I'm an undergraduate student who's thinking about a double major in computer science and applied math. I'd like to choose a math heavy, high demand computer science area to specialize in. I like the utility and elegance of math, and I'm trying to find a CS area that uses many different branches of math.<p>I'd also prefer if the area changes relatively slowly compared to other computer science areas (so maybe not security).<p>I'm currently thinking that graphics or machine learning would be high demand areas that use a lot of math, but I'm looking for more suggestions and advice. Thanks.
I think you're setting yourself up for disappointment with unrealistic expectations.<p>People that are in demand are programmers and good programmers code 90% of time. Some people, including yourself, suggested graphics but read the history of DOOM and read its source code: despite being a cutting edge technology at the time, 90% of the code is the non-math drudgery: reading and writing files, networking code, performant array and string classes, making the code cross-platform and cross-compiler, debugging code etc. Carmack certainly knows his math but he knows his C even more.<p>Math might be helpful/necessary in some fields but if you're thinking about being a programmer (as opposed to academic/researcher), don't expect math to be more than 10% of your time. The rest is the same drudgery that the rest of us has to deal with on a daily basis.
How about computational finance? Finance is what drives pretty much all modern economic activity around the world, and the fundamental concepts in finance haven't changed for hundreds of years, which takes care of both the stable/slow-changing and in demand criteria. It is also heavily mathematical if you choose to focus on the quant side, with probability, statistics, stochastic calculus, PDE's, and even measure theory being used in some form. Also, software is becoming an increasingly important part of modern finance, with things like electronic exchanges, high frequency trading, complex derivative/option pricing, prop trading, and actuarial science.<p>And the skills you'll learn in finance will be transferable to a range of other fields - advertising and cloud distribution, which pretty much drive most tech startups' revenue, heavily rely on concepts and techniques that you'll learn in finance.<p>Here's an interesting course you can take a look at: <a href="http://www.algorithm.cs.sunysb.edu/computationalfinance/" rel="nofollow">http://www.algorithm.cs.sunysb.edu/computationalfinance/</a>
Note that this is by steven skiena, the author of the well-known Alogrithmm Design Manual. Khan Academy also has a pretty extensive series on modern finance.
I think your thinking is right, machine learning is probably a good area as far as jobs go. Depending on your CS interests you'll find what areas of math they use most. AI / machine learning, learn lots of probability and statistics. High-performance, scientific computing, graphics, maybe more of a focus on linear algebra. Algorithms or theory or programming languages, lots of discrete math, logics, algebraic structures, etc. Of course it's good to get a good grasp of the basics of all these, since they're definitely not exclusive, and which you'll get from your degree, and beyond that, well explore, and enjoy!! Good luck!
Go for computer vision/pattern recognition. There seems to be a shortage of enough 'good' people for them - even in HN I read very low quality vision related blog news as if they are very good ones... (I remember an image rotation article a few months back...). There are very good CS guys here but vision/learning seems to be a bit underrepresented...<p>I'm getting pretty good contracts for doing research work from pretty big US companies/startups and I'm not even located in US...<p>I did electronics engineering in undergrad, signal processing in masters and computer vision in phd - also took some courses in physics. I sort of wish I did some formal math courses (optimization/graph theory/variational calculus) as well. Applied math will help you a lot in CV/ML domain so that's a pretty good idea for you to get it. ML is very hot and there are lots of people going after that but don't forget to check out the geometric part of the cv - finding camera calibrations, stereo, multiview stereo or the realtime stuff as cv is being used more and more in mobile apps... Computational photography is my new focus these days - I'm getting more queries about that... PM me if you like more detailed info...
