AXIOMS: Most basic mathematical structures are heavenly in the sense that they would be discussed in any part of the universe. In algebra these would be groups or rings. In topology these would be topological spaces or metric spaces, in probability theory it would be sigma algebras and stochastic processes. Of course, the notation and organization and names would be different but it would be no problem to translate into modern notions. Maybe, on Andromeda, one would rather focus on Magmas than Monoids or look at generalized rings, where the additive group does not have to be commutative. In any case, these are “heavenly structures”. One can apply this categorization to any thing. Also to open problems. The four Landau problems in number theory are all “heavenly”. Any mathematical culture developed in the universe deserving this name will eventually get to this. The Collatz 3k+1 problem on the other hand might be studied with other affine maps and other primes than 2. It is one of many similar problems. But the question whether infinitely many primes exist in the image of a given polynomial is too natural to be missed. The question whether n^2+1 has infinitely many primes is the smallest really difficult case and certainly be studied also because of other reasons like asking whether there are rows in the Gaussian integers free of primes. The picture to the right is from one of my my “outlines” I wrote for exam preparations in college (ETHZ).
GAMES: Most games are “man made” and would look completely different in an other part of the universe. Chess would almost certainly not played anywhere else. The reason is that one had to do many choices when designing chess. The axiom system which describes the motion of the figures is rather arbitrary. It almost certainly has evolved in an evolutionary way. Playing chess on a 4×4 board does not work most of the time for example. There are some small chess board problems on this math table talk of 2007 about AI (PDF). By the way, that was at a time, when almost nobody was interested in AI. See also the “toral chess” riddle from earlier this semester. For this talk I mentioned a chess game on a 8x8x8 board which I have never been able to play. I don’t have enough chess figures. The entire second and seventh plane is filled with pawns. The first and last is filled with more valuable figures. A natural choice is to arrange the known figures in a concentric ring shaped. One could take 2 kings, but more traditional is to have one king with 3 queens in the center, then a ring of bishops, then a ring of horses and finally a ring of rooks. I myself was first exposed to the idea of 3D chess when watching “Raumschiff enterprise” as kids. By the way, I once played in a tournament in Schaffhausen but lost badly (played heavily with “Gambits” but did not master them really) and lost interest.
MODELS: Models are important in physics. There are three major pillars which are fundamental, successful and simple in principle: the first is relativity GR, which is differential geometry in a pseudo Riemannian manifold (M,g). Matter moves on geodesics and mass determines the metric. While these are simple variational problems in theory finding explicit solutions is almost never possible. Geodesics on a surface of non-constant curvature can become very complicated. The problem to determine the metric from the mass tensor is a complicated partial differential equation for which existence theorems are only known in very special cases. All experimental verifications of GR are based on simple solutions of the Einstein equations. The second pillar is classical quantum mechanics QM in which a system is described by an operator H. The Schroedinger equation u’ = i H u, then gives a unitary evolution U(t)= exp(i Ht) and the wave moves as U(t) u(0). Since QM deals with very small systems, the laboratory itself plays a role. The Coopenhagen interpretation hacks this via “wave function collapse”. The third pillar is the standard model SM embedded in quantum field theory. The Lagrangian for this system looks bewildering complicated indicating that we do not yet understand more basic mechanisms which lead to this. The SM describes two major structures: Hadrons and Leptons. Hadrons are either Mesons or Baryons and are made of quarks. Leptons are either electrons or neutrini. All these particles interact via force particles like photons, gluons, vector bosons or Higgs boson. Fundamental symmetries are U(1), which topologically is the only commutative Lie group that is a sphere and SU(2) which is the only non-commutative Lie graph that is a sphere. SU(3) is an other symmetry which can also be seen as coming from symmetries within the unit quaternions SU(2). Fermions like quarks or leptons have half integer spin while Bosons like the force particles have integer spin.
