What are some of the Hardest Unsolved Mathematics Problems?

Question

At the moment, are there any major unsolved mathematical problems yet to be solved, and do they have any prize associated with the solving of them?

Furthermore, is there any particular reason that they have not yet been solved?

Answer 1

Any answer to your question will risk overgeneralizing. However, taking a look at the great problems of antiquity, one theme that occurs is the that we lacked the proper language to express what a solution would even look like.

For example, the Geometric Problems of Antiquity were insoluble given that they were expressed using the language of straightedge and compass. Once we moved to the more abstract algebraic approach, these ceased to be issues, and became, in a sense, trivial.

Another set of problems involved problems of infinite processes before calculus and real analysis. These are best described by Zeno's Paradoxes.

A quick glance at the Clay Millenium Problems shows they are a diverse set, so I doubt there is any one "thing" making them hard. However, again at the risk of overgeneralizing, they are unsolved because we cannot express the properties of a possible solution (note: not "the" solution, but the criteria that make it a possible solution.)

As a concrete example. In differential equations, we know the solution must be a function of the given arguments. A more abstract one is Wiles' proof of Fermat's Last Theorem: he re-cast it as a problem in the theory of modular elliptic curves, whose solutions require certain conditions, and, with a TON of ingenuity and intelligence thrown in (of which I cannot ever hope to comprehend) he shows that Fermat's Last Theorem is true. From what I can understand of his approach, it was the great insight the solution would be within the framework of elliptic curves.

I really can't make this more specific given your question and my limited experience with specific unsolved problems. My answer comes from my reading of various histories of mathematics, where previously "unsolvable" problems have now become standard fare.

However, given the number of very smart people working on them (unsuccessfully up to this point), it appears that the solutions are not "first order", where we know the outlines of what we need to do but lack a specific solution; instead, they are "second order" where we don't even know what a solution would imply or what properties it would have.

Answer 2

The Millennium Prize Problems | Clay Mathematics Institute $1 million allocated to each