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I wonder if anyone can give me some advice about how a computer science graduate shifts career to mathematics? I'm currently doing a Master's degree in computer science and i was thinking about doing a PhD in pure mathematics? so I was thinking is this transition feasible? how can I make this leap of faith?

Kindly note that currently -after graduation- I nearly don't do mathematics anymore, currently all what I am doing is writing software and administrating open source operating systems. In undergrad levels I took an introductory course in discrete mathematics and some calculus courses.

someone told me: Yes, the transition is feasible but it requires I take some courses and attend some math classes even as a special student (no degrees, no exams). But I thought I can get other opinions and views.

  • Did you have a specific country in mind where you want to do your PhD? – Carsten S Dec 31 '14 at 14:46
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The easiest transition is probably to focus on mathematics programs where you can study topics closely related to computer science. You didn't mention in the question which areas of mathematics you are interested in studying. There is a significant amount of overlap between "theoretical" computer science and mathematics, although you have to take time to find the schools where this is studied in the mathematics department.

Even at such schools, though, a mathematics PhD includes a significant amount of "general" coursework. For my PhD in mathematical logic and computability theory -- which is one of the closest mathematical topics to computer science - I passed exams in abstract algebra and real and complex analysis. The exact topics you would need to study vary by school.

One option that has not been mentioned -- which has advantages as well as disadvantages -- would be to take a master's degree in mathematics first. This would certainly give you the background needed to be prepared for PhD studies. The main disadvantages are that it takes a year or two of your life, and that the funding for master's programs is not as generous as for PhD programs.

  • Ah, i was going to mention taking a master's first, but I have no idea what Shazly's time frame looks like ! – galois Dec 31 '14 at 17:01
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It's always possible to go into another field, especially ones that are somewhat related (i.e., you'll have an easier time going to mathematics with CS than if you had a Psychology degree, for example).

The Bad

In undergrad levels I took an introductory course in discrete mathematics and some calculus courses.

That's about one or two classes more than even most humanities majors have to take. I'm assuming you've taken at most integral calculus (Calculus II in the US).

You're going to need to take quite a few courses to be prepared for a PhD program, and have a chance at doing some useful research afterwards, considering you want to do pure math.

Undergrad math majors usually take (to prepare for grad school) courses such as:

  • Real Analysis (at least two semesters)
  • Modern/Abstract Algebra
  • Vector and/or Complex Analysis
  • Geometry and/or Topology
  • Combinatorics/Graph Theory
  • Number Theory

Many schools also require a course on proofs/logic/set theory before you can do many of those, as well. It could behoove you to also know about functional analysis, and differential geometry/riemann surfaces before you enter as well, though you may be able to take those as graduate coursework.

So you're looking at a good amount of time and effort before you can even competitively apply.

The Good

Having a master's in CS is a very useful thing. After all is said and done, you will be a very big asset in industry and R&D. A computer scientist who knows mathematics is a very dangerous person ;^)

Good luck, and I'll be cheering for you.

  • Remember that graduate schools vary a lot in the material that they require for incoming students. For almost every school, a background in real analysis and abstract algebra is mandatory, along with mathematical maturity and proof ability. But topics like combinatorics, Riemann surfaces, and number theory are more specialized, and only a minority of schools would have those as part of their qualifying examinations. – Oswald Veblen Dec 31 '14 at 14:16
  • Sure, that's why I put that many math students 'usually' take courses like them. I probably should have not lumped them together – galois Dec 31 '14 at 16:58
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Take a look at Is it easy to change your major after starting your PhD or masters, if it is necessary?, which is somewhat similar.

In particular, unfortunately, it sounds like your current mathematics background is very far from what would be needed to enter a math PhD program - it's maybe 1/4 of the coursework that would be expected. You would need to have preparation equivalent to that of a BS in mathematics, and that sounds like it would be the equivalent of 2-3 more years of full time coursework for you.

To be honest, I don't think that the coursework you describe can even have given you much of a sense of what pure mathematics is, much less what it is like to do research in the area. One needs rigorous proof-based classes such as real analysis and abstract algebra.

I would not really advise you, or anyone, to even think about grad school in mathematics before they have enough of a background to reasonably understand what it involves.

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