hardest course I ever took for that very reason.

Same here. I took a course on linear operators in Hilbert space having taken prior courses in linear algebra and real analysis but because the courses didn't dovetail together well, I had to learn a lot of linear algebra on my own in order to understand the more abstract results of Hilbert space theory, which were often attempts to generalise finite-dimensional results about linear operators to the compact or bounded cases. It's interesting to see the way general topology blends with linear algebra in the infinite-dimensional case to produce proofs. One of the charms of functional analysis.

 

[chaoticrambler]

Combinatorics may or may not be. I have a friend at Lehigh getting his PhD in info sys engr. with a heavy quant finance application that respects combinatorics.

As for the rest of that stuff you listed...of course quants wouldn't touch it.

Difficult abstract math with no or very convoluted real-world applications is a quick way to get frustrated, stressed out, your GPA dragged through the mud, all while learning relatively little in terms of applied, marketable skills.

I guess that's the difference between engineers and theoretical mathematicians--engineers want to make something work, while theoretical mathematicians simply pursue the elegant and interesting, no matter its lack of actual societal meaningfulness.

I am pretty sure a lot of Math/Physics PhDs who are quants now are familiar with a lot of those areas.

I worked in classical mechanics/geometric mechanics myself and also took a course in topology/smooth manifolds (pretty hard btw).

 

[Theal]

Combinatorics may or may not be. I have a friend at Lehigh getting his PhD in info sys engr. with a heavy quant finance application that respects combinatorics.

As for the rest of that stuff you listed...of course quants wouldn't touch it.

Difficult abstract math with no or very convoluted real-world applications is a quick way to get frustrated, stressed out, your GPA dragged through the mud, all while learning relatively little in terms of applied, marketable skills.

I guess that's the difference between engineers and theoretical mathematicians--engineers want to make something work, while theoretical mathematicians simply pursue the elegant and interesting, no matter its lack of actual societal meaningfulness.

Taken to the extreme edge of pure mathematics, it's just about the opposite of practical, I'd agree. Well, I guess until someone finds a use, re: Knot theory and DNA.

However in fields that are fast evolving (I'm looking mainly to Computer Science, my feild of knowledge, though I'm sure there are others), the analytical rigours of a pure mathematics course keep you on your toes, and being adept at proofs is important. And again, who knows, maybe you'll be the lucky one find an application.

I'd have to say my introductory combinatorics and graph theory course. Could be the smart professor, but I really enjoyed it. The combinatorial proofs at that point were very new and interesting.

 

[IlyaKEightSix]

Oh--no question about that. An occasional pure math course just to keep you sharp on your analytical techniques is definitely good (and was one of my most favorite courses in my last semester of undergraduate studies), but the idea to just go overboard on this, that, and the other thing with no application to the real world seems to me a bit wasteful.

As for finding an application, why not simply attack a real world problem with whatever tools you have and see what works?

To paraphrase the flavor text of Goblin Piledriver, which reads:

"Throw enough goblins at the problem, and it'll go away. At the very least, there'll be fewer goblins."

Throw enough approaches at the problem, and it'll be solved. At the very least, you'll have saved a lot of other people a lot of time.

 

[Theal]

As for finding an application, why not simply attack a real world problem with whatever tools you have and see what works?

While I don't think we're really argueing much differant points at the moment, I have to say I am a bit confused with your attitude towards pure maths. I can't claim to be a historian of any sort, but I can think of a few examples off the top of my head where the application came after the math (a la graph theory, and a few bits of quantum stuff I know was built on previous math). In any case, subjective of course to your field, some of the tools you have are there because of pure mathematics. I mean, personally I wouldn't have the drive to bury myself too deply in that area of math, but for those that do, more power to them. I don't see how you can describe it as "useless" when it broadens our understanding of the world (again, maybe this is subjective as well), and has provided useful tools.

 

[chaoticrambler]

Oh--no question about that. An occasional pure math course just to keep you sharp on your analytical techniques is definitely good (and was one of my most favorite courses in my last semester of undergraduate studies), but the idea to just go overboard on this, that, and the other thing with no application to the real world seems to me a bit wasteful.

As for finding an application, why not simply attack a real world problem with whatever tools you have and see what works?

To paraphrase the flavor text of Goblin Piledriver, which reads:

"Throw enough goblins at the problem, and it'll go away. At the very least, there'll be fewer goblins."

Throw enough approaches at the problem, and it'll be solved. At the very least, you'll have saved a lot of other people a lot of time.

There is a whole lot of progress that has been made because of the abstract math people have come up with over the years. There is simply no end to the kind of stuff that has been find useful which was supposedly useless at the time of its creation.

Cryptography evolved from number theory, which is arguably the most abstract of the mathematics. Even knot theory has applications in chemistry, and genetics.
I will give you a related personal example: I am working with a group that uses the "theory of braids" to make very exact estimates regarding very complex fluid flows.
Braid theory is a super abstract geometric theory and was not supposed to have any applications. Turns out, it will be used some day to analyze nano-fluidic systems.

 

[DanM]

I agree with the previous two posters. I don't see how anyone can legitimately and seriously discredit the usefulness of pure mathematics, when a large part of the mathematics that is applicable to the real world is derived from it.

 

[Jedison]

complex analysis

 

[mastertrader]

I enjoy Probability and Statistics, because I'm very interested in Game Theory and finding optimal strategies. I enjoy Applied Math in general, but I haven't taken a course in Numerical Methods/Analysis. I've done all the others listed as well as Statistical Forecasting, Abstract Algebra, and Complex Analysis. Statistical Forecasting was interesting, because we got to learn about Time Series.