How to transition from theoretical mathematics to theoretical computer science

Question

Background: I started my undergraduate career in computer science and transitioned to mathematics leaving computer science as a minor. I have just recently completed my Bachelors in Mathematics with a research project in Galois theory, algebraic geometry, and additive number theory. For the next academic year, I am pursuing a Masters in Mathematics which I will complete in a year through a 5-year (BS/MS) program.

Although I did not complete more than a minor in computer science, I have been interested in having it play a vital role in my career goals in academia.

My goal after finishing my masters is to transition into a Ph.D. program in mathematics, but I have not yet decided what I want to focus my research on.

I would like to learn about what Theoretical Computer Science is for academia and what background someone needs to become a theoretical computer scientist.

Questions: Do most theoretical computer science professors have their Ph.D. in mathematics?

How much formal background does one need from computer science itself? Can this be achieved outside of a school setting?

What fields of mathematics play an important role in theoretical cs?

Answer 1

The answer to your question in "no": Most of people who call themselves Theoretical Computer Scientists have PhD in (T)CS.

And yes, you need to have a CS background because the word says it itself: Theoretical Computer Science which is the study of what a computer can(not) do and with what resources.

Math is a vital part and on different fields of TCS different fields areas of math play important role (number theory, linear algebra, combinatorics. probability theory are by far the most used ones). Generally speaking, the maths used are rarely "obscure" or too abstract/pure.

As on the question of if this can be achieved outside formal education the answer is "depends" on the motivation, circumstances etc. It is really hard to answer that.

Answer 2

Most theoretical computer scientists have PhDs in computer science. But there several math programs that also employ theoretical computer scientists and/or have a close relationship with the CS department at their university. These math PhD programs produce theoretical computer scientists.

The MIT math department is one such program --- plenty of computer scientists graduate with PhDs in math from MIT, although they are advised by people who would normally be considered (theoretical) computer scientists. Other programs include:

The ACO program at CMU: http://aco.math.cmu.edu/

The ACO program at Georgia Tech: http://www.aco.gatech.edu/

The AMCS program at Penn: https://www.amcs.upenn.edu/

I'm sure there are other such that I am missing.

These programs "leave your options open" in case you want to study some aspect of pure or applied math separate from TCS, but with the option to study TCS. Of course, if you are sure about what you want to do, a strong math background is usually good preparation for a PhD in (T)CS as well, even if you don't have an undergraduate CS major. (e.g. many of my own PhD students at Penn were math majors in college rather than CS majors)