What field am I describing?

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In summary: Universal algebra is more closely related to abstract algebra, while category theory has a wider range of applications in mathematics. In summary, the person is having trouble deciding on a field to pursue in graduate school, with interests in mathematical logic, abstract algebra, and theoretical/mathematical computer science. They are recommended to look into type theory, category theory, universal algebra, and algebraic logic, with specific resources suggested for each. The person is advised to explore and see which field interests them the most.
  • #1
SrVishi
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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?
 
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  • #2
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 [Broken]
 
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  • #3
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.
 
  • #4
SrVishi said:
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
 
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  • #5
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.
 
  • #6
micromass said:
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?
 
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  • #7
SrVishi said:
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.
 

What is the definition of "What field am I describing?"

The term "What field am I describing?" refers to a specific area of study or expertise within the scientific community. It can also refer to a particular subject or topic that is being discussed or researched.

What are some common fields that people describe?

Some common fields that people may describe include biology, chemistry, physics, environmental science, and psychology. These are just a few examples, as there are many different fields within the realm of science.

What is the purpose of describing a field in science?

Describing a field in science helps to categorize and organize different areas of study, allowing scientists to focus their efforts and collaborate with others who have similar interests. It also helps to establish a common language and understanding within the scientific community.

How do you determine which field a particular study or research falls under?

Determining the field of a study or research can depend on various factors, such as the topic being investigated, the methods and techniques used, and the overall goals and objectives of the study. It is often a collaborative process between scientists with different areas of expertise.

Can a study or research fall under multiple fields?

Yes, it is possible for a study or research to fall under multiple fields. This is especially true for interdisciplinary studies that combine different areas of science to address complex problems. In these cases, the researchers may have expertise in more than one field.

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