What's Happening in Mathematics?
An occasionally updated series of links to mathematical news — new applications,
new discoveries, problems, personalities, prizes...
new discoveries, problems, personalities, prizes...
The Kepler Conjecture states that the closest spherepacking in space is the face centred cubic lattice — essentially, how greengrocers stack oranges, extended to infinity. The 1998 proof by Thomas Hales involved extensive computer calculations, and the referees were unable to verify them. Now a team headed by Hales has published a proof in a strict logical form that can be, and has been, verified by computer.

Maryam Mirzakhani, the first woman to be awarded the prestigious Fields Medal for mathematics, has died aged 40. She made important breakthroughs in dynamics and the geometry of Riemann surfaces and their moduli spaces.

Liana Manukyan and coworkers show that the patterns formed by the scales of ocellated lizards are a very close match to a model by Turing reactiondiffusion equations, discretized to become a continuousstate cellular automation model on a hexagonal (or approximately so) lattice. The model can also be viewed as a coupled cell ODE on a network.

The Norwegian Academy of Science and Letters has awarded the Abel Prize for 2017 to Yves Meyer of the École normale supérieure ParisSaclay, France “for his pivotal role in the development of the mathematical theory of wavelets”. Wavelet analysis has been applied to data compression, noise reduction, medical imaging, digital cinema, the Hubble space telescope, and LIGO's detection of gravitational waves caused by two colliding black holes.

In 2014 Thomas Royen, a retired German statistician, was brushing his teeth when he realised how to prove the longsought Gaussian Correlation Inequality, conjectured in 1955. For a variety of reasons his proof went unnoticed until a few months ago, but it has now received proper recognition. The inequality states that if μ is ndimensional Gaussian measure, then for all convex sets E , F in ndimensional space that are symmetric about the origin, the inequality
μ(E∩F) ≥ μ(E) μ(F) holds. 
Plamen Stefanov, Gunther Uhlmann, and Andras Vasy have proved the boundaryrigidity conjecture for Riemannian manifolds — knowing the distance along geodesics between any two points on the boundary determines the metric (under suitable hypotheses). The 2dimensional case was proved in 2005; the new proof is general. There is an important application to seismology: the internal structure of the Earth can be inferred from travel times of seismic waves between any two points on the surface.

London's Science Museum has opened its new Winton Gallery, dedicated to mathematics and designed by architect Zaha Hadid.

Hydrant in Berlin

Images of numbers from all over Europe... and tomorrow, the world! Numbers on doors, numbers on walls... Numbers! Is your home town represented? If not, take a photo and send one in.

In 2012 the Japanese mathematician Shinichi Mochizuki published a 500page proof, online, of the ABC conjecture. This is one of the most important unsolved problems in number theory, and a correct proof would be revolutionary. Optimism is now growing that the proof is right. The conjecture relates the prime factors of three numbers a, b, c such that a + b = c. This looks very simple, but the consequences of the relationship are deep and powerful.

Marijn J. H. Heule, Oliver Kullmann, and Victor W. Marek have announced that it is impossible to colour the positive integers with two colours so that every Pythagorean triple uses both colours. They show this can be done for integers up to 7,824, but fails when 7,825 is included. The proof, intelligent casebycase checking, occupies 200 terabytes — a record.

John Cremona at the University of Warwick is the lead UK researcher on a 12nation database of mathematical objects that has been deployed to protect bank accounts and solve problems in physics and prime number theory. The team has catalogued over a billion mathematical items including elliptic curves, modular forms, Lfunctions, and nontrivial zeros, in the research initiative LFunctions and Modular Forms Database Project at: http://www.lmfdb.org/

How Alexander Huth and colleagues mapped the brain's semantic system — 'where the words are stored' — by using principal component analysis to extract from data 985 semantic dimensions, four of which explain most of the variance. Then the Bayesian algorithm PrAGMATiC is used to create a model of how the areas representing words tile the cortex. A case study in the maths of Big Data applied to neuroscience.

A new technique for Big Data. If the timing of ‘snapshots’ of a system is uncertain, it 's hard to extra dynamical information, and the sequence of events can get shuffled. To avoid this problem, R. Fung and coworkers use a battery of mathematical techniques: Riemannian measures, the LaplaceBeltrami operator, singular value decomposition, and phase space reconstruction methods from Chaos Theory. They apply their method to find new vibrational wavepackets in the Coulomb explosion of nitrogen molecules.

It's often claimed that mathematicians do their best when they're young, and lose their originality as they age. This article argues, with evidence, that this belief is largely mythical.

Derek Moulton's team at Oxford University has constructed a 20equation mathematical model of how a chameleon's tongue unrolls at high speed to catch insects. Newton's second law of motion is the most important ingredient. Nonlinear elasticity theory is another key feature. Applications could include the design of soft, elastic materials for robots, but the main motivation was curiosity.

Mathematician Eli Shlizerman has developed a model of the neural wiring of monarch butterflies, explaining how they carry out their annual migration to Mexico. The team recorded neurons in the butterflies' antennae and eyes, and showed that the input cues depend entirely on the Sun. The model shows how these cues are processed to keep them on track.

For spherepackings, 8 and 24 dimensions are known to be special. The kissing number is known in both cases, for example. Now the analogue of the Kepler problem has been solved.

Building on Viazovska's work in 8 dimensions, the analogous theorem has been proved in 24 dimensions by Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, Maryna Viazovska. The Leech lattice is the densest packing of congruent spheres in twentyfour dimensions, and also the unique optimal periodic packing. Density is π^(12)/12!

Maryna Viazovska has announced a proof that the largest packing density for identical spheres in eightdimensional space is π^4/385,
which is approximately 25%. This is the 8dimensional analogue of the Kepler Problem in 3 dimensions. The proof, only 23 pages long, shows that the previously known E8 lattice packing is densest. Experts are now checking. 
The Norwegian Academy of Sciences and Letters has awarded the 2016 Abel Prize to Andrew Wiles “for his stunning proof of Fermat’s last theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in number theory”. The prize is worth $700,000.

Colin Hegarty from Preston Manor School in Wembley was among the 10 teachers reaching the final of a competition for the world's most exceptional teacher. The Global Teacher Prize was created by the Varkey Foundation. Its aim is to bring greater public recognition to the importance of teachers. The winner, announced in Dubai in March 2016, was Hanan al Hroub.

Perrin Schiebel at Georgia Tech has observed how snakes move in unprecedented detail. They contort their bodies into "a very specific, stereotyped waveform", which travels along the length of the snake. Mathematics, robotics, biology, and experiment all combine to provide new insights into a longstanding biological puzzle.

Google's Deep Mind AlphaGo program beats Go world champion Lee SeDol 41, winning the fivegame series. Although the contest is often described as 'algorithm versus intuition', AlphaGo's algorithm is a generalised learning 'deep neural network', and human intuition works in a similar, though probably less structured, way.

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How can the integers Z be recognised among the rationals Q? Sounds trivial: 1 is the unique multiplicative identity and the ring it generates is Z. But can Z be defined by a universal firstorder formula? Jochen Koenigsmann proves that Q\Z is diophantine: there is a polynomial g in several variables that is nonzero precisely on Z. He defines Z using a formula with only one universal quantifier.
