Of the topics you mentioned, perhaps Representation Theory (of Lie (super)algebras) has been the most useful. I realise that this is not the point of your question, but some people may not be aware of the extent of its pervasiveness. Towards the bottom of the answer I mention also the use of representation theory of vertex algebras in condensed matter physics.
The representation theory of the Poincaré group (work of Wigner and Bargmann) underpins relativistic quantum field theory, which is the current formulation for elementary particle theories like the ones our experimental friends test at the LHC.
The quark model, which explains the observed spectrum of baryons and mesons, is essentially an application of the representation theory of SU(3). This resulted in the Nobel to Murray Gell-Mann.
The standard model of particle physics, for which Nobel prizes have also been awarded, is also heavily based on representation theory. In fact, there is a very influential Physics Report by Slansky called Group theory for unified model building, which for years was the representation theory bible for particle physicists.
More generally, many of the more speculative grand unified theories are based on fitting the observed spectrum in unitary irreps of simple Lie algebras, such as $\mathfrak{so}(10)$ or $\mathfrak{su}(5)$. Not to mention the supersymmetric theories like the minimal supersymmetric standard model.
Algebraic Geometry plays a huge rôle in String Theory: not just in the more formal aspects of the theory (understanding D-branes in terms of derived categories, stability conditions,...) but also in the attempts to find phenomenologically realistic compactifications. See, for example, this paper and others by various subsets of the same authors.
Perturbative string theory is essentially a two-dimensional (super)conformal field theory and such theories are largely governed by the representation theory of infinite-dimensional Lie (super)algebras or, more generally, vertex operator algebras. You might not think of this as "real", but in fact two-dimensional conformal field theory describes many statistical mechanical systems at criticality, some of which can be measured in the lab.
In fact, the first (and only?) manifestation of supersymmetry in Nature is the Josephson junction at criticality, which is described by a superconformal field theory. (By the way, the "super" in "superconductivity" and the one in "supersymmetry" are not the same!)
Freeman Dyson's proposal is online, based on a talk he gave at MSRI.
Lillian Pierce's senior thesis gives a summary of Peter Sarnak's program to use properties of Gaussian Unitary Ensemble to study the zeros of the Riemann Zeta function.
N. G. Debrujin wrote about Penrose tilings and their Fourier transforms.
Crystalline structures on the line are pretty boring. They are just evenly spaced lattices, like $\mathbb{Z}$, which might appear on different scales.
--o---o---o---o---o---o---o--
---o-----o-----o-----o-----o-
However, there are many quasi-periodic structures on the line, for example $\lfloor n\sqrt{2}\rfloor
= \{ 1, 2, 4, 5, 7, 8, 9, 11, 12, 14,\dots \}$ which we can draw on the line.
--o--o-----o--o-----o--o--o-----o--o-----o--
Many of these have special recursive properties. Consider the line $y = \frac{1 + \sqrt{5}}{2} x$ which Golden ration slope. Mark "0" if it crosses a horizontal line and "1" if for a vertical line. You get the Fibonacci Word
Of course in 2D you get more interesting quasicrystals, which have interesting number theoretic and recursive structures.
Freeman Dyson wishes the zeros of the Riemann Hypotheses have structure like these.
Best Answer
Not a direct answer, but maybe worth pointing out:
An apparently substantial insight into a relation between gravity and the zeros of the Riemann zeta-function (hence the Riemann hypothesis) was recently found via $p$-adic string theory by Shing-Tung Yau et al.:
Seems quite remarkable to me.