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I couldn't think of a better forum to ask this question. I am a first year cs graduate student. I couldn't really pursue theoretical cs during my undergrad, but as part of our PhD qualifiers, i had to take computability and complexity theory, advanced algorithms, and i loved those courses. More than the computational aspects, i enjoyed the theoretical aspects and rigorous proofs. This kindled a lot of interest in me and I checked out Enderton's set theory, which i liked. Right now I am going through Spivak's Calculus in order to refresh my knowledge for a real analysis course next semester. Lately, i have been feeling that i would love learning and doing pure math more than applied CS. I know theoretical CS is an option, but i feel i would enjoy pure math more.

So my question is, whether introductory graduate level courses on algorithms and complexity theory involve enough rigor and abstract concepts (computability and complexity theory) that i am not underestimating pure math. I would really appreciate sincere inputs. I am thinking of taking standard analysis, algebra, topology sequence next year and find a professor to work with.

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    $\begingroup$ Depends. In terms of rigor & abstract concepts, sure. In terms of standard background that is assumed of graduate students at your university before they take the standard analysis/algebra/topology sequence, that depends on the university and on what other math classes you've taken. Probably you'll get a better sense once you take the standard analysis/algebra/topology sequence. And, if you take that sequence and do decide to stay in TCS, the additional math background will surely come in handy. $\endgroup$ – Joshua Grochow Jun 3 '18 at 6:08
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    $\begingroup$ Thanks :-). Doing courses on algorithms, complexity and computability theory, I've also gone through elementary set theory, number theory. I have calculus/linear algebra/probability background since i work in machine learning. I took a cursory look at analysis, abstract algebra books for 400 level courses, and they don't look too bad. I will be a year or two behind other folks though, who might have already taken introductory graduate level courses in pure math. But I feel I have to take this shot. The only issue is how do i do this without going off funding from my current advisor who is CS. $\endgroup$ – student Jun 3 '18 at 17:05
  • $\begingroup$ @student I think the point about funding is important. Stipends, funding for research, and summer appointments could differ between departments. In addition, depending on the school, it could be much more (or less) competitive to become a funded math PhD student vs cs PhD student. $\endgroup$ – Michael Wehar Jun 4 '18 at 17:02
  • $\begingroup$ I could see many possibilities, but it would really depend on your circumstances, advisor, school, and departments. (1) Your advisor could support you for a thesis that overlaps between math and cs while you remain in the cs department, (2) you could seek a co-advisor within the math department, (3) select schools offer programs that are joint between math, cs, philosophy, etc such as ACO, (4) you could leave your advisor after finding support from the math department, or (5) you could remain in the CS department and just pursue math on the side until you publish in a math conference/journal. $\endgroup$ – Michael Wehar Jun 4 '18 at 17:12
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    $\begingroup$ @MichaelWehar Thank you :-) 1/2/3 are not possible because my advisor's work is very applied CS and bio. 5th could be possible, but i would be making half hearted attempts at two things. I wish to dedicate myself completely to a subject i love. It seems to me that 4th is the only option. It's that it would take me at least the fall semester to find a math prof (after taking their courses), and I do not want to waste my summer, and also do not want to dishonestly eat my current advisor's money. But there is no other option it seems. Maybe I can genuinely work for him until i find a math prof. $\endgroup$ – student Jun 5 '18 at 20:49

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