Today appeared on arxiv a set of lecture notes by Strominger, called Lectures on the Infrared Structure of Gravity and Gauge Theory. They review the progresses of the last few years in connecting different parts of physics, where the same mathematical object had been studies for decades but under very different guises. This is illustrated by the "infrared triangle", taken from the notes :

The introducting section of the notes is definitely worth reading. Let me quote it extensively (references removed) :

The first corner is the soft theorems'. These originated in QED in 1937 [...]. Soft theorems characterize universal properties of Feynman diagrams and scattering amplitudes when a massless external particle becomes soft', i.e. its energy is taken to zero.
They tell us that a surprisingly large --- in fact infinite --- number of soft particles are produced in any physical process, but in a highly controlled manner which is central to the consistency of quantum field theory.

The second corner is the subject of asymptotic symmetries'.
This is the study of the non-trivial exact symmetries or conserved charges of any system with an asymptotic region or boundary. One of the earliest examples appears in the pioneering work of Bondi, van der Burg, Metzner and Sachs (BMS), who sought to recover the Poincaré group of special relativity as the symmetry group of asymptotically flat spacetimes in general relativity (GR). Instead, in a spectacular failure of their original program, they discovered the infinite-dimensional BMS group whose deep implications are still being unravelled today.
Analogous asymptotic symmetries in QED and nonabelian gauge theory, on the other hand, were discovered only recently and are a subject of ongoing research.

The third corner of the triangle is the memory effect, investigated in the context of gravitational physics in 1974 [...]. This is a subtle DC effect, in which the passage of gravitational waves produces a permanent shift in the relative positions of a pair of inertial detectors. Detection of the memory effect has been proposed at LIGO or via a pulsar timing array. It is an exciting experimental prospect for the coming decades.
Again, the gauge theory analog appeared only recently.

[...]

The bigger picture emerging from the triangle is that deep IR physics is extremely rich, perhaps richer than previously appreciated. Every time we breathe, an infinite number of soft photons and gravitons are produced. Quantum field theory analyses tend to treat this as a technical problem to be overcome
by resorting to carefully constructed inclusive cross sections. IR regulators are often used which explicitly break the symmetries, and it is difficult to see them emerge as the regulator is removed. GR analyses tend to treat the BMS discovery that the deep IR limit of GR is not special relativity as a problem' to be avoided by the ad hoc imposition of extra boundary conditions at infinity. In fact, BMS discovered the classical version of infinite graviton production, an equivalent indication of the richness of the deep IR. Here we will see that, far from being technicalities, these IR phenomena are associated with fundamental symmetries of nature whose fascinating consequences we are just beginning to unravel.

This fascinating triangle is then seen to pervade a seemingly infinite number of areas of physics, as illustrated in the next picture, also taken from the notes, which you should read to have the comments :)