# Examples of hard but accessible problems in mathematics

There are some very old problems in mathematics, whose solutions involve reasonably accessible tools, by which I mean that (assuming knowledge of the foundations of the subject) it does take one or two semesters to introduce them. Examples include

1. Galois Theory can be used to explain
• why there are no formulas for solving polynomials of degree higher than four
• why it is impossible to construct a square of area $$\pi$$ by using compass and straight edge
2. singular homology (or maybe even cellular homology, which is easier to define) can be used to show
• the Fundamental Theorem of Algebra
• the Brouwer Fixed Point Theorem
• the Jordan Curve Theorem, which was clear a long time until it wasn’t clear, why its clear
3. I recall Cauchy‘s Integral Theorem (translated from German, might be the wrong name) to be a major tool in complex analysis, giving new ways to calculate indefinite integrals and leading to other consequences like the Fundamental Theorem of Algebra
4. Bezout‘s theorem is a result in projective geometry, which implies many well known classical theorems like for example Pascal‘s theorem.
5. The classification of closed surfaces is a fairly hard theorem to have. I don’t really know of applications though.
6. Undecidability of the word problem. I read that it gives obstructions to classifications of higher dimensional manifolds. It has also some consequences when working with group presentations.
7. Dirichlet‘s theorem was suggested by Anthony Saint-Criq.

On the other hand there are some problems that (though solved) don’t seem accessible for „normal“ mathematicians

• Fermat‘s Last Theorem seems to lie within the reach of a master program dedicated to showing it, but otherwise seems to involve a proof, which is too technical to be understood within a reasonable time frame
• the Classification of Finite Simple Groups involves a handful of books, spanning thousands of pages

It feels to me like currently most problems, which are of interest, involve very deep techniques, which take years to learn and master. So my question is

What are some (harder) problems, whose solutions are accessible in the sense that they involve machinery, which can be introduced in one or two semesters time?

Edit It is hard to pin down what accessible should mean precisely, so I think any problem of the form „assuming basic knowledge of $$X$$ and taking one or two semesters worth of time to develop $$Y$$ we can proof $$Z$$“ applies to my question. Personally I would prefer $$X$$ to be in the toolbox of a broadly educated master student (e.g. $$X\in$$ {differential equations, commutative algebra, smooth manifolds, euclidean geometry, finite groups, point set topology,...})

• You tagged this open-problem; did you want to include open problems in this big-list? If you do want open problems, then how can we warrant that the techniques necessary to solve them are accessible? Jul 13, 2020 at 15:13
• @user804886 ups, you are right. I edited accordingly... Jul 13, 2020 at 15:14
• I like Dirichlet's theorem, which involves a wide variety of notions, yet being accessible. Jul 13, 2020 at 15:21
• You should probably specify which level your audience is: undergraduate or master or phd? The latter two classes tend to be more specialized in certain domains, so that your choices are further narrowed. Also note that the square of area $\pi$ problem involves not only Galois theory but also Lindemann's theorem (in fact more on the latter, from my point of view, as it's easy to show that constuctible points are algebraic, even without Galois theory). Jul 13, 2020 at 15:28
• @WhatsUp Indeed. I added a paragraph intended to clarify the audience. I feel like PhD is too specialized to count as being accessible... Jul 13, 2020 at 15:54