The top 10 mathematical achievements of the last 5ish years, maybe

Posted by richardelwes on June 18, 2015
Posted in: Maths. Tagged: , , , . 11 Comments

I have recently been going through my book Maths 1001 making updates for a forthcoming foreign edition (of which more in future). So I have been looking over mathematical developments since approximately 2009. Thus, I present ten major developments in the subject since around then, arranged arbitrarily in ascending order of top-ness.

"Socolar-Taylor tile" by Parcly Taxel - Own work. Licensed under CC BY-SA 4.0 via Wikimedia Commons - https://commons.wikimedia.org/wiki/File:Socolar-Taylor_tile.svg#/media/File:Socolar-Taylor_tile.svg

“Socolar-Taylor tile” by Parcly Taxel – Own work. Licensed under CC BY-SA 4.0 via Wikimedia Commons – https://commons.wikimedia.org/wiki/File:Socolar-Taylor_tile.svg#/media/File:Socolar-Taylor_tile.svg

 

10. Mochizuki’s claimed proof of the abc conjecture.   The countdown kicks off on an awkward note. If Shinichi Mochizuki’s 2012 claimed proof of the abc conjecture had gained widespread acceptance, it would definitely top this list. As it is, it remains in limbo, to the enormous frustration of everyone involved.

 

9. The weak Goldbach conjecture. “From 7 onwards, every odd number is the sum of three primes.” We have known since 1937 that this holds for all large enough odd numbers, but in 2013 Harald Helfgott brought the threshold down to 1030, and separately with David Platt checked odd numbers up to that limit by computer.

 

8. Ngô Bảo Châu’s proof of the Fundamental Lemma. Bending the rules to scrape in (time-wise) is this 2009 proof of a terrifyingly technical but highly important plank of the Langlands Program.

 

7. Seventeen Sudoku Clues. In 2012, McGuire, Tugemann, and Civario proved that the smallest number of clues which uniquely determine a Sudoku puzzle is 17. (Although not every collection of 17 clues yields a unique solution, their theorem establishes that there can never be a valid Sudoku puzzle with only 16 clues.)

 

6. The Growth of Univalent Foundations/ Homotopy Type Theory. This new approach to the foundations of mathematics, led by Vladimir Voevodsky, is attracting huge attention. Apart from its inherent mathematical appeal, it promises to recast swathes of higher mathematics in a language more accessible to computerised proof-assistants.

 

5. Untriangulatable spaces. In sixth position is the stunning discovery, by Ciprian Manolescu, of untriangilatable manifolds in all dimensions from 5 upwards.

 

4. The Socolar–Taylor tile. Penrose tiles, famously, are sets of tiles which can tile the plane, but only aperiodically. It was an open question, for many years, whether it is possible to achieve the same effect with just one tile. Then Joan Taylor and Joshua Socolar found one (pictured above).

 

3. Completion of the Flyspeck project. In 1998, Thomas Hales announced a proof of the classic Kepler conjecture on the most efficient way to stack cannon-balls. Unfortunately, his proof was so long and computationally involved that the referees assigned to verify it couldn’t complete the task. So Hales and his team set about it themselves, using the Isabelle and HOL Light computational proof assistants. The result is not only a milestone in discrete geometry, but also in automated reasoning.

 

2. Partition numbers. In how many ways can a positive integer be written as a sum of smaller integers? In 2011, Ken Ono and Jan Bruinier provided the long-sought answer.

 

1. Bounded gaps between primes. It’s no real surprise to find that the top spot is taken by Yitang Zhang’s wonderful 2013 result that there is some number n, below 70 million, such that there are infinitely many pairs of consecutive primes exactly n apart. The subsequent flurry of activity saw James Maynard, and a Polymath Project organised by Terence Tao, bring the bound down to 246.

 

But, but,…?!

Where’s Hairer’s work on the KPZ equation? What about Friedman’s new examples of concrete incompleteness?! What can I say? It’s just for fun, folks. If you think I’ve got it horribly wrong, then feel free to compile your own lists. (The real answer for such things being left out is that I couldn’t easily update my book to include them.) And now…

 

Bonus feature! Progress in computational verifications and searches

In no particular order:

  • The simple continued fraction of π has now been computed to the first 15 billion terms by Eric Weisstein, up from 100 million.

 

  • The decimal expansion of π has been computed to 13.3 trillion digits, up from 2.69999999 trillion.

 

  • The search for the perfect cuboid: if one exists, one of its sides must be at least 3 trillion units long, up from 9 billion.

 

  •  Goldbach’s conjecture has been verified for even numbers up to 4 × 1018 by Oliveira e Silva, up from 1018.

