Possible research directions for a beginner in TQFT

[math-physicist]

I am highly interested in Topological Quantum Field Theory (TQFT) and am currently planning on doing a project on this topic this year. Some of my relevant background: Algebra (Groups, Rings, Fields, basics of Categories and Modules), Topology (Munkres), Smooth Manifolds (John Lee's book, first 17 chapters - up to integration on manifolds and basics of deRham Cohomology), 2 semesters of undergrad Analysis and finally some introductory algebraic topology (though I'll be studying this more completely this coming Fall). For physics, I'm currently starting to learn QFT from Peskin-Schroeder's popular book.

Now, TQFT recently has been developing rapidly and turned into a vast subject for a newcomer like me to find a specific topic to work on. I'm interested in aspects of TQFT concerning Category theory, knot theory, quantum groups and the likes. I'm also interested in existence of smooth structures. My understanding is that each one of these are themselves vast enough.

So, my question is:

• What are some TQFT research topics a beginning grad student can pursue that relates to higher categories and/or knot theory? Can someone suggest some possible sub-topics to look at? Also, what "extra" background is needed?

Feel free to suggest if there are other sub-topics that are more suited for a beginner in the field. Ultimately, I want to get a feel for actual research in TQFT through a project. TIA.

 

[CrysPhys]

You will probably get more useful answers if you clarify your situation. Your profile says US, and your post says that you are a beginning grad student. So are you a grad student at a university in the US? If so, are you in a Master's or PhD program? If you are in a PhD program, have you passed the qual exam?

You say that you are interested in a topic for a project, not for a thesis. What is the nature of this project? Will you have an advisor or supervisor? If yes, have you discussed topics with them?

 

[Haborix]

In addition to CrysPhys' questions, are you on the math or physics side? I don't know much about this area, but I recall having the impression that there were a lot of mathematicians working on TQFT. The only thing even remotely sounding like physics in your post is the invocation of P&S's book.

 

[math-physicist]

You will probably get more useful answers if you clarify your situation. Your profile says US, and your post says that you are a beginning grad student. So are you a grad student at a university in the US? If so, are you in a Master's or PhD program? If you are in a PhD program, have you passed the qual exam?

You say that you are interested in a topic for a project, not for a thesis. What is the nature of this project? Will you have an advisor or supervisor? If yes, have you discussed topics with them?

For the moment I'm just an independent research. I said beginning grad just to give an idea of my academic preparation so far. So to answer your question, no, I won't have an advisor. That's one of the big reasons of posting here to see if someone could suggest looking into something.

 

[math-physicist]

In addition to CrysPhys' questions, are you on the math or physics side? I don't know much about this area, but I recall having the impression that there were a lot of mathematicians working on TQFT. The only thing even remotely sounding like physics in your post is the invocation of P&S's book.

My current academic training has been more focused on the theoretical math side of things. This is mainly because my undergrad institution didn't have a strong theoretical physics program and I wasn't really into the experimental side of things. However, I had taken all the "theory" based classes offered, like, classical mechanics (Goldstein), e&m(Griffiths), stat mech (blundell) and of course QM 1&2. I only mentioned P&S's QFT to mean that I've done almost everything that's assumed as a prereq for that QFT text.
One other book I've recently finished reading is Baez's Gauge Fields, Knots & Gravity, which was a wonderful read.