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Mathematics can come across as a sterile, dead subject - a catalogue of techniques long-ago decided, and forever relearned by each successive generation of students.

This is approximately true for elementary and secondary mathematics, and for the standard progression of undergraduate courses (eg, Calculus 1,2,3, discrete math / combinatorics, ODE + Vector Calc, Analysis and Algebra).

Of course, the subject is alive and kicking, with many thousands of active researchers learning, creating, refining, and publishing every day. But the vast majority of fresh research requires considerable expertise to understand, and are therefore inaccessible to younger students of the subject.

What, then, are some recent results that are interesting and accessible to students at (say) a secondary school level, which might exemplify that the subject remains active?

A couple of examples that come to mind (which could be fleshed out as answers) are the recent progress against the Twin Prime Conjecture, and the surprising observation that primes ending in $X$ 'favor' being followed by a prime ending in $Y$, for various $(X,Y)$ pairs.

Where it's appropriate, please include links to any media treatment of the result.

Let's roughly define 'recent' as being within the lifetime of some collection of students.

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Perhaps: The discovery a year ago of a new tiling of the plane by a convex polygonal tile, found by Mann, McLoud, and Von Derau (the latter of whom was an undergraduate at the time of the discovery):


PentagonalTiling
Here is a nice article on the discovery in The Guardian, by Alex Bellos. As Alex says, the problem has been studied for $100$ years now, since Reinhardt in 1918. It remains unknown if the current list of $15$ such tilings is complete.

Not only is there no algorithm to determine if a candidate tile can indeed tile the plane, it is not even known that the problem is decidable.

Mann, Casey, Jennifer McLoud-Mann, and David Von Derau. "Convex pentagons that admit $ i $-block transitive tilings." arXiv abstract (2015).

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    $\begingroup$ +1, I mentioned this discovery/Guardian article in a comment back in Aug 2015 & was concerned it would go unseen! $\endgroup$ – Benjamin Dickman Nov 15 '16 at 4:35
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The 2016 result about Unexpected biases in the distribution of consecutive primes.

This is really pretty simple to understand - the distribution of primes had been supposed to be unconditionally random, but this interesting bias is suddenly discovered. The amazing thing about it is that the observation could have been made by pretty well any beginner computer programmer in the past 30 years, but only now had anyone noticed.

The story had a number of treatments in a range of media:

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Not a single answer but rather a resource:

The snapshots of modern mathematics from the mathematical research institute Oberwolfach (http://www.mfo.de/math-in-public/snapshots/) aim to provide pretty much exactly what you ask for. Research mathematicians present some "recent developments" in their field in a way that is understandable for high school students/ math teachers/ undergraduates (the actual level varies quite a bit between the various snapshots).

The snapshots are freely available on the homepage of the institute and cover quite a lot of topics.

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    $\begingroup$ Great resource! Thanks for the pointer. $\endgroup$ – Joseph O'Rourke Nov 23 '16 at 13:04
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There are elementary results in mathematics being produced regularly either in the sense that the ideas don't need a lot of background to understand them or sometimes that even the "details" of the proofs are comprehensible. An example, is that of the de Bruijn graph, which has ties to the theory of Eulerian circuits in directed graphs and has found many applications, in particular, to assisting with genome sequencing.

https://en.wikipedia.org/wiki/De_Bruijn_graph

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