There's been a lot of talk recently about complex mathematical ideas such as the amplituhedron. The nonlocal aspect of this is interesting, but the fact that this geometrical shape is simple is misleading, since the mathematics itself is still complex and it has no physical justification (it reminds me of Kepler's erroneous Platonic solid model of the Solar system). Also, it uses supersymmetry, whose predictions have not been seen. This kind of thing is very common in modern physics (I remember also Weinstein's 14 dimensional maths) and although supersymmetry looks like it is being finally tested, many of these ideas are presented without making any testable new predictions about nature and rely solely on their agreement with the standard models.
To be fair, few anomalies from the standard model have been seen in particle accelerators, it seems that physics is successful so far at predicting things in the narrow regime that we call 'high' energy and high acceleration. The huge anomalies in physics are at low accelerations, for example for spacecraft in deep space (maybe), objects in cryostats (low thermal acceleration), for stars at the edges of galaxies (the galaxy rotation problem) and the acceleration of distant supernovae (cosmic acceleration). Dark matter and dark energy have been devised to explain these, but these hypotheses are arbitrary and unpredictive. For example, given the light distribution of a galaxy you cannot predict the motion of its stars with dark matter. You have to first assume that general relativity (GR) is right and then work out the dark mass distribution that makes GR agree with the velocity you see. You have not predicted the velocity, you have used the velocity and the assumption that GR is right, to predict the dark mass distribution, and you can't test your result since you can't detect dark matter! So dark matter is unpredictive and untestable. Safe from disproof, but completely useless.
The problem, as always, is that old theories are respected more than new data. There is no reason for this: quantum mechanics and general relativity are incompatible with each other, so they are demonstrably not the final word, and yet they are extrapolated from the scale of our experience (Solar system scale) to scales at least 10 orders or magnitude upwards to galaxies and the cosmos. The last time that happened was when classical physics, designed for the human scale, was extrapolated ten orders of magnitude down to the atomic scale but didn't work, so the strange ideas of quantum mechanics had to be invented. In the modern case our theories don't work when extrapolated up to these huge scales or low accelerations, and arbitrary patches are applied.
What is desperately needed for progress in physics is a more liberal attitude to strange new results. These are controversial at the moment and they should not be! Publish a paper with the word "Podkletnov" in it and you'll will seriously damage your career. This is against the spirit of science. The experiment may have been wrong, but it passed peer-review and it may be nature telling us something very new (as I argue here, and I am about to submit another paper on this). An honest study of controversial anomalies has always been the best way to new science (there is the danger of being wrong too).
Examples of anomaly-driven science are first class: Galileo saw the moons of Jupiter orbiting and believed Copernicus' model of the Solar system, Newton split up white light with a prism, and was surprised when he couldn't split coloured light, the early Einstein was puzzled by the photoelectric effect and the anomalous Michelson-Morley experiment which failed to detect the aether. Darwin saw dissimilar finches on seperate Galapagos islands and wondered why.
This is why I do not trust hypotheses like the amplituhedron, string theory et al., that utilise hugely complicated maths and agree nicely with standard models, but say nothing new and testable about nature. Give me a solid anomaly anyday!