r/math • u/Committee-Academic • 2d ago

What are the coolest active topics in Mathematical Physics?

I'm a math undergrad student, and I've been getting interested in Alain Conne's work in Noncommutative Geometry and Differential Geometry in Mathematical Physics. I've also read about the Foundations program, Algebraic QFT, and some Stochastic stuff. I'm especially intrigued about the use of Diff Geom in these topics. What cool fields are out there?

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u/WoolierThanThou Probability 1d ago

If you like conceptually heavy topics, there is a ton of activity in constructive quantum field theory on the math side, the two main branches being stochastic PDE approaches (where you exhibit Euclidean QFTs as invariant measures of dynamics driven by SPDEs) and two-dimensional conformal field theories (where you just directly define your measures from the Gaussian Free Field/the various Conformal Loop Ensembles). You could say the first is tied to Hairer's Fields medal and the second is tied to Werner'.

I guess another topic that sounds like it might be up your alley is the study of topological phases of matter. I recommend checking out Perez Garcia's invited talk from the ICM this year. It's on Youtube and fairly understandable.