What are some recent, interesting, accessible pieces of mathematics

Question

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.

Answer 1

Perhaps: The discovery a year ago in 2015 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):

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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. (But see update below.)

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).

Update. Rao, Michaël (2017), Exhaustive search of convex pentagons which tile the plane, arXiv:1708.00274.

We present an exhaustive search of all families of convex pentagons which tile the plane. This research shows that there are no more than the already 15 known families. In particular, this implies that there is no convex polygon which allows only non-periodic tilings.

Answer 2

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:

• http://www.nature.com/news/peculiar-pattern-found-in-random-prime-numbers-1.19550
• http://www.independent.co.uk/news/science/maths-experts-stunned-as-they-crack-a-pattern-for-prime-numbers-a6933156.html
• https://www.quantamagazine.org/20160313-mathematicians-discover-prime-conspiracy/
• https://www.scientificamerican.com/article/peculiar-pattern-found-in-random-prime-numbers/

Answer 3

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.

Answer 4

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

Answer 5

I think a lot of results are "accessible" in terms of a pop science-y explanation, rather than following a detailed proof. And since you're looking for emotional motivation, I actually think the former is the more important!

For example, see this (quite engaging) video about moving sofas around corners.

https://www.youtube.com/watch?v=rXfKWIZQIo4

Many of the mathologer and numberphile and such videos are very accessible, motivating, and quite well done (content and production and engaging presenters).

Heck, I would even include the Andrew Wiles doc by Horizon. It has some real connection to school age children (Wiles even says he was motivated at age 10). It doesn't matter so much that I don't understand the details of the proof itself. The problem is describable and a topical (literally naming "topics" in advanced math) explanation of the achievement is possible.

https://www.dailymotion.com/video/x1btavd

Probably a key thing is that the problem itself is describable (even if the details of the proof are not). For instance the Poincare Conjecture as a problem is tricky to understand, just as a concept. I mean I don't get it...so that must mean it's hard, right? ;)

You can maybe still mention that "they bagged a recent big problem" if someone solves the Riemann hypothesis tomorrow, but harder to really talk about the topic. And also note that it was the less general, but more understandable FLT problem that really motivated Wiles to his seven-year attic-sweat, not the more math-important (because of generality) Taniyama conjecture.

But I also would not underestimate the appeal of older "aha"s. After all, they are new to the student and are thrilling achievements of Man. I remember thinking Archimedes must have been a freaking time traveler when we learned his method for area under the curve of a parabola. Sure smells like integral calculus to me!

E.g. here is an engaging mathologer video on solving cubics:

https://www.youtube.com/watch?v=N-KXStupwsc

Or this Veritasium video on Newton's advances in calculating pi (I might wait for a BC calc class, not do it in geometry though as the methods are sort of calculus and "algebra two").

https://www.youtube.com/watch?v=gMlf1ELvRzc

So I wonder if the desire to highlight research results is more a desire of the teacher than the students. (After all the teacher is probably a math researcher or at least was one during grad school.) I did chem research during grad school, but was a bit skeptical (as a TA) of big wheel tenured profs at an R1 who wanted to connect in recent research results to a service course in undergrad chem (mostly non-major students, vast swath of pre-meds). Felt like they were trying to justify themselves and the whole concept of an R1 (which I do not agree is really beneficial for undergrad training). And they were falsely deprecating the importance to the student of learning concepts, tricks that are not new research results, but are new to the student.

P.s. But I would try to keep it light and motivational and not derail your course time with too much external content. Unless the kids are very strong to start with (thus tolerating more enrichment as the basic content itself is easier on them).

P.s.s. Also take a look at the American Scientist columns (or books) by Brian Hayes.

https://www.americanscientist.org/author/brian_hayes

They are good popularizations. I dated an R1 prof who was doing cutting edge research in one of the topics (it was graphs or queuing or Hamming cubes or something like that) and she was impressed that I basically had an overview of her area from such a Hayes column. She took a look at the article in her area and said it was a lot better than she expected.