While there is tremendous progress happening in mathematics, most of it is just accessible to specialists. In many cases, the proofs of great results are both long and use difficult techniques. Even most research topologists would not be able to understand the proof of the virtually fibering conjecture or the Kervaire problem, to name just two recent breakthroughs in topology, without spending months on it.
But there are some exceptions from this rule. As a topologist, I think here mostly about knot theory:
- The Jones and HOMFLY polynomials. While the Jones polynomial was first discussed from a more complicated context, a rather simple combinatorial description was found. These polynomials help to distinguish many knots.
- The recent proof by Pardon that knots can be arbitrarily distorted. This might be not as important as the Jones polynomial, but quite remarkably Pardon was still an undergrad then!
Or an example from number theory are the 15- and 290-theorems: If a positive definite integer valued qudadratic form represents the first 290 natural numbers, it represents every natural number. If the matrix associated to the quadratic form has integral entries, even the first 15 natural numbers are enough. [Due to Conway, Schneeberger, Bhargava and Hanke.] [Edit: As mentioned by Henry Cohn, only the 15-theorem has a proof accessible to undergrads.]
My question is now the following:
What other major achievements in mathematics of the last 30 years are there which are accessible to undergraduates (including the proofs)?