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Advanced Mathematics for Computer Science

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What do you feel are some of the more important mathematical courses / topics that one should learn for computer science?

For example:
Abstract algebra? Number theory? Chaos theory/nonlinear dynamics? Combitorics? Graph Theory? Optimization?

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alvaro    21246
What WaryVirus described all falls under Discrete Mathematics, and it is indeed important.

Although perhaps not very specific to CS, I would add Linear Algebra because it is important for everybody.

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Cornstalks    7030
Yeah, Discrete Mathematics is huge, and I'd put Linear Algebra in there too (especially for graphics, though I use Linear Algebra concepts all the time in various ways). I'd also say a good knowledge of Calculus (and maybe some Differential Equations) can be very handy at times. It seems to me like Discrete Mathematics helps to decompose the problem properly, and then often Linear Algebra or Calculus can be useful in solving some of the decomposed problems.

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Anything else? Discrete mathematics isnt really what I would consider upper division math, but I know its important. Has anyone found applications for Abstract algebra? Number theory? or Chaos theory? If so what are they?

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alvaro    21246
I have used a few things from Abstract Algebra and Number Theory for hashing and pseudo-random number generators. I've used statistics quite a bit (Are those numbers really random? Is this version of my chess program stronger than the previous one?). Probability is really important for almost anything in Artificial Intelligence. If you do graphics, Euclidean and Projective Geometry are pretty important.

I only know a little bit about Chaos Theory, but my understanding is that it's pretty useless. It's just one of these things that have a sexy name and produce pretty pictures, but I don't think you can really do a whole lot with it.

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clb    2147
I enjoyed watching the videos at [url="http://aduni.org/"]ADUni.org[/url]. It shows an example structure of what undergrad CS education is like.

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[quote name='clb' timestamp='1332030366' post='4922930']It shows an example structure of what undergrad CS education is like.
[/quote]
This is a nice link, but I am ideally looking to go into upper level CS stuff such as theortical computer science, algorithm design, hpc, etc.

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alvaro    21246
[quote name='Azh321' timestamp='1332179502' post='4923361']
On top of what everyone said, without a doubt take a graduate course in Neural Networks.
[/quote]
Why?

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daviangel    604
At a minimum IMO:
Discrete math covers such a potporrui of stuff you will see in CS that any decent CS program will require it.
More Statistics than covered in Discrete.
Linear Algebra for graphics and matrix work you will see popup
and
if you can
2nd course in logic preferablly covering HOL(higher order logic) you will see if you ever mess with Lisp, Haskell, and theorem proving.
Abstract algebra if you plan on doing any crypto stuff since a lot of advance number theory is used.
Public key cryptography draws on many areas of mathematics, including number theory, abstract algebra, probability, and information theory.
numerical analysis if you plan on doing any scientific programming or otherwise work with very large or small numbers, etc where results have to be very precise.

Actually, chaos theory comes into play in numerical analysis:
In [url="http://en.wikipedia.org/wiki/Numerical_analysis"]numerical analysis[/url], the [url="http://en.wikipedia.org/wiki/Newton-Raphson"]Newton-Raphson[/url] method of approximating the [url="http://en.wikipedia.org/wiki/Root_of_a_function"]roots[/url] of a function can lead to chaotic iterations if the function has no real roots

bottom line is that you can never take/have enough mathematics as someone once said I'm sure [img]http://public.gamedev.net//public/style_emoticons/default/biggrin.png[/img]

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[quote name='daviangel' timestamp='1332200485' post='4923456']bottom line is that you can never take/have enough mathematics as someone once said I'm sure [img]http://public.gamedev.net//public/style_emoticons/default/biggrin.png[/img]
[/quote]
thats why im majoring in math, I just want to be sure i dont take useless courses [img]http://public.gamedev.net//public/style_emoticons/default/wink.png[/img]

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Emergent    982
The three most useful math classes:

1. Linear Algebra
2. Ordinary Differential Equations
3. Convex Optimization

...and they work best if taken in that order.

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Crowley99    194
In addition to what has been mentioned:
- computational geometry (a must!)
- differential geometry
- topology (primarily because it improves your understanding of the math you already know)
- signal processing

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alvaro    21246
Let's see, what math classes are left? You guys haven't mentioned Algebraic Geometry, Algebraic Topology and Complex Analysis. Oh, but he needs Algebraic Geometry to learn how to manipulate polynomial expressions using Gröbner bases, and Algebraic Topology is important if you want to really understand global features of Differential Geometry, and Complex Analysis is essential for signal processing.

[/sarcasm]

Computer Scientists should learn Differential Geometry? Really? Why? If you are a physicist, sure... but for CS?

Of course, the more Math you learn, the easier it will be for you to think mathematically, and that can be very useful for a computer scientist, but I don't think every CS student should get a Ph.D. in Math to do his job.

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Crowley99    194
@Alvaro: I have found all of these useful in my career - some of them don't often have specific application (like I said for topology), but really help one internalize the concepts taught at the less abstract level. The OP wanted "upper division math" - there aren't a ton of things there that are directly and obviously applicable.

I am comIng at this from the perspective of a graphics/computational-geometry phd with an undergrad math major. I don't think that this math is something that everyone would benefit from directly, but i definitely believe that it would be beneficial for anyone who works in computer graphics at a reasonably sophisticated level - not necessaily because they will use it every day, but because it helps to see the bigger picture (except computational geomety - that you really may use every day :-) ).

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Maze Master    510
I second discrete mathematics, and linear algebra as a background for computer science. However I'm an applied mathematics grad student who also programs, not a true "computer scientist", so take it with a grain of salt..

