This month's topics:
Random voltage improves math learning
On May 16, 2013 Science Now, the web arm of Science magazine, posted "Trouble With Math? Maybe You Should Get Your Brain Zapped" by their staff writer Emily Underwood. Underwood refers to an article in Current Biology reporting the effect of transcranial random noise stimulation (TRNS) on learning; particularly on the learning of mathematical-type data. As Underwood presents it: "Now, a new study suggests that a gentle, painless electrical current applied to the brain can boost math performance for up to 6 months. Researchers don't fully understand how it works, however, and there could be side effects." The study is by Roi Cohen Kadosh (Oxford) and seven collaborators. Underwood asked one of the co-authors to describe
Turing instabilities, hallucinations, and the beginnings of symbolic culture
A paper published on July 8, 2013 in Adaptive Behavior ("Turing instabilities in biology, culture and consciousness? On the enactive origins of symbolic material culture", by T. Froese, A. Woodward and T. Ikegami) found a lurid echo in the Daily Mail online, where Victoria Woollaston posted "Were cave-painters on DRUGS? New study claims paintings show prehistoric man was 'high' on psychedelic plants"(7/15/13). The authors' thesis is more complex. At the base is the recognition that Turing instabilities, which are understood today as playing an essential role in the development of biological patterns, e.g. the markings of animal coats, may also manifest themselves in the nervous system. Turing's 1952 paper ("The chemical basis of morphogenesis") states
"It is suggested that a system of chemical substances, ... reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis. Such a system, although it may originally be quite homogeneous, may later develop a pattern or structure due to an instability of the homogeneous equilibrium, which is triggered off by random disturbances. Such reaction-diffusion systems are considered in some detail ..." (Such an instability is now called a Turing instability).
According to Froese and his collaborators, "Turing's idea can be readily transposed to the functioning of the nervous system. For example, action potential propagation along a neuron's axon can be directly described by reaction-diffusion equations ..." It is not too big a leap to surmise that instabilities in those equations can produce neural patterns with visual consequences analogous to the stripes, dots and dapples manifested by the Turing instabilities in morphogenesis. The authors point to similarities between the earliest patterns known in human decoration (which go back perhaps 100,000 years) and patterns that seem to occur universally in visual hallucinations: lattices, honeycombs and spirals, and suggest a connection.
Mathematical models of cancer therapy
"Calculated treatment," a News and Views piece in Nature (by N. Komarova and C. R. Bolland, July 18, 2013) reviews a study published on June 25 in eLIFE: "Evolutionary dynamics of cancer in response to targeted combination therapy," by Ivana Bozic (Harvard) and 14 collaborators. Bozic et al. used a mathematical model to prove that combination therapy (using two or more anti-cancer drugs simultaneously) has better outcomes than a sequential approach, where one drug is used until resistance appears, and then another is substituted. Significantly, the analysis depended on the recent exact solution of the multitype branching process involved in their model. That solution, due to Tibor Antal (one of Bozic's co-authors) and P. L. Krapivsky, was published in the Journal of Statistical Mechanics: Theory and Experiment on August 30, 2011 ("Exact solution of a two-type branching process: models of tumor progression").
"The Rosetta Stone of Mathematics"
is Edward Frenkel's characterization of André Weil's intuition of the profound connection between topology and number theory. Frenkel explains it in a guest blog on the Scientific American website. The occasion for the blog (May 21, 2013) was the awarding of this year's Abel Prize to Pierre Deligne, one of whose most important achievements was the proof of the last of the conjectures in which Weil sketched out his view of that connection. Frenkel describes some of the elements that Weil considered: "given an algebraic equation, such as $x^2+y^2=1$, we can look for its solutions in different domains: in the familiar numerical systems, such as real or complex numbers, or in less familiar ones, like natural numbers modulo N. ... the same equation has many avatars, just like Vishnu .... The avatars of algebraic equations in complex numbers give us geometric shapes like the sphere or the surface of a donut; solutions in natural numbers modulo N give us other, more elusive, avatars." Weil used this observation to "to come up with what became known as the Weil conjectures, organizing the solutions modulo N in a way that made them look similar to geometric shapes." Frenkel goes on to bemoan the lack of general appreciation for the magnitude of this discovery, which "did for mathematics what quantum theory and Einstein's relativity did for physics, and what the discovery of DNA did for biology." As he reminds us, mathematics is still a blind spot in our culture, despite its growing economic importance. This should not be. "There is nothing in this world that is so deep and exquisite and yet so readily available to all."
PBS tackles: "Does math exist?"
PBS' Idea Channel has been running a series on the top ten unanswered scientific questions of all time; one of them turns out to be "Is Math a Feature of the Universe or a Feature of Human Creation?" An 8-minute segment on the topic, narrated by the Idea Channel host, Mike Rugnetta, was posted on June 3, 2013. The style is breathless; in fact most pauses between sentences have been edited out. Images, often posted for less than a second, flash on and off beside Rugnetta's head to illustrate the words he's using, or to provide a droll commentary (including, but not limited to, LOL cats). In short, extremely up-to-date. The series is clearly aimed at bringing an intellectually sound discussion of ideas to the generation raised on Sesame Street, Monty Python and the Electric Company. With regard to whether mathematics exists or not, the discussion is really more philosophical than mathematical. It might have been enriched by a discussion of some of what we consider intrinsic phenomena, prime numbers for example, instead of focussing as it does on $5+5=10$.
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