6 Replies - 1516 Views - Last Post: 30 November 2013 - 05:16 PM

#1 htoruen  Icon User is offline

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which upper division math courses should I take for math minor

Posted 30 November 2013 - 02:19 AM

I'm a CS major planning minoring in Math. The math minor requires 20 units ( 5 courses ) of upper division math but doesn't restrict you to choosing any specific set of courses. I've bolded the courses which I'm strongly considering. I guess I'm interested in courses which will compliment computer science and prepare me for grad school.

Introduction to Abstract Mathematics
Number Theory A,B
Differential Geometry
Convex Geometry
Partial Diff Eq. A,B,C
Ordinary Diff Equations A,B
Real Analysis A,B,C
Numerical Analysis A,B,C
Fourier Analysis
Mathematical Finance
Probability A,B
Euclidean Geometry
Combinatorics
Algebraic Combinatorics
Discrete Mathematics
Modern Algebra A,B,C
Mathematical Foundations of Database Theory, Design and Performance
Mathematics and Computers
Applied Linear Algebra
Optimization
Complex Analysis A,B

The thing is that I'm not certain what I want to do with it. I am leaning towards graphics and visualization with a some extra focus in high performance computing, parallel algorithms, scientific computing. But I'm starting to wonder whether I can have such a broad focus without lacking some aspects of individual areas.

But I'm also interested in theoretical computer science. The only thing holding me back from focusing on theoretical computer science, is that I am not sure what job prospects would be like, or whether or not I have what it takes to do high quality theoretical research at the PHD level.

Any opinions on the matter?

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Replies To: which upper division math courses should I take for math minor

#2 baavgai  Icon User is offline

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Re: which upper division math courses should I take for math minor

Posted 30 November 2013 - 05:51 AM

Neat list. For the programmer, Discrete Mathematics should have directly applicable stuff.

"Mathematical Foundations of Database Theory, Design and Performance" looks like mental, mathematical, masturbation more than anything else. "The larger the system the less likely clean theoretical models approach reality. ( Baavgai's Axiom #42 )" ;)
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#3 macosxnerd101  Icon User is online

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Re: which upper division math courses should I take for math minor

Posted 30 November 2013 - 12:36 PM

What is your motivation for taking Real Analysis? I'm in the class now. It's a tough, tough class. I easily put in 20+ hours a week into the course on homework, and I consistently score below 80% (74-85 is a B ). If you're not doing math for grad school, I'd pass on Reals.

Probability is good, but you'll probably get what you need from the course in Combinatorics. Unless you get additional statistics (inference testing, regression, continuous distributions), I'd pass on probability.

Number Theory is always a good course to take, and it's very CS applicable. Modern Algebra is a good course to look at as well. You can study Graphs with either Groups (Modern Algebra techniques) or Vector Spaces (Linear Algebra techniques). Modern Algebra also has applications in Number Theory and Cryptography, if you look at AES, Pollard Rho, or Elliptic Curve Cryptography. You will get into topics like Finite Fields and Galois Fields, which are very important.

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Neat list. For the programmer, Discrete Mathematics should have directly applicable stuff.

Look at the syllabi and course descriptions. baavgai is right here; but in a semester, Discrete may be a rehash of Number Theory, Combinatorics, and Graph Theory. If it covers additional topics like automaton theory, natural/formal languages, and computability theory, I might be inclined to look more closely at it.
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#4 jon.kiparsky  Icon User is online

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Re: which upper division math courses should I take for math minor

Posted 30 November 2013 - 02:27 PM

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in a semester, Discrete may be a rehash of Number Theory, Combinatorics, and Graph Theory.



This seems to be the typical Discrete syllabus - though since it's usually taken before those others, it's more of a survey than a rehash. Basically, it's a set of useful tools for computer science. Probably will also cover the basics of proofs and some set theory as well. This is easily the most common math course for CS majors, from what I can see.

I would imagine Applied Linear Algebra would cover some machine learning, which might be interesting for you. I find linear to be a very boring sort of math - there's nothing beautiful about it, that's for sure - but it's very useful if you're dealing with problems involving large sets of mutually independent data points. (for example, image processing, machine learning)

The most beautiful stuff there is probably going to be found in number theory, if that makes any difference to you. Unfortunately, they're going to want to spend the whole time talking about crypto, because people think that's what's interesting about number theory, but that's just a little piece of number theory.

