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Related to this question. I am a mathematician interested in doing a PhD in areas such as categorical logic, homotopy type theory, constructive math, etc. In the UK and Europe, most of the people studying this kind of stuff live in Theory groups in CS departments, rather than math departments. As a result, I would have to study in a CS department, and work towards a PhD in computer science, rather than mathematics.

What impact would this have on my intended future career plans as a research mathematician? e.g. would it be harder to secure jobs in mathematics departments with a PhD in computer science?

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  • What evidence do you have for "most of the people..."? Seems a bit suspect.
    – Buffy
    Nov 30, 2020 at 22:45
  • The answer is in the question: It is a rarer path to a job in a math department, so it is probably harder. Nov 30, 2020 at 22:52
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    Why don't you want to do this research in a computer science department after your PhD? I suspect the answer to this matters. Nov 30, 2020 at 23:07
  • @Buffy: just my own research - I've found many in CS departments but barely any in math. In North America it seems to be the other way round. Nov 30, 2020 at 23:11
  • @AlexanderWoo: I suppose I could, but I really consider myself a mathematician (logician) rather than a computer scientist, and would want to be appointed as such for teaching, supervising, etc... Nov 30, 2020 at 23:12

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First-rate people attract other first-rate people, but second-rate people tend to hire third-raters, and third-rate people hire fifth-raters.

This is a quote from Paul Halmos, who attributed the idea to Andre Weil.

Outside of the top universities, relatively few US research universities have significant Theory groups in CS departments at all, because people doing theory research don't tend to bring in million dollar grants.

Outside of a few clusters, mostly at top universities, relatively few US research universities have logicians in mathematics departments, because the perception amongst most mathematicians is that logicians work in a completely disconnected area of no interest to mainstream mathematicians. (Logicians have not helped themselves in this regard.)

EDITED TO ADD: My impression is that categorical logic, homotopy type theory, and related areas have been somewhat exempted from the difficulties logic has as a subfield because a lot of the interest in it comes from people who were trained, and in many cases first earned tenure, as algebraic topologists or algebraic geometers. However, this doesn't help someone coming into the area whose background is in logic rather than algebra.

Outside of top universities, it is hard to get hired into a permanent position at a mathematics department in the US without some North American teaching experience. You can get this experience as a postdoc, but you'd have to first get a postdoc.

If it turns out that you really are first-rate, and first-rate in logic usually means being at least arguably the single best logician on the job market that year, then I think it doesn't matter much. Part of being a first-rate department and hiring first-rate people is understanding that the title on your degree doesn't mean much.

If it turns out that you're not first-rate, then the job market for logicians in the US has been terrible for many years - it's easily been the worst among the major subfields of mathematics - and the economic crisis associated with the pandemic can only have made it worse. In the general mathematics job market, doing a PhD in Europe and not having US teaching experience as part of your PhD puts you at a disadvantage, and not having mathematics teaching experience (as part of your PhD or afterwards) also puts you at a disadvantage. (CS works somewhat differently because administrators love those million dollar grants.)

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  • Thanks, this is all good to know. Seems I certainly haven't picked the easy path - oh well. Perchance, does anyone know what the average mathematician thinks of reverse mathematics? It is effectively logic applied to "normal mathematics", so they might see it as more useful / interesting / relevant to their work. Dec 1, 2020 at 4:24
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    Exaggerating a bit, I have particular problems I am interested in, and I am interested in the work of other mathematicians only if this work gives insight that helps solve the problems I am interested in. There isn't much history as far as I know of reverse mathematics helping to solve mainstream mathematics problems. Model theory is probably the subfield of logic with the most applications to mainstream mathematics. Dec 1, 2020 at 5:17

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