For a glimpse into machine learning, check out Professor Yaser Abu-Mostafa's "Learning From Data" course from Caltech. The videos are online for free (<a href="http://work.caltech.edu/telecourse.html" rel="nofollow">http://work.caltech.edu/telecourse.html</a>, <a href="https://www.edx.org/course/caltechx/cs1156x/learning-data/1120" rel="nofollow">https://www.edx.org/course/caltechx/cs1156x/learning-data/11...</a>), and its corresponding book is on Amazon (<a href="http://www.amazon.com/Learning-From-Data-Yaser-Abu-Mostafa/dp/1600490069/" rel="nofollow">http://www.amazon.com/Learning-From-Data-Yaser-Abu-Mostafa/d...</a>).<p>Also Professor Ng's course from Stanford (<a href="http://cs.stanford.edu/people/ang/?page_id=22" rel="nofollow">http://cs.stanford.edu/people/ang/?page_id=22</a>).
Actually computer security doesn't change very fast compared to other industries, and is probably a very good fit if you have a math background. But machine learning would be fun too. Stay far, far away from graphics if you are worried about rapid change.<p>But really, find what you're interested in and do that. It may involve trying them all out, or reading up some reference works on each. Making an important life decision based on “what's in demand” is a very poor choice.
Quick question- are you aiming for a computer science/software job with lots of math, or a math job that involves computing?<p>It's an important distinction. The people who are doing math with the help of computers (rather than doing software that uses math) are <i>much</i> more involved in mathematics.
Let me offer a different advice: go for computer science and a major with practical application, e.g. biology, chemistry, medicine, economy, or even linguistics.<p>In our age we are increasingly seeing technology changing other fields. But there are relatively few CS majors who are proficient in another domain and there are relatively few non-CS majors who can engineer/program well. There is a lot of demand for people who can program and have domain-specific knowledge, e.g. in computer vision, market prediction, natural language processing, etc.<p>The other advantage is that, even if the demand for computer science majors collapses, there may be opportunities in the other field (well, perhaps not in linguistics ;)).
I did a dual degree in Computer Science and Math. My recommendation is to take the classes in both areas that you find the most interesting, with no concern for your career. Your career will span decades and unlikely to involve a steady focus on a single area, as either the environment or you will change. Personally, I've been working for 11 years in a variety of roles (PM at Microsoft, Product Analyst at a SF startup, Quantitative Developer in finance).<p>You never know where the most valuable lessons will come from and how they will pay off. For example, my second Real Analysis class leveled up my ability to communicate clearly and precisely in a way that no writing class could have. Graph theory, automata theory, numerical methods, abstract algebra, and statistics have each made their way into my work, sometimes in ways that I never would have expected.<p>If you want to ignore this, Machine Learning.
I graduated two years ago with this exact degree from University of Waterloo. My favourite subjects were Quantum Mechanics, General Relativity, and Computer Graphics. Throughout my undergrad I was somewhat hoping to go into game dev to work on graphics or physics engines. I ended up getting a ton of internship experience doing web dev, and now I work at Google. The math degree ended up being more of a hobby, and I'm ok with that. I am pretty lazy and so I found it hard to stick to my guns about utilizing both degrees on the job. You have to be prepared to move a lot, and to pass on very good opportunities. I prefer using my math knowledge for side projects and keeping it from becoming a job, though maybe thats just sour grapes.
There's a machine learning path that fits this: study math/stats/algorithms, get a graduate degree, work on research in industry or academia.<p>Your question is somewhat vague, though. Do you want to spend the majority of your time working on math? Even machine learning researchers only spend a majority of their time on annoying data cleanup issues, model coding, or data infrastructure. Further, as you become more successful you worry about grant-writing, lab management, or stressing about tenure (or, if industry, department cuts). What's your motivation for entering a "slow" field? Regression is going nowhere but ML's research frontiers are expanding rapidly right now.<p>Note also that you won't land this work with just an undergrad degree, so you should add another 2-5 years of schooling if considering ML.
