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First of all let me begin with saying that I have seen there is a question quite similar to this one, but I thought I'd ask again so I could get more specific about my situation since I am guessing it's different for every major.

I am currently doing a master in mathematics, thinking about possibly pursuing a phd after that. During my undergraduate studies in mathematics, I mainly took pure math classes with an extra emphasis on algebra related courses(group theory, field theory, applied algebra). So far I thought that if I were to do a phd it would be in algebra.

During my masters I also continued with pure math classes, such as real analysis, algebra etc. But I am currently taking a class in theory of computation and it has really caught my attention. That along with the fact that during the last year of my undergraduate studies I took a class in design & analysis of algorithms which I really enjoyed, has made me thinking that maybe if I were to do a phd, it wouldn't necessarily be in algebra after all but maybe in something in the interface of mathematics and theoretical computer science or in foundations of mathematics,or maybe even something that combines algebra with theoretical computer science if there's even something like that. Not really sure yet.

My questions are: 1. Can my master thesis area be different than my phd area? If yes, how different?Say if I were to do a master thesis in pure or applied algebra, would it be possible to then pursue a phd in something that maybe combines math and theoretical computer science? 2. What kind of background is needed for a phd in a topic somewhere between pure math and theoretical computer science? Would maybe a second masters in computer science be needed? (Also I guess a good 3rd question would be if maybe having taken only 2 classes in theoretical computer science that I enjoyed is not enough for me to be able to tell if I would actually like something like that?)

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  • I think this really is a duplicate and doesn't depend on field. There is no restriction whatever on changing topics or even fields, provided you get accepted to a program.
    – Buffy
    Commented Mar 26, 2021 at 21:55
  • @Buffy I guess my main concern is ,even if I can get accepted to some program,if I will have the necessary background going into it considering the courses I have taken so far, that's why I asked
    – sanbrie
    Commented Mar 26, 2021 at 22:01
  • You either need the background for acceptance, or the program provides a way to attain it if they accept you without it. Your concern is actually moot. They won't accept you if they can't predict your success with some confidence.
    – Buffy
    Commented Mar 26, 2021 at 22:06
  • @Buffy so would my background be considered enough? considering I have mainly only taken pure math classes and only very few computer science related courses?
    – sanbrie
    Commented Mar 26, 2021 at 22:13
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    I'm not on your admissions committee. In the US, a lot of things are considered. I have a PhD in pure math and learned CS on my own. Math is a pretty good foundation for theoretical CS, but the committee decides.
    – Buffy
    Commented Mar 26, 2021 at 22:15

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A big part of the Theoretical Computer Science (language theory, automata theory, Turing machines and other computational devices) is actually a part of algebra so if you switch to TCS, you won't change the area too much. But in general changing the area when going from Masters to PhD is not a problem at all. People sometimes even have multiple PhDs in different totally disjoint areas.

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