## "Strange topology" in physics

"The strange topology that is reshaping physics" is the title of a News Feature piece by Davide Castelvecchi in Nature (July 19, 2017). The sub-head: "Topological effects might be hiding inside perfectly ordinary materials, waiting to reveal bizarre new particles or bolster quantum computing." The occasion for Castelvecchi's piece is "Topological Quantum Chemistry", an article in the same issue by Andrei Bernevig (Princeton) and seven collaborators: "The researchers created an atlas of topological matter by looking at all 230 different symmetries that can exist in a material's crystal structure. Then they systematically predicted which of these symmetries could, in principle, accommodate topological states, without having to first calculate all their energy levels. They think that between 10% and 30% of all materials could display topological effects, potentially amounting to tens of thousands of compounds."

"Topological states" and "topological effects" refer to a class of physical phenomena, typified by the Quantum Hall Effect (QHE), whose mathematical description involves algebraic topology. The QHE was discovered in 1980 but, as Castelvecchi tells us, "only in the past few years have researchers begun to realize that [these effects] could be much more prevalent and bizarre than anyone expected." Today "it seems increasingly rare to see a paper on solid-state physics that doesn't have the word topology in the title."

Explaining these effects in simple terms has proved difficult, because the topology in question is not the doughnut/coffee-cup variety, but rather the topology involved in the classification of fibre bundles: these objects are intrinsically high-dimensional and resist direct visualization. Castelvecchi's solution is to work with vector fields (cross-sections of tangent bundles of surfaces) which are much more familiar and can be represented graphically.

• "Just as combing a hairy ball invariably produces 'cowlicks' at each pole, a sphere will always host some swirls, or vortices, in the patterns. But that is not always the case for a doughnut."
• "Each vortex has a 'winding number' that capture how many times it loops around a point."
• "The number of vortices can change, but the sum of their winding numbers cannot." Castelvecchi's article has a nice animated GIF illustrating how a $+1$ and a $-1$ vortex can cancel, or can be born together.
• "This invariance underpins the behaviour of topological insulators"

Here's where Castelvecchi's approach leads him into trouble. "This effect [the QHE] sees the electrical resistance in a single-atom-thick layer of a crystal jump in discrete steps when the material is placed in magnetic fields of different intensities." "... in this case, the underlying shape is ... the surface of a doughnut. As the magnetic field ramps up and down, vortices can form and disappear on the surface ... ." The problem is, as he has mentioned, that the sum of the winding numbers of the vortices on a surface cannot change (it equals the Euler characteristic of the surface). "Vortices" are a phenomenon in the tangent bundle, which is just one possible bundle over a surface; the changes in topology caused by large variations in the magnetic field are changes from one bundle to another. With small variations the bundle stays the same, leading to the quantization in the QHE.

## The wrong automaton

The distinguished mathematician John Horton Conway, the inventor among many other things of the finite automaton Life, is a graduate of Cambridge University. So when Britain's National Rail planned a new station building at Cambridge North it must have seemed like a nice idea when the architects announced they would use patterns from the game to decorate the exterior.

Unfortunately the architects chose output from Stephen Wolfram's automaton Rule 30 instead, as reported by Corinne Purtill in Quantz (June 12, 2017): "A UK train station's tribute to a famous mathematician got everything right except his math," apparently because the output from Wolfram's automaton was "more aesthetically pleasing." Quantz contacted Conway, now Professor Emeritus at Princeton: "That's not mine," Conway said of the pattern. "I have had an influence on Cambridge, but not apparently on the new railway station." According to Purtill, Wolfram attended Oxford.

## Millennium Prizes at the box office

From Frank Sheck's review of the indie film "Gifted," on Hollywoodreporter.com (March 30, 2017): "Chris Evans stars in Marc Webb's comedy-drama about a man fighting to maintain custody of his 7-year-old niece. ...  Frank shares a modest home with his 7-year-old niece Mary, the daughter of his sister, who committed suicide when Mary was just six months old." It turns out that his sister was a gifted mathematician, and that Mary herself is a math prodigy. Frank wants Mary to have "a normal little girl's life unlike her mother, who was driven by her and Frank's wealthy mother Evelyn to cultivate her math skills whatever the emotional cost." Evelyn comes into the picture and "takes Frank to court to fight for custody of the little girl she recognizes as another prodigy."