These subjects can be taught at a low or high level in a class, but the subjects themselves run quite deep. Linear algebra becomes functional analysis, operator algebras, etc - fields of current research. Discrete math branches into combinatorics, graph theory, etc. You don't have to look far in discrete mathematics to stumble on unsolved problems.

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Maze Master    510
If you're interested in abstract Algebra, you might want to check out the harvard video lectures by Benedict Gross; they're really good:
[url="http://www.extension.harvard.edu/open-learning-initiative/abstract-algebra"]http://www.extension...bstract-algebra[/url]

For (convex) optimization, there are two great video lecture series by Steven Boyd at stanford:
[url="http://academicearth.org/courses/convex-optimization-i"]http://academicearth...-optimization-i[/url]
[url="http://academicearth.org/courses/convex-optimization-ii"]http://academicearth...optimization-ii[/url]

For numerical analysis and more advanced numerical linear algebra, I really liked Gilbert Strang's (MIT) computational engineering videos,
[url="http://academicearth.org/courses/computational-science-and-engineering-i"]http://academicearth...d-engineering-i[/url]
[url="http://academicearth.org/courses/mathematical-methods-for-engineers-ii"]http://academicearth...or-engineers-ii[/url]

For a lot of the topics mentioned (topology, differential geometry, nonlinear dynamics, etc), basically anything where there is a continuum instead of just finite structures, it will be difficult to make much progress without a solid grounding in real analysis. There's a great set of video lectures by Francis Su from Harvey Mudd where I did my undergrad,
[url="http://beta.learnstream.org/course/6/"]http://beta.learnstream.org/course/6/[/url]
(or http:/ /www.youtube.com/watch?v=sqEyWLGvvdw and click through to the other videos)

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daviangel    604
[quote name='Maze Master' timestamp='1332305063' post='4923837']
For a lot of the topics mentioned (topology, differential geometry, nonlinear dynamics, etc), basically anything where there is a continuum instead of just finite structures, it will be difficult to make much progress without a solid grounding in real analysis. There's a great set of video lectures by Francis Su from Harvey Mudd where I did my undergrad,
[url="http://beta.learnstream.org/course/6/"]http://beta.learnstream.org/course/6/[/url]
(or http:/ /www.youtube.com/watch?v=sqEyWLGvvdw and click through to the other videos)
[/quote]
Cool beans [img]http://public.gamedev.net//public/style_emoticons/default/biggrin.png[/img]
Been looking for an easy i.e. video intro to real analysis. I tried to read Mandelbrot's Fractal book and Kip Thorne's gravitation books a while back and both of them quickly lost me since right off the bat they both go into metric spaces [img]http://public.gamedev.net//public/style_emoticons/default/sad.png[/img]
I can also brush up on monoids too now [img]http://public.gamedev.net//public/style_emoticons/default/laugh.png[/img]

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taby    1265
[quote name='alvaro' timestamp='1332017510' post='4922884']
I have used a few things from Abstract Algebra and Number Theory for hashing and pseudo-random number generators. I've used statistics quite a bit (Are those numbers really random? Is this version of my chess program stronger than the previous one?). Probability is really important for almost anything in Artificial Intelligence. If you do graphics, Euclidean and Projective Geometry are pretty important.

I only know a little bit about Chaos Theory, but my understanding is that it's pretty useless. It's just one of these things that have a sexy name and produce pretty pictures, but I don't think you can really do a whole lot with it.
[/quote]

Alvaro, surely you know that chaos theory applies to any system with more than two parts. A very interesting article is "On the nature of turbulence". Another interesting book is "Galactic Dynamics". Surely Poincare would be offended if he had heard you disparaging his theory of chaos as such. [img]http://public.gamedev.net//public/style_emoticons/default/smile.png[/img]

Anyway, otherwise inexplicable behaviour in physics, chemistry, and biology that was once labeled as noise and swept under the rug is now labeled as chaos. It's similar to how lightning was once labeled as the anger of the gods, but is now labeled as electrons and photons. I personally have no interest in knowing what makes up a lightning bolt, but science isn't about subjective opinion.

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alvaro    21246
[quote name='taby' timestamp='1332374893' post='4924136']
Alvaro, surely you know that chaos theory applies to any system with more than two parts. A very interesting article is "On the nature of turbulence". Another interesting book is "Galactic Dynamics". Surely Poincare would be offended if he had heard you disparaging his theory of chaos as such.
[/quote]

I am not saying that you won't encounter chaotic systems. If you do pretty much anything that involves iterating a function, you'll get there pretty soon. However, there isn't all that much that is useful that you can do with "Chaos Theory".

[quote name='taby' timestamp='1332374893' post='4924136']
Anyway, otherwise inexplicable behaviour in physics, chemistry, and biology that was once labeled as noise and swept under the rug is now labeled as chaos. It's similar to how lightning was once labeled as the anger of the gods, but is now labeled as electrons and photons
[/quote]
We understand lightning and we have the lightning rod to show for it. What has Chaos Theory given us? And if you do find some application, is it relevant to a CS professional?

[quote name='taby' timestamp='1332375169' post='4924137']
Category theory is basically object oriented programming on steroids.
[/quote]
They are both forms of abstraction, and the people that invented OOP borrowed lots of terms from category theory, although I don't think you can push the parallels beyond superficial similarities.

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taby    1265
While googling for topological mixing and bifurcation diagrams, I ran across this PhD thesis "CHAOTIC COMPUTATION By ABRAHAM MILIOTIS" from 2009:
http://etd.fcla.edu/UF/UFE0024234/miliotis_a.pdf

References 25-28, and the entire thesis actually, seem to be about the marriage of chaos theory and computation.

Then again, I really don't put any faith whatsoever in people with university educations, so it might be as much bunk as anything else. I'm sure there are many university educated people in this thread who have a subjective opinion on the matter. Hopefully they'll grace us with their knowledge.

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