Stephen Hawking, disabled English physicist and perhaps the most well-known scientist since Albert Einstein, died on 14 March 2018. He spent a large part of his career attempting to reconcile the differences between the two great scientific theories of our time - General Relativity and Quantum Mechanics. Just like Einstein (who devoted his last few years to the same problem) he failed; these discrepancies remain the single biggest unsolved problem in the entire realm of science.
So what is the problem exactly?
Well, there are four fundamental forces in the Universe, three of which are described by Quantum Mechanics:
(1) The Strong Nuclear Force
(2) The Weak Nuclear Force
The remaining force is described by General Relativity:
The two theories overlap and are generally in agreement, but in certain extreme situations they are contradictory and predict different results. No problem, I hear you say - one (or both) of them is wrong, all we have to do is conduct experiments to prove which of the theories is correct.
In normal circumstances you would be right - experiments are always the ultimate test of a scientific theory. Indeed, over the past hundred years, many thousands of experiments have been performed to test the predictions of both Quantum Mechanics and General Relativity, and in every instance both theories have been proven to be uncannily accurate.
But the situations where the theories differ are so extreme that we have been unable to create the environment needed to perform the required experiments. For example, how do you create a Black Hole and remain in control of it for long enough to see what happens right on the edge of the event horizon?
However, this has not stopped generations of mathematicians and physcists from trying to find a Grand Unified Theory, or Theory of Everything. There have been several strong candidates, such as String Theory, M-Theory, Loop Quantum Gravity and Causal Sets. But none of these have proven satisfactory and very often the mathematics becomes so complex that it is hard to imagine that it bears any relation to reality. A scientific theory that contains beautiful and elegant mathematics is no use unless it matches how things actually work in nature.
For example, one of the problems with String Theory is that it requires the existence of several extra dimensions; if the theory is correct, where are these dimensions? In fact, what exactly does it mean to have extra dimensions beyond the four we know about (three spatial dimensions and one time dimension)? No experiment has ever turned up the slightest trace of something that could be regarded as a fifth, sixth or tenth dimension. The mathematics of M-Theory requires no less than 11 dimensions!
So a grand unified Theory of Everything remains as elusive as ever. Indeed, some theorists are of the opinion that such a theory will never be found. Famed mathematician Kurt Gödel published an "Incompleteness theorem" in 1931 that suggests that any attempt to construct a Theory of Everything is bound to fail. Stanley Jaki and Freeman Dyson used slightly different logic to derive the same conclusion.
So if you're working on this problem I wish you the very best of luck - you're either doomed to years of frustration and wasted effort, or everlasting fame and a Nobel Prize ...