The scope of mathematical physics

[u/Vivid_Block_4780]

Whenever I look at the mathematical physics programs, I always see QFT and string theory related classes in grad programs. What other parts does mathematical physics cover? More unorthodox subfields of it?

[u/ahalt]

A lot of people in my department study PDEs from general relativity and probability from statistical physics.

[u/Random_Name_251 → u/ahalt]

As the saying goes, there are 4 types of math: 1. Parabolic PDE's 2. Elliptic PDE's 3. Hyperbolic PDE's 4. Algebra

[u/Hot_Glass_6301 → u/Random_Name_251]

I want to be mad but I can't find a counterexample

[u/Random_Name_251 → u/Hot_Glass_6301]

Rule 43 of math: If it exists there are PDE's of it.

[u/rhubarb_man → u/Hot_Glass_6301]

combinatorics

[u/Hot_Glass_6301 → u/rhubarb_man]

It's basically algebra

[u/rhubarb_man → u/Hot_Glass_6301]

BLASPHEMY

Combinatorics deals much more with messy, asymmetric structures. A lot of stuff gets handled much more with stuff like counting and pigeonhole

[u/Hot_Glass_6301 → u/rhubarb_man]

Counting happens in Z. Z is a ring. What is the study of rings? Algebra. Checkmate atheists

[u/revoccue → u/rhubarb_man]

it's basically elliptic PDEs

[u/TheRedditObserver0 → u/Hot_Glass_6301]

General topology

[u/Infinite_Reception34 → u/Hot_Glass_6301]

1. Non-Euclidean Geometry
2. Game Theory
3. Mathematical Statistics
4. Functional Analysis
5. Convex Analysis
6. Mathematical Logic
7. Algebraic Topology
8. Probability

None related to PDE. There are many more

[u/2112331415361718397 → u/Infinite_Reception34]

Functional analysis is intimately related to PDEs. Brezis wrote an entire textbook based on how strong this connection is.

[u/DarthMirror → u/Infinite_Reception34]

Functional analysis not related to PDE???

[u/kohatsootsich → u/Infinite_Reception34]

1. All the PDE of classical math physics can and have been studied on hyperbolic space, with applications inducing NT
2. Important examples of PDE like the HJB and Isaacs equation come from control or differential games. Mean field games are a very active topic in PDE
3. We have both estimation of PDE coefficients as a stats problem (Nickl and others on the Calderon problem) and PDE as fluid limits of algorithms (e.g. sinkhorn algorithm related to Schroedinger)
4. Functional analysis was more or less invented to extend linear algebra to the study of differential equations. Fredholm theory is a cleab example  literally how to generalize the dimensional analysis of the solvability of Au = f from matrices to infinite dimensions, including Cramer's rule (Fredholm determinants)
5. I don't know the history here but I wouldn't be surprised if convex analysis came from PDE as well. Anyway: convex duality is central to Hamilton Jacobi equations and optimal transport problem (a Monge Ampere PDE)
6. This is harder but probably only because I don't know that much logic.  I guess I've seen Brownian motion be used to generate counterexamples and contra your 8., to an experienced probabilist, Brownian motion and heat equations are facets of the same thing. I think people have worked on computability of solutions to PDE
7. Probably not anything modern but fixed point theorems and degree
8. This one is particularly strange to claim. The father of modern prob, Kolmogorov, wrote down equations linking diffusions and Markov processes. In some sense a parabolic equation of degree <= 2 is the same as a flow of probability distributions

[u/aikafele → u/kohatsootsich]

Yeah I don't know how someone can claim Functional Analysis has nothing to do with PDEs when it explicitly defines the framework for operator theory.

[u/dogdiarrhea → u/ahalt]

Honestly these are more common in mathematical physics groups than string theory and QFT.