I often receive questions from students such as:
“I am interested in the intersection between mathematics and physics. How should I study?”
“I like geometry, and I am fascinated by the fact that physics makes non-trivial predictions in geometry. But quantum field theory looks difficult. How should I begin?”
“How can I enter research in mathematical physics?”
To be honest, I find these questions hard to answer in a concrete way, because they are all vague. “Mathematical physics” is an extremely broad field. The relationship between mathematics and physics has many facets and a long history, and each researcher has their own perspective on how the two subjects are related.
For instance, Yuji Tachikawa lists 20 distinct topics (written in Japanese, but AI translation will work well) regarding the relation between math and physics (such as Knot invariants/Chern–Simons theory and Quantum anomalies/Index theorems). These examples are only the tip of the iceberg. There are many more topics that lie at the interface between mathematics and physics.
In what follows, I offer some general advice based on my own experience and perspective while this approach may not be optimal for every student.
The study of quantum field theory uses a wide range of mathematics. I am often asked,
“What mathematics should I study in order to research mathematical structures of quantum field theory?”Since quantum field theory itself is vast, the answer depends on which part you wish to pursue.
If you are a physics student, my advice is simple: begin by studying quantum field theory itself—especially two-dimensional conformal field theory, supersymmetric theories and string theory. Then, learn whatever mathematics turns out to be missing as the need arises. This “learn on demand” approach is often the most efficient and natural.
It is hard to overstate the influence of Edward Witten on modern theoretical physics and mathematics. In particular, his work initiated the subject now known as topological quantum field theory (TQFT). For anyone entering the field, reading his original papers will be one of the best starting points.
A good example of a bridge between physics and mathematics is the book Mirror Symmetry:
It is a massive volume; do not feel pressured to read it cover-to-cover. Working through some of its physics-oriented sections already provides an excellent starting point.
If you are a mathematics student, the following books are very useful for future research at the interface between mathematics and physics. They are written as pure mathematics, without explicit reference to physical models, yet they form the conceptual and technical foundations of modern mathematical physics.
Although these books describe pure mathematics, they underpin important parts of modern mathematical physics—from geometric representation theory to supersymmetric theory and string theory. Understanding them will later prove invaluable for research at the interface between mathematics and physics.
Another excellent approach is to read lecture notes written by researchers working at the intersection between mathematics and physics. I can list only a few of them from my own taste:
Students often ask how to find research problems. A good starting point is to pick a research paper that genuinely attracts you and begin reading it carefully. It is often very helpful to attend a seminar talk on the paper, or to watch its recording, in order to grasp its motivation and broader context. Research papers often contain open problems, conjectures, or unexplored directions—try to engage with them seriously. If you find that you cannot solve these problems or develop them into a paper, simply move on to another work that captures your interest and repeat the same process. This cycle of reading, struggling, and moving on is a natural part of research. It is also valuable to talk with experts: discussions with advisors and senior researchers can help you identify promising directions, clarify what is important, and refine vague interests into concrete research topics.
“I am interested in category theory and its applications to TQFT. How can I find research related to TQFT?”
For example, Theo Johnson-Freyd has done brilliant work in this direction: Pick one of his papers and start reading it. Useful keywords in this direction include: En-algebras, factorization homology, AKSZ, holomorphic-topological twists, and related topics.
“I am interested in symplectic or algebraic geometry and its connection to physics. How can I find research topics?”
For instance, homological mirror symmetry is a particularly rich area. Personally, I am interested in applying the Gross–Siebert program to character varieties (or Hitchin moduli spaces) related to the Deligne exceptional series, with the goal of understanding DAHA representation theory via homological mirror symmetry.
A good starting point is:
Other topics I am personally interested in include:
If you read and understand these papers, I can suggest concrete research problems related to them.
What is written here is, again, only the tip of the iceberg. Mathematical physics is vast, and this page could easily grow without bound. I will therefore stop here.
Stay curious, stay open-minded, and keep studying.
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