### Tony Phillips' Take Blog on Math Blogs

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# This month's topics:

His life was the subject of full-scale obituaries in Science (June 19, 2015) and in Nature (June 25, 2015). Both focused on his contribution to economics, but each of the authors contributed personal details of his own.

• In Science, Martin Shubik, a lifelong friend of Nash's, starts back at Princeton in 1949 when he, Lloyd Shapely and Nash shared a suite in the Graduate College. "John, Lloyd, and I were competitive in our research, which John enhanced with practical jokes such as removing the light bulb from a fixture in our joint bathroom and filling it with water, poised to drench his roommates." There is also the account of a game of So Long Sucker (that John had helped invent) played at tea time in Fine Hall, at the end of which John reportedly said to John McCarthy, in essence, "I do not understand why you are mad at me; you could do the backward induction to have seen that it was completely rational for me to double-cross you, and it was not personal."

• Martin Nowak, who wrote for Nature, did not know Nash nearly as well. The anecdote he contributes, set at the Institute for Advanced Study, may say more about the place and the profession than the man. "One summer's day, when the usual sitting arrangements for lunch were disrupted by the closure of the main kitchen, I noticed John, the physicist Edward Witten and Andrew Wiles, the British mathematician who proved Fermat's last theorem, sitting down together at a small table. I wondered which of them would start the conversation. None of them did. I seem to remember that they ate their meal in silence."

## Möbius strips of light

"Observation of optical polarization Möbius strips" appeared in Science, February 27, 2015. The authors, an international team led by Thomas Bauer (Erlangen), report in their Abstract that they construct a q-plate, "a liquid crystal device that modifies the polarization of light in a space-variant manner," tightly focus the emerging beam of light, and then, "using a recently developed method for the three-dimensional nanotomography of optical vector fields, ... fully reconstruct the light polarization structure in the focal region." The reconstruction reveals that this structure contains Möbius strips. As the authors remark: "Despite being easily realized artificially, the spontaneous emergence of [Möbius strips] in nature is exceedingly rare."

Experimentally observed polarization topologies in the focal plane of the beams produced by two different $q$-plates. The image shows how the major axis of the polarization ellipse varies around a circle of 150 nm radius centered on the beam axis. With $q=-1/2$ the axis sweeps out a Möbius strip with three half-twists; with $q=-3/2$ there are five half-twists. Image courtesy of Ebrahim Karimi, Department of Physics, University of Ottawa.

## The mathematics of cortical folding

A report in Science for July 3, 2015 has the title: "Cortical folding scales universally with surface area and thickness, not number of neurons." The authors, Bruno Mota and Suzana Herculano-Houzel of UFRJ in Rio de Janeiro, begin: "Larger brains tend to have more folded cortices, but what makes the cortex fold has remained unknown. We show that the degree of cortical folding scales uniformly ... as a function of the product of cortical surface area and the square root of cortical thickness." They add: "This model also explains the scaling of the folding index of crumpled paper balls."

These two sheets, crumpled with the same amount of pressure, started with equal surface area, but the exposed area is larger on the thicker one. Mota and Herculano-Houzel determine that the total area $A_{\rm T}$ and the exposed area $A_{\rm E}$ are related to the thickness $T$ by $T^{1/2}A_{\rm T} = kA_{\rm E}^{1.1055\pm 0.022}$ "as a single, universal power function across all paper balls of different surface areas and thicknesses."

When the authors examined the problem of cortex folding, i.e. the relation between the total surface area $A_{\rm G}$ of the cortex and its exposed surface area $A_{\rm E}$ they discovered an analogous relation: "$A_{\rm E}$ scales across all lissencephalic [smooth-brained] and gyrencephalic [convoluted-brained] mammals (and even across species usually regarded as outliers such as the manatee and cetaceans [porpoises, whales]) as a single power law of $T^{1/2}A_{\rm G}$." They continue: "The finding that cortical folding scales universally across clades, species, individuals, and parts of the same cortex implies that the single mechanism based on the physics of minimization of effective free energy of a growing surface subject to inhomogeneous bulk stresses applies across cortical development and evolution. This is in stark contrast to previous conclusions that different mechanisms regulated folding at different levels."

## Lives of mathematicians, reviewed in Nature

Amir Alexander reviewed two personal views of a mathematician's life for Nature, March 5, 2015. The mathematicians are Cédric Villani (Birth of a Theorem, A Mathematical Adventure, Bodley Head/ Faber and Faber, 2015) and Michael Harris (Mathematics Without Apologies: Portrait of a Problematic Vocation, Princeton Univ. Press, 2015). Alexander characterizes Villani's book as "the personal record of a single-minded quest." This was Villani's search for a full mathematical account of Landau damping, which ultimately led him to the Fields Medal in 2010. Harris' book, by contrast, is "a kaleidoscope of philosophical, sociological, historical and literary perspectives on what mathematicians do, and why. Do they pursue their work for the public good?"--a pose. "Is it the absolute truth of mathematical demonstrations that drives the field?"--a conceit of philosophers. "What about the lauded beauty of mathematics?"--"Perhaps, Harris concedes, but when mathematicians talk about beauty, what they mean is pleasure."

"Perhaps more than any other field, mathematics pulls the practitioner away from the 'normal' world of things and people into a strange alternate universe, in which we catch glimpses of beauty and coherence, but spend most of our time groping in the dark. ... Villani offers one way of straddling that divide; ... Harris presents a very different one. Together, they provide an unmatched perspective on life in this 'problematic vocation' by two of its leading practitioners."

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