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I am graduating from mathemetics bachelor program and I've been accepted for masters degree in mathematics. But I am not sure to pursue math, hence I am looking for other field I can do research. I am going to study potenial theory in the masters degree in math so I am good at analysis, particularly in measure theory. I also took some statistics courses, java and matlab. Moreover, I really like computers so I educated myself on some other stuff(I do some game modding and photoshop)

Some friends of mine suggested machine learning but they are undergrad and they don't really know the subject well. So my question is what fields/areas of computer science I can work on for PhD and what should I learn.

Or in this respect can I work on CS(Phd) as a math grad?

(I've read the similar questions but they are only similar. So I appreciate any help.)

Edit: In Math M.S program I only have one none-technical elective and I will get some undergrad CS lectures(additional). So, as @RB suggested, I am asking for the areas that require realtively less CS and more math. But note that I am willing to study/work hard.

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    $\begingroup$ Can the downvoters please explain their vote? this seems like a legitimate question to me.. $\endgroup$
    – R B
    May 27, 2014 at 16:12
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    $\begingroup$ I don't think there is enough in the question to give any specific answer. All of TCS is mathematical and I don't see why one area would be better for a math major than another. I think this question needs to be a lot more specific to be a good one. One possibility is to ask in what areas of TCS measure theory can be useful (e.g. resource-bounded measures). As the answers in this question show, analysis is also useful all over TCS: cstheory.stackexchange.com/q/10128/4896. $\endgroup$
    – Sasho Nikolov
    May 27, 2014 at 19:45
  • $\begingroup$ @SashoNikolov - while every field in CS could be interesting for math majors, I'd assume that programming languages for example will require more catch-up to do, as compared to many more math-oriented areas. While I agree that the question needs editing and specification, I think it's better to leave a comment than to dw, as I think it's not a bad question. OliverEmerson - perhaps rephrase the question to asking what are the CS areas a math graduate has to make only minimal effort to get into? $\endgroup$
    – R B
    May 27, 2014 at 22:25
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    $\begingroup$ I have a similar background, and have found Spectral Graph Theory and Complexity Theory to be the most welcoming. $\endgroup$
    – GMB
    May 28, 2014 at 2:57
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    $\begingroup$ agreed this is a bit broad & unfocused but there are indeed areas of TCS that are more accessible to mathematicians based on the way the fields have developed historically. one particular area of strong overlap is graph theory/graph algorithms as recent cs.se question asked/touched on. $\endgroup$
    – vzn
    May 28, 2014 at 15:05

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It had been argued that Theoretical Computer Science is a branch of mathematics, so it seems to me that any answer to this question would necessarily be primarily opinion-based. That said, in your place (and this is an uninformed opinion) I would steer clear of subfields in which a tremendous volume of technical work has been done, and in which coding plays a greater role. The barrier to entry in such subfields will be quite high for a math major. So for example, I would be cautious of anything related to computer vision and speech recognition. I would also be cautious of subfields in which the objective is perhaps a bit fuzzy, such as AI and machine learning. More controversially, I think that perhaps you should steer clear of quantum computation for this reason.

Conversely, subfields in which the objective is well-defined and in which coding plays a lesser role have a lower entry barrier for a math major. For example, complexity theory, graph algorithms, information theory (perhaps especially for you!), concurrency, cryptography, and maybe even things like clustering and compressed sensing.

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    $\begingroup$ Actually for someone who likes real analysis, probability and statistics, machine learning could be a good fit. Statistical learning theory in the style of Vapnik studies well-defined problems and uses a lot of analytic tools. $\endgroup$
    – Sasho Nikolov
    May 28, 2014 at 2:59
  • $\begingroup$ Good point! When I wrote "machine learning" I was thinking more about things like SVM and similarity learning. $\endgroup$
    – Daniel Moskovich
    May 28, 2014 at 3:10
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    $\begingroup$ I think SVMs are also built on a rigorous theory, again due to Vapnik. (BTW the downvote was not from me.) $\endgroup$
    – Sasho Nikolov
    May 28, 2014 at 3:14
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    $\begingroup$ Thanks! I'm not an expert, but my impression was that, in SVM, how to represent a data point as a point in Euclidean space (or whether such a representation makes sense) is not an exact science. The same objection could be raised, of course, in regard to statistics itself, where we assume variables to be IID, or directly comparable, when there is in fact no reason to believe this to be the case. Another objection to SVM for a math major, again speaking from ignorance, is that the mathematical content is just linear algebra, therefore "uninteresting". $\endgroup$
    – Daniel Moskovich
    May 28, 2014 at 4:07
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    $\begingroup$ I am also not an expert, but AFAIK to justify all that is happening you would need to understand convex programming duality, various uniform convergence results and concentration of measure techniques, VC theory, some basic functional analysis (e.g. the Riesz representation theorem, Mercer's theorem) to understand kernels. More generally, there is some art to machine learning (especially in practice), but there is very interesting, rigorous and mathematically rich science as well. In fact it's the algorithms in ML that are often heuristic, with no running time analysis. $\endgroup$
    – Sasho Nikolov
    May 28, 2014 at 5:07

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