 

 

  • The largest known prime, and the 48th known Mersenne prime (up from 47), is 257885161-1, up from 243112609-1.

 

 

  • The longest known Optimal Golomb Ruler is now 27 notches long (up from 26):
    (0, 3, 15, 41, 66, 95, 97, 106, 142, 152, 220, 221, 225, 242, 295, 330, 338, 354, 382, 388, 402, 415, 486, 504, 523, 546, 553)

 

  • The most impressive feat of integer-factorisation using classical computers, is that of the 232-digit number RSA-768:

    1230186684530117755130494958384962720772853569595334792197322452 1517264005072636575187452021997864693899564749427740638459251925 5732630345373154826850791702612214291346167042921431160222124047 9274737794080665351419597459856902143413

    into two 116 digit primes

    3347807169895689878604416984821269081770479498371376856891243138 8982883793878002287614711652531743087737814467999489
    and
    3674604366679959042824463379962795263227915816434308764267603228 3815739666511279233373417143396810270092798736308917.

    (The previous record was the 200 digit semiprime RSA-200.)

 

  • The most impressive feat of integer-factorisation using a quantum computer is that of 56,153=233 × 241. The previous record was 15.

 

  • The Collatz Conjecture has been verified for numbers beyond 2 × 1021. The previous record was 5.76 × 1018. (However, this has happened via a patchwork of distributed computing projects, and I have not been able to establish with any certainty that every number up to the new higher limit has been checked. I encourage someone in this community to organise all the results in a single location.)

Euler’s Partition Theorem

Posted by richardelwes on March 7, 2015
Posted in: Number theory. Leave a comment

euler I’ve recently been revisiting Euler’s Theorem. Not that one. No, not the one on this commemorative stamp either. No, no, not the totient function one. (Good guess though.)

I mean the one about partitions.

What is a partition? It’s simply a way of splitting up a number (a positive integer) into smaller pieces. For instance 2+2+1 is a partition of 5. The question is: how many ways of doing this are there?

Well there’s just one partition of one, namely 1.

There are 2 partitions of two: 2 and 1+1.

There are the 3 partitions of three: 3, 2+1, 1+1+1. (Notice that for these purposes, we count 2+1 and 1+2 as the same partition.)

So far the pattern looks rather easy. But there are the 5 partitions of four:
4, 3+1, 2+2, 2+1+1, 1+1+1+1

And then 7 partitions of five:
5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+, 1+1+1+1+1

Continuing the sequence, we find 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010,…

What’s the pattern now? Finding an explicit formula for the number of partitions of n is seriously hard. Even Euler couldn’t manage it. But he was able to make some progress: he found a generating function.

Next question: what is a generating function? To answer with an example, the generating function for the sequence 1,2,3,4,5,6,\ldots is the power series x+2x^2 +3x^3 +4x^4 +\ldots .

More generally, the generating function for the sequence a_0, a_1, a_2, a_3, \ldots is a series: a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \ldots .

Next question: what’s the point? Often the resulting series can be radically compressed. For instance, a bit of mathematical magic tells us that the series x+2x^2 +3x^3 +4x^4 +\ldots can be rewritten[1] as \frac{1}{(1-x)^2} . Now the whole infinite sequence has been encapsulated by a short, simple algebraic expression. So, if you want to know the nth element of the sequence, you can extract it (if you know how) fairly easily from the generating function.

Well, there is not a lot of mystery in the sequence 1,2,3,4,\ldots – certainly there are no prizes for guessing its nth term. But what of the partition sequence: 1, 2, 3, 5, 7, 11,\ldots? Euler’s great insight was that there is a generating function for this too:
\left( \frac{1}{1 - x} \right) \times \left( \frac{1}{1 - x^2} \right) \times \left(\frac{1}{1 - x^3} \right) \times \left(\frac{1}{1 - x^4} \right) \times \ldots

You can see that this is not as nice as the function above – and it’s certainly not nice enough to close the book on the whole problem of computing partition numbers. All the same, this function is hugely easier to work with than scrabbling about with partitions with bare hands. Euler’s generating function can also be written as
\left(1+x+x^2+x^3+\ldots \right) \times \left( 1+x^2+x^4+ \ldots \right) \times \left(1+x^3+x^6+ \ldots \right) \times \ldots

On first sight, this looks like an algebraic nightmare: you have to open up an infinite number of brackets, each of which contains an infinite series! However, if you want to compute the nth partition number, you only care about the first n brackets, and only the first few terms in each: from the first bracket you care about the first n terms, from the second bracket approximately the first \frac{n}{2}, from the third approximately the first \frac{n}{3}, and so on. In fact the total number of terms you need to worry about is approximately n \ln n, which is a manageable number when n is not too big.