With Evelyn, quite a bit of mathematics enters the movie: she takes Mary to MIT where a professor is stewing over an identity he can't work out: Mary, at the blackboard, stands on tiptoe to put in a crucial minus sign and carries through iterated improper integrals, etc. to get the desired answer: $\sqrt{2\pi}|\sigma|$; she adds "Q.E.D." with a flourish. Then Evelyn takes her to a majestic Millenium Problems Hall of Fame and says, approximately, "That's Grigory Perelman. He proved the Poincaré conjecture and won the Fields Medal. He'll be famous forever. And here is the Navier-Stokes problem. Your mother was close to a solution when she died. You have it in you to be famous forever, too."

Who will win? Hint: the plot resolution involves a missing one-eyed cat. Sheck's evaluation: "... despite its recycled tropes, the comedy-drama manages to be both funny and moving even if its emotional manipulations are fully apparent."

## Australia vs. mathematics

"The laws of mathematics are very commendable, but the only law that applies in Australia is the law of Australia," according to Australian Prime Minister Malcolm Turnbull (as reported by James Titcomb in The Telegraph, July 14, 2017). As Titcomb explains the context, "Messaging apps like WhatsApp and iMessage would be forced to hand over the contents of encrypted messages under laws being proposed by the Australian government." Turnbull's comment comes as a response to the observation that "tech companies could only give authorities access to the messages by weakening the cryptographic protection that applies to them, which would risk making them available to hackers and rogue states."

## Math and the solar eclipse

Stephen Wolfram's blog posting "When Exactly Will the Eclipse Happen? A Multimillenium Tale of Computation," dated August 15, 2017, traces the mathematics behind solar eclipse prediction from the Babylonians to Chaos Theory with everything in between. The wonderful graphics include an animation of the Antikythera Mechanism.

## Plimpton 322 is back in the news

"Plimpton 322 is Babylonian exact sexagesimal trigonometry," by Daniel Mansfield and N. J. Wildberger, will appear in Historia Mathematica; it can be read online. The sensational press release from the University of New South Wales: "Mathematical mystery of ancient clay tablet solved. UNSW scientists have discovered the purpose of a famous 3700-year-old Babylonian clay tablet, revealing it is the world's oldest and most accurate trigonometric table." (Deborah Smith, August 25, 2017) and an accompanying promotional video were picked up enthusiastically by media around the world. For example:

Some reporters checked with the experts.
• This ancient Babylonian tablet may contain the first evidence of trigonometry (Ron Cowen, Science, August 24). "However, other experts on the clay tablet, known as Plimpton 322 (P322), say the new work is speculative at best." Cowen contacted Jöran Friberg (Chalmers University of Technology, Sweden), Mathieu Ossendrijver (Humbolt University, Berlin) and Christine Proust (CNRS, Paris).
• Ancient Tablet May Show Earliest Use of This Advanced Math (Sarah Gibbens, National Geographic website, August 24.) "A new study claims the tablet could be one of the oldest contributions to the the study of trigonometry, but some remain skeptical." Gibbens contacted Duncan Melville (St. Lawrence University) and Donald Allen (Texas A&M).
• Hints of Trigonometry on a 3,700-Year-Old Babylonian Tablet (Kenneth Chang, New York Times, August 29). Chang contacted Alexander Jones (ISAW, New York) and Eleanor Robson (University College, London), and walks us through a sample calculation -the height of a ziggurat- from the text. He seems to be the only reporter who actually read the paper.
• The Mesopotamians, trigonometry champions? (Stéfane Foucart, Le Monde, August 29). Foucart consulted Christine Proust, and quotes her at length. "I personally think one must be prudent but that this idea should be considered. For decades, mathematicians and information theorists have been proposing hypotheses about this tablet and while some were off the wall, others turned out to be well founded." [My translations -TP]
A more negative take comes from Evelyn Lamb on her Scientific American blog: "Don't Fall for Babylonian Trigonometry Hype." She starts: "I'd like to help separate fact from speculation and outright nonsense when it comes to this new paper." She zeroes in on Mansfield's "claims that this table is 'superior in some ways to modern trigonometry' and the 'only completely accurate trigonometry table.'" At the heart of them she finds an agenda: "A little digging shows that Wildberger has a pet idea called 'rational trigonometry.' ... It's hard not to see their work on Plimpton 322 as motivated by a desire to legitimize an approach that has almost no traction in the mathematical community."

Tony Phillips
Stony Brook University
tony at math.sunysb.edu