math.MP Mathematical Physics: this subject already uses several of the other already mentioned categories, like group theory, functional analysis, applied to physical theories like quantum field theory. But since the intention is to see concrete applications to the real world taken from a mathematically rigorous framework, then we could mention
- In the first place Noether's theorem: "every differentiable symmetry of the action of a physical system has a corresponding conservation law".
- Onsager reciprocal relations: They "express the equality of certain ratios between flows and forces in thermodynamic systems out of equilibrium, but where a notion of local equilibrium exists". Onsager's contribution was to demonstrate that not only is $L_{\alpha \beta}$ positive semi-definite (the Onsanger matrix of phenomenological coefficientes), it is also symmetric, except in cases where time-reversal symmetry is broken.
- The issues about proving thermodynamical statements strictly from statistical mechanics. This became one the biggest discussions in mathematical physics on its time, between Ludwig Boltzmann and Ernst Zermelo (see the book of the colleted works of Ernst Zermelo, volume II, edited by Herausgegeben von, Heinz-Dieter Ebbinghaus and Akihiro Kanamori, Springer, 2013).
- The existence of anti-particles by P.A.M. Dirac, which was first a mathematical result. Dirac realised that his relativistic version of the Schrödinger wave equation for electrons predicted the possibility of antielectrons, themselves discovered four years latter, by Carl D. Anderson.
- The Universality of the Feigenbaum constants, proven by Landorf ( Lanford III, Oscar (1982). "A computer-assisted proof of the Feigenbaum conjectures". Bull. Amer. Math. Soc 6 (3): 427–434.), with some corrections by Eckmann and Wittwer (Eckmann, J. P.; Wittwer, P. (1987). "A complete proof of the Feigenbaum conjectures". Journal of Statistical Physics 46 (3–4): 455).
- The very concept of Universality, together with scaling, introduced by Kadanoff (see for example, Physica A 163 (1990) 1-14 "Scaling and Universality in Statistical Physics"). Roughly speaking, scaling is about the description of changes in the behavior of physical phenomena, in terms of adimensional constants, and how do they scale with them, as is the case of the Reynolds number. Universality concerns the invariance of properties for different dynamical systems, independently of some other physical details. For example, (see the aforementioned paper of Kadanoff), any perturbation which does not drive away a Hamiltonian near a critical point is deemed irrelevant, and all the Hamiltonians with any such kind of perturbations are said to belong to the same Universality class.
- In the speculative region (as of today), the prediction of a possible second Island of Stability, where some new stable chemical elements might be found, with possibly interesting physical properties (see, for example, Zeitschrift für Physik 1969, Volume 228, Issue 5, pp 371-386 "Investigation of the stability of superheavy nuclei around Z=114 and Z=164", by Jens Grumann, Ulrich Mosel, Bernd Fink, Walter Greiner).
- Even more speculative, but still valid mathematical results, some solutions from Einstein's General Relativity equations, allowing for closed timelike curves (i.e. time machines to the past), such as Gödel Spacetime (see, Review of Modern Physics, volume 21, Number 3, July 1949, "An example of a new type of cosmological solutions of Einstein's field equations of gravitation", Kurt Gödel).
- Not yet published as mathematically solved, the strict mathematical derivation of turbulence, from the Navier-Stokes equation, or otherwise (turbulence does not have a complete mathematical theory, to the extent of my knowledge). Navier-Stokes does not even have a theorem for the following problem: "Prove or give a counter-example of the following statement: In three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier–Stokes equations."
- Along similar lines for mathematical completeness, is the Yang-Mills existence and mass gap: "Prove that for any compact simple gauge group $G$, a non-trivial quantum Yang–Mills theory exists on $\mathbb{R}^4$ and has a mass gap $\Delta > 0$. Its importance comes from the fact that it is the simplest Quantum Field Theory available (as far as I know), and it does not need to assume the existence of quarks.
