Where to specalize in Applied Mathematics?

[MathWarrior]

I am currently enrolled to become a Applied Mathematics major, the school I will be going to has a relatively small number of classes for upper division math so I am thinking I should try and specialize in something. Even though their is a limited amount of courses to choose from being as the math program is rather new.

I can choose for upper division math courses: Optimization, Linear Optimization, Partial Differential Equations, Mathematical Models in Biology, Intro to Chaos Theory and Nonlinear Dynamics. Their are also some other courses they have but I am not sure if they count as upper division such as number theory and abstract algebra.

I am mainly studying math so I can pursue more complex topics in computer science that have mathematical basis to them. I realize this would be more discrete mathematical structures, and things like combinatorics or even complex analysis.

So given that theirs only so many courses really offered, thus far, which would be best suited for someone who is intending to do graduate level computer science topics? I also am somewhat interested in scientific computing and high performance computing.

 

[chiro]

Hey MathWarrior

From your post it seems that you want to focus on an algorithmic approach to mathematical topics. By algorithmic I mean that you focus on the algorithms, data structures and specific implementation details on various architectures.

Is this a good characterization or is it way off?

 

[MathWarrior]

Is this a good characterization or is it way off?

Yes this is pretty close to what I was thinking of going into later on. I rather enjoy algorithms in a computer science perspective. However, since most of that comes from mathematical concepts I am pursuing math for this reason, especially in upper division algorithm design.

 

[chiro]

Yes this is pretty close to what I was thinking of going into later on. I rather enjoy algorithms in a computer science perspective. However, since most of that comes from mathematical concepts I am pursuing math for this reason, especially in upper division algorithm design.

In that case, the course selection will most likely be one of preference since each course will have its own particular focus.

In terms of practicality I can see pretty much every course in that list being very high in this regard. I don't know what's involved in mathematical biology but I imagine it has just as much application especially with the growth in the biotechnology industry as well as the other scientific applications.

The thing is that because what you're looking to do is very interdisciplinary, what you will probably be looking to do is get into some kind of graduate program or be hired by the right company to get some experience to work on this kind of thing.

There are definitely companies out there that take this kind of thing very seriously. One example is in finance where computations have to be as fast as possible.

In fact anywhere where the amount of computational power is huge will no doubt have an area where people actively research the problem, or have people that they can go to for advice who do the research and have experience in their own capacity.

So yeah if you can't get any real interdisciplinary experience in your undergraduate degree whether that is the form of coursework or otherwise, then apart from the right graduate programs, also consider looking at companies that have huge investments in a lot of computing power because they will have every advantage of shaving a few machine cycles off every loop if that loop has to be run millions or billions of times (maybe even more!).

 

[MathWarrior]

In terms of practicality I can see pretty much every course in that list being very high in this regard. I don't know what's involved in mathematical biology but I imagine it has just as much application especially with the growth in the biotechnology industry as well as the other scientific applications.

The thing is that because what you're looking to do is very interdisciplinary, what you will probably be looking to do is get into some kind of graduate program or be hired by the right company to get some experience to work on this kind of thing.

I was thinking of taking the optimization and the non linear dynamics and chaos theory course. Even though I am not that fond of differential equations. I figured optimization would at least introduce common algorithms like the simplex method/convex hulls, and non linear dynamics and chaos theory might be useful for something in computer science not quite sure what though.

 

[chiro]

I was thinking of taking the optimization and the non linear dynamics and chaos theory course. Even though I am not that fond of differential equations. I figured it would at least introduce common algorithms like the simplex method/convex hulls and non linear dynamics and chaos theory might be useful for something in computer science not quite sure what though.

The computer science part will focus more or less on analyzing existing problems and existing algorithms usually in the context of run-time under different situations as well as for different architectures if you want to describe implementation specifics that might change run-time attributes slightly.

It's not to say that the domain knowledge of non-linear dynamics or chaotic systems won't be useful: it will be when you have to analyze run-time and order information and suggest improvements if you know conditions that you can exploit to reduce these characteristics.

This is the reason why interdisciplinary work is important because a computer scientist will probably not be aware of certain conditions and other information to exploit that results in a better algorithm and the applied mathematician probably won't be aware of the different architectures, processing models, and other details that would otherwise give an optimal implementation.