I do Computer Vision/Machine Learning for a living, but would caution a little against super-focusing on a narrow area. AI in general is known for its AI winters, where jobs dry up and opportunities are far fewer and you could have a lot of people with undergraduate in ML (or whatever that era's AI is called). Now that said, who knows may be you might find it interesting enough that you may decide to go for graduate school. If I was giving advice to my younger self, it would be to learn Algorithms, Linear Algebra, and Probability really well. Get exposure to Machine Learning, and a little bit of Linear Programming. But if you really want to be able to apply machine learning or math heavy subjects to work, it might be a reasonable idea to do a Masters.
You should definitely check "Operational Research".<p>Although it is not as fun as machine learning and computer graphics, there is a strong industrial demand for strong mathematicians.
<a href="http://en.wikipedia.org/wiki/Operations_research" rel="nofollow">http://en.wikipedia.org/wiki/Operations_research</a><p>Many of the work do not involve much coding but require advanced mathematical skills to transform the original problem into something that can be send to a "solver".
I agree with adrock: take classes you think are interesting without a career in mind. Things will change way too much and way too quickly for you to plan a career around courses in college. The main thing you need to learn as an undergrad are thinking and communicating skills.<p>Also, here's a good video that might give you some ideas:<p><a href="http://www.youtube.com/watch?v=0tuEEnL61HM" rel="nofollow">http://www.youtube.com/watch?v=0tuEEnL61HM</a>
For a good overview from one of the big math guys: <a href="http://blog.stephenwolfram.com/2013/03/talking-about-the-computational-future-at-sxsw-2013" rel="nofollow">http://blog.stephenwolfram.com/2013/03/talking-about-the-com...</a><p>Otherwise, I'd suggest taking a bit of time to think about the sector like a entrepreneuring hacker. Look beyond the well worn paths and take advantage of your current naivite.
This article, "The Shape of Math to Come",talks about applying topology to large data sets. <a href="https://www.simonsfoundation.org/quanta/20131004-the-mathematical-shape-of-things-to-come/" rel="nofollow">https://www.simonsfoundation.org/quanta/20131004-the-mathema...</a>
Might stimulate your thinking.
Machine learning.<p>If you also happen to be interested in finance, or think it might interest you at some point later, you could transition to quant finance (where machine learning will also be very usefull).<p>See if this piques your interest for instance: <a href="http://janestreet.com/technology/" rel="nofollow">http://janestreet.com/technology/</a>
In my view, I vote for machine learning. In order to study better in machine learning area, you need to have better understanding in statistics, probability, matrix, optimization and numerical computation. machine learning just like a model, the important thing is that know how to build, it exactly mathematics can help us.
DSP related jobs are pretty math heavy, albeit in a very particular focused area of math. These jobs are in demand and highly specialized, so you aren't going to see a thousand job postings, but when you find one, if you are good, they will want you.
Formal verification of software and hardware systems relies heavily on mathematics, as well as an obvious core of computer science. (Not 'number' math, per se, but logic math.) Static analysis of programs, proofs of correctness, et cetera.
Concurrent processing and database design.<p>But for second major I would recommend some 'soft' science, such as finance, economics or accountancy. Most people on those fields do not know high math.
Stay away from math.<p>I would caution you that you haven't seen `real math` as an undergraduate who still has time to decide on your major - you will find that real math is not elegant. Math is baroque, infinitely deep and you success will entirely depend on the community and the perspective you get from your mentors. For example, conversations I had with math professors were able to frame problems so that I could look past the equations and understand the big picture. Then I had to describe it in the precise language of mathematics. I quickly realized that math wasn't that precise of a language - just esoteric hand waving. I then realized that Mathematics is a language that is unintelligible without context.<p>When I did my CS algorithm classes, I skipped all the lectures and spent 8 hours doing homework from a textbook - my school is rated 3rd in the US.<p>If you only take Math classes you will not find a job or find yourself in a situation where you have not learned the creative skill necessary to extend upon existing solutions.
Go and be a quant at a hedge fund. All the maths (statistics, probability) and programming you could ask for (and people in the back office to do the boring bits for you).