So although Euler didn’t find a formula for partition numbers, his generating function does provide a manageable procedure for computing these numbers. But what of an actual formula?

Further progress came courtesy of a pair of mathematical superstars to rival Euler: Hardy and Ramanujan. The nth one is given approximately by the formula
\frac{1}{4 n \sqrt{3}} e^{\pi \sqrt{\frac{2n}{3}}}.

As n grows bigger and bigger, this number gets proportionally closer to the right answer. But what about an exact formula? That’s something there has been huge progress on in recent years… but perhaps that’s a story for another day.

 

[1] There are issues of convergence here, which I will ignore for now.

Math Frolicking

Posted by richardelwes on March 1, 2015
Posted in: Elwes Elsewhere. Leave a comment

As an ambitious young researcher, you can miss the wood for the trees. When you’re presented with a theorem, your inclination is to dive straight into the proof, and start grappling with the toughest ideas in there. But when you’re addressing a general audience, you have to step back, pause, and ask “What is the point of this theorem? Where did it come from? Why should we care?” You’re forced to adopt a different perspective on the subject, and that’s refreshing.

Read more of my interview over at Math Frolic.

The Grothendieck Song

Posted by richardelwes on January 2, 2015
Posted in: Elwes Elsewhere, Geometry, Music, Nonsense. 2 Comments

There are many songs in the world about love and loss, heartbreak and heart-ache. There are altogether fewer about algebraic geometry in the style of Alexander Grothendieck. Here is my attempt to fill that gap:

 

Disclaimer: I hope none of this needs saying, but just in case of misinterpretation:

1. No views expressed therein are attributable to any organisation to which I am affiliated.

2. Most of the views expressed therein are not attributable to me either, but are a deliberate exaggeration, for comic effect, of a common initial reaction to one’s first meeting with Grothendieck-style algebraic geometry.

3. It is not intended as a serious critique of any mathematician or school of mathematics!

 

Lyrics

When I was a young boy doing maths in class
I thought I knew it all.
Every test that I took, I was sure to pass.
I felt pride, and there never came a fall.

Up at university, I found what life is for:
A world of mathematics, and all mine to explore.
Learning geometry and logic, I was having a ball.
Until I hit a wall…

For I adore Euler and Erdős,
Élie Cartan and Ramanujan
Newton and Noether. But not to sound churlish
There’s one man I cannot understand.

No, I can’t get to grips with Grothendieck,
My palms feel sweaty and my knees go weak.
I’m terrified that never will I master the technique
Of Les Éments de Géométrie Algébrique.

He’s a thoughtful and a thorough theory-builder, sans pareil.
But can anybody help me find the secret, s’il vous plaît
Of this awe-inspiring generality and abstraction?
I have to say it’s driving me to total distraction.

For instance… A Euclidean point is a location in space, and that we can all comprehend.
René Descartes added coordinates for the power and the rigour they lend.
Later came Zariski topology, where a point’s a type of algebraic set
Of dimension nought. Well, that’s not what I thought. But it’s ok. There’s hope for me yet!

But now and contra all prior belief
We hear a point’s a prime ideal
In a locally ringed space, overlaid with a sheaf.
Professor G, is truly this for real?

No, I can’t make head nor tail of Grothendieck
Or Deligne, or Serre, or any of that clique.
I’ll have to learn not to care whenever people speak
Of Les Fondements de la Géométrie Algébrique.

But don’t take me for a geometrical fool.
I can do much more than merely prove the cosine rule.
I’ll calculate exotic spheres in dimension 29
And a variety of varieties, projective and affine.

I’m comfortable with categories (though not if they’re derived)
I’ll tile hyperbolic space in dimension 25
I can compute curvature with the Gauss-Bonnet law
And just love the Leech Lattice in dimension 24.

But algebro-geometric scheming
Leaves me spluttering and screaming.
And in logic too, you may call me absurd
But I wouldn’t know a topos, if trampled by a herd.

I’ve tried Pursuing Stacks but they vanished out of sight,
I’ve fought with étale cohomology with all my might.
And Les Dérivateurs. It’s 2000 pages long.
I reach halfway through line 3, before it all goes badly wrong.

No, I’ll never get to grips with Grothendieck
And I’m frightened that I’m failing as a mathematics geek.
All the same, I can’t deny the lure and the mystique
Of Le Séminaire de Géométrie Algébrique.

– Richard Elwes, 2015

 

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Points and lines

Posted by richardelwes on November 4, 2014
Posted in: Geometry, Maths. Leave a comment
From https://royalsociety.org/awards/sylvester-medal/

Last week, we Leeds mathematicians were treated to a talk by Professor Green Professor Green with the delightfully elementary title of Points and Lines. It reported on joint work of Ben Green and Terence Tao. Here I’m going to talk about the background to their work. As always, any errors are mine.

