Which Graduate Field Combines Math, Logic, and Computability?

[SrVishi]

Hello everyone. I'm having some trouble deciding on what field I should pursue in graduate school. I really like any branch of math, but I'd have to say that both mathematical logic (and meta-mathematics in general) and abstract algebra might have to be my favorites. I am also interested in theoretical/mathematical computer science, such as questions of what is computable (or beyond like superturing). Is there any way to combine any of these fields?

 

[The Bill]

You'd probably enjoy studying type theory. Type theory is a form of category theory applied to languages and sentences. It allows formal definitions of what constitutes a proof, and is used for structuring computer languages designed for error checking and debugging. Category theory, in turn, is an offshoot of abstract algebra which has applications all over mathematics. Of those applications, perhaps categorical logic would be of particular interest to you.

Emily Riehl wrote the book Category Theory in Context recently. You might find it helpful. It's available free on her website, and will be published by Dover later this year: http://www.math.jhu.edu/~eriehl/727/context.pdf

 

[SrVishi]

Thanks for your response, I was going to look into category theory anyways, but now I have even more motivation to do so! I was wondering if recursion theory also fit the bill, or maybe I should just stick with category theory. Thanks again for the help.

 

[micromass]

Hello everyone. I'm having some trouble deciding on what field I should pursue in graduate school. I really like any branch of math, but I'd have to say that both mathematical logic (and meta-mathematics in general) and abstract algebra might have to be my favorites. I am also interested in theoretical/mathematical computer science, such as questions of what is computable (or beyond like superturing). Is there any way to combine any of these fields?

I'm not all that convinced that you're looking for category theory, although it is definitely worth looking into.
Other things you might want to try are universal algebra. For example see http://www.math.hawaii.edu/~ralph/Classes/619/univ-algebra.pdf
You also might want to check out algebraic logic which establishes a link between logic and algebraic structures. See Rasiowa and Sikorski: https://www.amazon.com/dp/B005JGKZXW/?tag=pfamazon01-20

As for category theory, take a look at Aswodey's very neat book: https://www.amazon.com/dp/0199237182/?tag=pfamazon01-20 or try to get into topos theory with Goldblatt's beautiful book: https://www.amazon.com/dp/0486450260/?tag=pfamazon01-20

 

[chiro]

Hey SrVishi.

You might want to look at specific applications of algorithms in a mathematical context (i.e. - based on abstract algebra and theoretical computer science).

Mathematical cryptography in a very rigorous form might be up your alley since they have to understand all of these issues and have some idea of how to enforce the computational complexity of things like one way functions (i.e. easy to do, hard to undo without the necessary piece of information which is the basis for a lot of asymmetric cryptography).

If you can find an abstract algebraic treatment of cryptography along with the computer science (theoretical) treatment and combine them yourself (or find another who can combine it for you or has already done so) then it might meet your needs.

 

[SrVishi]

I'm not all that convinced that you're looking for category theory, although it is definitely worth looking into.
Other things you might want to try are universal algebra. For example see http://www.math.hawaii.edu/~ralph/Classes/619/univ-algebra.pdf
You also might want to check out algebraic logic which establishes a link between logic and algebraic structures. See Rasiowa and Sikorski: https://www.amazon.com/dp/B005JGKZXW/?tag=pfamazon01-20

As for category theory, take a look at Aswodey's very neat book: https://www.amazon.com/dp/0199237182/?tag=pfamazon01-20 or try to get into topos theory with Goldblatt's beautiful book: https://www.amazon.com/dp/0486450260/?tag=pfamazon01-20

Would universal algebra really be what I'm looking for, more than category theory?

 

[micromass]

Would universal algebra really be what I'm looking for, more than category theory?

Don't know. Take a look at both fields and see what you like best.