What's Happening in Mathematics?
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An occasionally updated series of links to mathematical news — new applications,
new discoveries, problems, personalities, prizes...
new discoveries, problems, personalities, prizes...
Was the moon landing fake? David Grimes has modelled how long a big conspiracy can be kept secret, by setting up an equation for the probability of a conspiracy being deliberately uncovered by a whistleblower or accidentally revealed by a bungler. The more people are in the know, the sooner word gets out. A hoax moon landing would have been revealed in 3 years 8 months, climate change fraud in 3 years 9 months, and a vaccination conspiracy in 3 years 2 months. Nothing of the kind has happened.

David Silver and coworkers have trained a computer to play Go, beating a human professional — the first time this has been achieved. The method is published in Nature. They use deep neural networks, trained on a library of games and by playing against itself, and Monte Carlo tree search. This approach is not tailored specifically for Go, so may be applicable to a wide variety of hard AI problems.

Massive primes have no current practical value, but searching for them does. GIMPS's prime95 software and members of a German computing community uncovered a flaw in Intel's latest Skylake CPUs. 
The Great Internet Mersenne Prime Search has found a new Mersenne prime, 2^74,207,2811. It has 22,338,618 decimal digits and is the largest known prime. (Euclid proved there is no largest prime: there are infinitely many of them.) Discovered by Curtis Cooper at the University of Central Missouri, it is the 49th Mersenne prime.
Scroll through its digits here. 
The Science of Beauty
Reports, plus videos of some talks, from a conference at the Royal Society of Edinburgh in November 2015. Speakers were Sir Michael Atiyah, Sir David Attenborough, Angela Breitenbach, Nicky Clayton, Clive Wilkins, Robbert Dijkgraaf, Cinzia Di Dio, and Beatrice de Gelder. 
Semir Zeki, John Romaya, Dionigi Benincasa and Michael Atiyah have used fMRI scans to investigate whether a sense of mathematical beauty correlates with activity in the part of the brain that handles more conventional emotional responses. It does—and the results show that Euler's formula relating e and π really is perceived as unusually beautiful.

How researchers at MIT joined forces with mathematician Thomas Brun to control 'ropes' of liquid glass. The machine can be used to construct complex threedimensional objects. Wobbles in the glass rope, previously considered to be a fault, can be controlled precisely using a dynamic model of flowing glass.

Toby Cubitt, David PerezGarcia, and Michael Wolf have proved that the existence, or not, of a spectral gap for a quantum system is algorithmically undecidable. There is a spectral gap if the energy levels of all excited states do not lie arbitrarily near the energy of the ground state. That is, a first excited state, one having the lowest energy, exists.
Their proof links the existence or otherwise of a spectral gap to the halting problem for Turing machines, which is known to be undecidable. In effect, this problem asks for an algorithm which, given a computer 
program, can always decide whether the program eventually stops.
Each program can be encoded into the Hamiltonian of a suitable quantum spin system on the sites of a 2dimensional lattice, using a construction based on a special tiling of the plane, the Robinson tiling. This tiling is aperiodic: unlike wallpaper, it does not repeat the same pattern indefinitely. With some extra clever tweaks, the 
quantum system has a spectral gap if and only if the program halts. Therefore the existence of a gap is undecidable. Indeed, given any axiomatisation of mathematics, there exists a quantum system for which the existence of a gap is independent of the axioms—neither provable nor disprovable.
This result does not rule out proofs or disproofs for specific quantum systems, for example the mass gap hypothesis for the YangMills equation, which is one of the Clay Millennium Prize Problems. But it does put the whole issue in a radically new light. 
Amol Aggarwal, PhD student at Harvard, wins the 2016 Frank and Brennie Morgan Prize for Outstanding Research in Mathematics by an undergraduate. He is honoured for his outstanding research in combinatorics: four published research papers while still an undergraduate, all of 'postdoc' quality.

Constantine M. Dafermos will receive the 2016 Norbert Wiener Prize in Applied Mathematics. Presented by the AMS and SIAM for 'foundational work in partial differential equations and continuum physics.' His main work is on conservation laws, which arise in continuum mechanics, gas dynamics, and elasticity.

Potentially a major advance, and a surprising application of the finite simple group classification theorem

László Babai has announced a proof that there is an efficient algorithm to determine whether two graphs are isomorphic. Technically, it runs in quasipolynomialtime (2^polylog n). Remarkably, the method uses the classification theorem for finite simple groups. If the proof holds up, it will be a very significant advance on this basic problem in combinatorics and computer science.

Marc Lackenby's proof that any diagram of the unknot with c crossings may be reduced to the trivial diagram using at most (236c)^11 Reidemeister moves now published in the Annals of Mathematics. Originally posted to the arXiv in 2013, revised 2014. Recognising the unknot is class NP. This bound does not prove it lies in P, but is still a major advance in knot theory.

An original application of symmetry and functional analysis, with significant medical potential

Two powerful new methods by Liebi et al and Schaff et al to obtain unprecedentedly highresolution images of bones and teeth in three dimensions. Methods combine 'computed tomography, smallangle xray scattering, and serious computing power'. Actually, the key ideas are mathematical: rotational symmetry of collagen fibrils in bone, used to reduce the dimension of the computation to make it feasible, and expansion in spherical harmonics.

Christoph Koutschan, Manuel Kauers, and Doron Zeilberger will receive the 2016 AMS Robbins Prize for their proof of George Andrews's and David Robbins's qTSPP conjecture on plane partitions. Both the conjecture and its proof were derived from 'experimental mathematics', that is, computer experiments informed by human mathematical insight.

Fernando Codá Marques (Princeton) and André Neves (ICL) will receive the 2016 AMS Oswald Veblen Prize in Geometry, for work on variational problems in differential geometry including the proof of the Willmore conjecture. This states that a particular shape of torus, the Clifford torus, minimizes bending energy. The proof uses Almgren–Pitts minmax theory for minimal surfaces.

or
What's Not Happening in Mathematics 
Naive media reports and sloppy journalism spread rumours of a solution to the Riemann Hypothesis and the award of $1M. The Clay Prize rules (which the journalists concerned didn't bother to check) require publication and widespread acceptance after two years, neither of which has happened. A proof has been claimed, but the circumstances reported reveal there is no substance to the claim.

Matteo Massironi and others have used geometry and statistics to show that the 'rubber duck' comet 67P/ChuryumovGerasimenko, observed by ESA's Rosetta and Philae, is made from two rounded bodies that gently collided. The method is to fit planes to 103 geological features indicating ice strata. The geometry of these planes distinguishes theories of the comet's formation.