The idea is that we will be presented with some points, specifically a finite collection of points in the plane. We want to predict the answer to questions of the form: how many lines are there through exactly (n) many of these points? We want to do this while knowing as little as possible about the details of what we’ll be given. Actually, on a first pass, these questions are pretty trivial:

  • There are guaranteed to be infinitely many lines through zero points since when you’re positioning a line, a finite set of points is very easy to miss.
  • Similarly, there will be infinitely many lines which hit exactly one point, since once you’ve decided which to hit, you can easily tweak that line to miss the rest.
  • Must there be any lines which hit exactly two points? Not necessarily. If all the points lie on a single straight line (and if there are more than two of them), then any line either goes through 0, 1, or all of them.
  • Arranging the points around a circle illustrates that there also need not be any lines which hit exactly 3 points. The same argument works for n= 4,5,6…

 

So far, so straightforward: we can predict the answer exactly for (n=0,1) and not at all otherwise. But let’s think again: in the case of (n=3), there’s obviously nothing special about a circle. There are numerous other curves which would do just as well: a parabola, hyperbola, squircle, catenary, … all of these and many others share the property than any straight line hits them at most twice.

However, for the case (n=2), there is something undeniably special about demanding that all our points lie along a single straight line. So the question arises: is this configuration the only obstacle? Must it be the case that either all the points lie along a line, or we will be able to find a line hitting exactly two of them? This problem was first posed, so far as we know, by J.J. Sylvester in the back pages of the Educational Times in 1893, under the unassuming title of mathematical question 11851. (The Royal Society’s Sylvester Medal is depicted above, of which Ben Green is the most recent winner, for another famous result of his jointly with Tao.)

The question was later re-asked by Paul Erdős and then solved in 1940 by Tibor Gallai, who established that the answer was yes: if a set of points do not all lie on a line, then there must be some line through exactly two of them.

A later proof by Leroy Kelly is, as Ben put it, a `classic of mathematics’ and actually induced an audible gasp from the audience. So let’s see it:

KellyProof1 Kelly’s proof (sketch)

We assume that our points don’t all lie along a single line. That means we can definitely find two of them which lie on a line L, along with some other point X lying off L, as shown. In fact, there are likely to be numerous such configurations; we choose the one where the perpendicular distance from X to L is the least possible.

KellyProof2 The claim is then that L contains only those two points. Suppose not. Then there must be two points A & B both lying on L and on the same side of the perpendicular line XL. Now, let M = XA. This line also contains two points (obviously), but the distance from B to M is less than that from X to L, which is a contradiction. QED

Clever, right? Anyway, for reasons unexplained, lines which pass through exactly two of our specified collection of points are known as ordinary lines. The Sylvester-Gallai theorem above therefore says that (so long as our points are not colinear) an ordinary line will always exist. But the next question is: how many must there be? In most cases there will be lots: if you try this for a random scattering of (m) points, then in all likelihood, whenever you draw a line through two points it will miss all the others. Thus there are around (frac{1}{2}m^2) ordinary lines.

But it is not hard to come up with sets with fewer. For instance, take (m-1) points all lying along a line, and add in one extra point (X) off it. Then the only ordinary lines are those which connect (X) to another point, giving (m-1) many.

A cunning construction by Károly Böröczky brings this down even further: a set of (m) points with only (frac{1}{2}m) many ordinary lines.[1]

Now we come to the theorem of Green and Tao. In their paper, they have shown that Böröczky’s examples are as good as it gets. For large enough values of (m), they prove that every non-colinear set of (m) points must have at least (frac{1}{2}m) many ordinary lines.

How large is `large enough’? The bound they establish is in the vicinity (10^{10^{10}}), but the true answer may be as low as 14. For smaller sets, it is possible to do better. The left-hand picture here shows a set of 7 points with only 3 ordinary lines.

And how is it proved? Well, I’m not going to go into details, but the key step is the unexpected (to me) connection of the property `has few ordinary lines’ with the property `can be covered with a single cubic curve’.
 

[1] For the cognoscenti: it achieves this by moving the action to the real projective plane, arranging (m) points symmetrically around the circle, and placing another (m) on the line at infinity at positions corresponding to the directions determined pairs of points from the first set. Then the only ordinary lines are the (m) many tangents to the circle at points. Finally, since no-one mentioned anything about lines at infinity being permitted, the whole set-up needs to be transported back to the plain old Euclidean plane, which can be achieved via a projective transformation. This however will distort the circle into an ellipse. See page 11 of their paper for a picture. )