Use the Feynman method: Richard Feynman was fond of giving the following advice on how to be a genius. You have to keep a dozen of your favorite problems constantly present in your mind, although by and large they will lay in a dormant state. Every time you hear or read a new trick or a new result, test it against each of your twelve problems to see whether it helps. --Gian-Carlo Rota


I regard as quite useless the reading of large treatise of pure analysis: too large a number of methods pass at once before the eyes. It is in the works of applications that one must study them; one judges their ability there and one apprises the manner of making use of them. --Jospeh-Louis Lagrange (1736-1813)


If there's a book that you want to read, but it hasn't been written yet, then you must write it. --Toni Morrison

In one sentence, my research interest lies at the crossroads of geometry and physics. Zooming in a bit, I am mostly interested in the interaction of topology with quantum field theories. Other than that I follow Lagrange's advice of learning in the context of a problem. At present, the projects I am working on are related to the following three fields:

I do not claim to be an expert in any of these as I am still learning them by getting my hands dirty with projects. However, my preparation includes a decent understanding of (extended) topological field theory, defects, cellular gauge theory, lattice TFT, higher categories, categorification, quantum invariants of knots and links, Khovanov homology, stratified spaces, microlocal sheaf theory, and obstruction theory.

Resources:

My general understanding of mathematics owes to the following classics (thanks to Verma brothers):

Weyl's book was a turning point which pulled me to Mathematics (from Physics). After migrating to divinity for some time, I filled (or have been filling) gaps in my knowledge of Physics using the following (thanks to Anton Zitlin for his semester long course on Geometry and Physics.):

alongside with many lecture notes by Dan Freed (check in particular 'Quantum theory from a geometric viewpoint, Part I') and David Tong. I supplemented the chapters on QFT and supersymmetry from Mirror Symmetry (Hori, et-al) with notes on these topics by David Skinner.

The lecture videos by Kevin Costello on Renormalization and Effective Field Theory is also my favourite (modulo the bad camera work).

I also supplemented Lancaster-Blundell with An Introduction to Quantum Field Theory by George Sterman and What Is a Quantum Field Theory? by Michael Talagrand.

I use this list by Prof. Mikhail Khovanov for the resources and references on Representation Theory. This list is only about online resources and references, so does not contain books like Fulton and Harris, which is my personal favorite.

For people curious about the connection of representation theory to physics, Hermann Weyl's The Theory of Groups and Quantum Mechanics is a great place to look at.

Quantum Topology: Q-cohomology, Spectral Networks, BPS States etc

I owe the following list to Sergei Gukov, which I have ordered in increasing order of difficulties.

  1. Lectures on Knot Homologies and Quantum Curves
  2. Homological Algebra of Knots and BPS States
  3. 3-Manifolds and Vafa-Witten Theory
  4. BPS Spectra and 3-Manifold Invariants
  5. Vertex Algebras and 4-Manifold Invariants
  6. VOA(M_4)

This lecture notes 'seems' an excellent place to learn about Spectral Networks, in a broad sense.

Topological (Quantum) Field Theory, global symmetry, higher structures, etc

***Under Construction***


Other Useful Resources

Homotopical and Algebraic QFT by Donald Yau

Lecture notes on 2-dimensional defect TQFT by Nils Carqueville.

One dimensional topological theories with defects: the linear case by Mikhail Khovanov.

Field Theories with defects and the center functor by Davydov, Kong, and Runkel

Cohomological Field Theory Resources:

Cohomological Field Theory Calculations by Rahul Pandharipande

Stacks and Deformation Theory resources:

The unbearable lightness of deformation theory by Balazs Szendroi

Deformation Quantization Resources:

Operads and Motives in Deformation Quantization by Maxim Kontsevich

*** Under Construction ***