You should keep this in the back of your mind. In other words, it's not so much learning standard algorithms even though it is really important to do this initially when you start your training: it's more so having enough depth in both areas in the right ways so that you can exploit the necessary details to achieve your goal which will most likely be introducing better algorithms that run faster. You might have other objectives, but if you are going to focus on the algorithmic side which is what you seemed to imply, then this is going to be a critical thing to keep in mind.

 

[MathWarrior]

What class would be most relevant to computer science in terms of math from the list provided? I am trying to find out, would one be more beneficial long-term then the others? I just want to make sure I take relevant courses. I realize its probably largely based on which field you ultimately end up in, or what you end up doing. However, would any of these be large scale, in that topics gained from learning the math behind it be extremely useful in the future in all sorts of situations.

 

[bpatrick]

From reading the thread so far, I'd suggest:

Algebra, Number theory from that list (both are more "pure" math but loads of pure and discrete areas have applications to computer science), and also Optimization and Linear Optimization would be the next most valuable to you on that list.

I'd also look for some electives in:

Discrete Mathematics, Combinatorics, Graph Theory, Algorithm Analysis.

I don't see PDEs nor the dynamics/bifurcation/chaos being super useful to you, same with math bio ... but who knows, if you have the time and there aren't any better electives, then why not.

Most of the math-bio courses use JD Murray's Mathematical Biology I & II books, and if it's strictly an undergrad class (no grad students also taking it as a first part in a 2-3 semester sequence), you will probably spend the great majority of the course only doing the population dynamics chapters from the first book. You might get some exposure to reaction kinetics at the end, but the pace of the class is probably not going to be that fast. Either way, none of this stuff is terribly applicable to computer science, so it's probably best to find more pure math or computer science electives instead of math-bio.

 

[MathWarrior]

From reading the thread so far, I'd suggest:

Algebra, Number theory from that list (both are more "pure" math but loads of pure and discrete areas have applications to computer science), and also Optimization and Linear Optimization would be the next most valuable to you on that list.

So should probably look into number theory, abstract algebra and the optimization courses then? Where does chaos theory appear in the real world applications then, I've always wondered.

 

[bpatrick]

So should probably look into number theory, abstract algebra and the optimization courses then? Where does chaos theory appear in the real world applications then, I've always wondered.

Well, one thing I use elements of it for is analysis of stochastic processes in statistical mechanics, physical biochemistry, and molecular kinetics (going to school for an MD/PhD in biophysics). I'm not sure, but I'd imagine it's used a lot in particle physics, mechanical engineering, geophysics, and very high end computer / electrical engineering. My knowledge of chaos is limited to what I got from Taylor's 'Classical Mechanics' and Hale/Kocak's 'Dynamics and Bifurcations', which hardly makes me an expert.

PDEs are used a lot in physics, engineering, finance, and mathematical biology. Just as there are some computer science applications of chaos and dynamics, there are surely applications of PDEs in the computer fields, but I'd imagine it's not as useful as courses on optimization, algorithms, number theory, etc...

So overall I think you'll be better off taking more pure and/or discrete math vs the applied/engineering courses. Just keep in mind that this advice is based on me being familiar with the "applied" courses and what I was presented with as their applications, rather than me actually being a computer science mathematician ... so if one of those guys actually comments, heed their advice more carefully than mine.

Adding diversity to your schedule isn't a bad idea either if you do end up taking any of those courses (possibly due to scheduling conflicts / professor sabbatical / etc...). You may find you enjoy those topics more than your other classes ... which may inspire a change in direction for grad school, who knows.

When I was an undergrad, my main science interest was differential equations and modern physics (mainly for hobby considering I was a musicology major). I had no interest in combinatorics, graph theory, algebra, topology, numerical analysis, and statistics ... but those are the things I'm working with now, go figure.

 

[MathWarrior]

Thanks, I kind of have an idea of where I might want to kind of go into now. I would however like to hear from anyone else who has taken chaos theory, optimization, number theory, abstract algebra, and so forth maybe get an idea where it leads.