That being said, though, I have no idea what you like in math or what you want to do with what you learn, or what background you have - I really don't know anything that would allow me to make any recommendations. I suggest that you pick a math prof who seems to be on your wavelength - or else the oldest one, on the grounds that they have the most experience - and ask them to help you work through the list. They've seen this before, and they know the courses, and they'll have good advice for you. Whether you take it or not is your decision, of course, but you should listen to it for sure.
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#5 jon.kiparsky  Icon User is online

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Re: which upper division math courses should I take for math minor

Posted 30 November 2013 - 02:34 PM

View Postmacosxnerd101, on 30 November 2013 - 02:36 PM, said:

Look at the syllabi and course descriptions. baavgai is right here; but in a semester, Discrete may be a rehash of Number Theory, Combinatorics, and Graph Theory. If it covers additional topics like automaton theory, natural/formal languages, and computability theory, I might be inclined to look more closely at it.



Jeff Ullman's course on coursera is doing a very good job of covering automatons. If you work through his textbook, with his lectures, you should have automaton theory covered.

(I actually got a lot of the basics when I took compilers, but Ullman's doing all of the proofs and obviously there's a lot more extensive coverage of the ways the ideas unfold)


Another suggestion: if you haven't done so, just go ahead and memorize the Greek alphabet. It'll be useful.
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#6 htoruen  Icon User is offline

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Re: which upper division math courses should I take for math minor

Posted 30 November 2013 - 03:15 PM

View Postmacosxnerd101, on 30 November 2013 - 12:36 PM, said:

Probability is good, but you'll probably get what you need from the course in Combinatorics. Unless you get additional statistics (inference testing, regression, continuous distributions), I'd pass on probability.


Here is the description.

Probability space; discrete probability, combinatorial analysis; independence, conditional probability; random variables, discrete and continuous distributions, probability mass function, joint and marginal density functions; expectation, moments, variance, Chebyshev inequality; sums of random variables, random walk, large number law, central limit theorem.

It says Real Analysis is a pre-req, but the instructor said I could take it anyways.

View Postmacosxnerd101, on 30 November 2013 - 12:36 PM, said:

Look at the syllabi and course descriptions. baavgai is right here; but in a semester, Discrete may be a rehash of Number Theory, Combinatorics, and Graph Theory. If it covers additional topics like automaton theory, natural/formal languages, and computability theory, I might be inclined to look more closely at it.


I'm already taking D.M. for CS, but the upper division D.S. offer in the Math department seams a little different.

Here is the description,

Coding theory, error correcting codes, finite fields and the algebraic concepts needed in their development.
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#7 macosxnerd101  Icon User is online

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Re: which upper division math courses should I take for math minor

Posted 30 November 2013 - 05:16 PM

Your probability class sounds like a standard senior level statistics class. I definitely think it would be beneficial. When I took a similar class, the prerequisite was multivariable calculus. We needed double integrals for joint distributions. Real analysis is strictly a theory class, and you will most likely cover the real numbers (Dedekin cuts, why study the reals, suprema and infima, ordering), point set topology, series and sequences, continuity, and differentiability. I don't see why you would need a ton of that theory for a stat class. Double check with the probability professor before delving into it more. He may let you skip the prereq, but make sure you don't pay for it later.

The Discrete Structures class sounds really neat! Modern Algebra will definitely help you here.

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I would imagine Applied Linear Algebra would cover some machine learning, which might be interesting for you. I find linear to be a very boring sort of math - there's nothing beautiful about it, that's for sure - but it's very useful if you're dealing with problems involving large sets of mutually independent data points. (for example, image processing, machine learning)

Linear Algebra is incredibly useful in graph theory, looking at vector spaces of graph vertices, edges, cycles, cuts, and flows. Eigentheory and linear algebra is also incredibly useful in graph theory. Some of the applications include electricity and magnetism, numerical methods, and computational chemistry. Applied Linear Algebra may get a little dry or engineering-like, but it is still useful. An abstract linear algebra course will be more along the lines of number theory.

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The most beautiful stuff there is probably going to be found in number theory, if that makes any difference to you. Unfortunately, they're going to want to spend the whole time talking about crypto, because people think that's what's interesting about number theory, but that's just a little piece of number theory.

A lot of in-depth number theory is elegant. I found the class to be useful in learning to form combinatorial proofs. I didn't really glean that when I took Applied Combinatorics and Graph Theory. A first semester number theory class will probably be fairly introductory, and the big takeaways will be the Euclidean Algorithm, Chinese Remainder Theorem, and the Pidgeonhole principle. Fermat's Little Theorem is incredibly important, as is Euler's Totient Function. You'll definitely get other material as well, but those are the highlights. If you've had Discrete for CS majors, you've probably covered quite a bit of number theory already. Extensions of Fermat's Little Theorem are useful in finding multiplicative inverses in finite fields. I've been using it in my homework due this coming week regarding the AES cryptosystem.
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