Open problems in mathematical physics

Abstract

We present a list of open questions in mathematical physics. After a historical introduction, a number of problems in a variety of different fields are discussed, with the intention of giving an overall impression of the current status of mathematical physics, particularly in the topical fields of classical general relativity, cosmology and the quantum realm. This list is motivated by the recent article proposing 42 fundamental questions (in physics) which must be answered on the road to full enlightenment [1]. But paraphrasing a famous quote by the British football manager Bill Shankly, in response to the question of whether mathematics can answer the Ultimate Question of Life, the Universe, and Everything, mathematics is, of course, much more important than that.

Theoretical cosmology

“…All numerical results, and current observational data, provide evidence that the Kerr (and Kerr-Newman) black holes are non-linearly stable [43]. 5 Often erroneously called the 'Horizon'. It has nothing to do with causality, i.e.…”

“…Noted problems have always been of importance and part of the culture in mathematics [5]. The twenty-three problems by Hilbert [6] are perhaps the most well known problems in mathematics.…”

“…However, the way this happens depends on the comoving wavelength of the perturbation relative to the scale set by the Hubble expansion rate at that time. 5 These studies hold while the perturbation is linear, and have been heroically extended to the non-linear case (see later). However numerical simulations are required for the strongly non-linear case [76].…”

Light, the universe and everything – 12 Herculean tasks for quantum cowboys and black diamond skiers

“…The use of questions to focus thinking (293), (294), (295), (296) is an honorable tradition that extends back at least to Newton's Opticks, where he posed 31 queries including the following: We can only hope that the questions raised here remain equally meaningful after the passing of centuries, or at least have some significance during the course of the present century. But it is clear that quantum (and classical) optics has already undergone a historic blossoming into realms of understanding and application that are almost magical, and that the research programs of the authors in this article (and their colleagues) promise equally exciting developments in the future.…”

Mathematical general relativity

“…In addition, their problem (CGP11) asks whether the existence of Cauchy horizons, not necessarily compact or analytic, implies the existence of local isometries. Another concise survey on the Cauchy problem was given in [10], a number of problems in mathematical GR were discussed in [11], and a survey on strong cosmic censorship is anticipated soon by Dafermos and Luk.…”