### Tony Phillips' Take Blog on Math Blogs

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February 2002

Polyhedron Man is the title of Ivars Peterson's piece in the December 22 & 29 2001 Science News about the mathematical sculptor George W. Hart and his work. Hart, who teaches in the Computer Science department at Stony Brook, devotes the bulk of his time ... to his varied polyhedral pursuits,'' including his large and rich website and his production of intricate polyhedrally based sculptures, often using familiar objects (plastic knives, forks and spoons, compact discs, pencils) in unfamiliar ways. Battered Moonlight, shown here, is one of his simpler creations.

 Battered Moonlight by George W. Hart. Paper mache over steel, approximately 21 inches across. If you like, you can think of it as an 11-holed, 6-edged, 1-sided surface, with chiral icosahedral symmetry.'' Image reproduced with permission.

The Differential Equations of Pathogen Virulence. An imperfect vaccine can lead to increased virulence in a pathogen to the point where overall mortality rates are unaffected, or even increase, with the level of vaccination coverage.'' This in a letter to Nature (Imperfect vaccines and the evolution of pathogen virulence, December 13 2001) from an Edinburgh team led by Sylvain Gandon and Margaret Mackinnon. Gandon, Mackinnon and their collaborators drew their conclusions from the long-term behavior of a system of differential equations, which were set up to analyze the long-term effect of vaccines designed to reduce pathogen growth rate and/or toxicity (as opposed to infection-blocking'' vaccines). The equations are nonlinear but simple in form. The population has two classes of hosts: those that are fully susceptible to the pathogen (density of uninfected x and infected y) and those that are partially immune (density of uninfected x' and infected y'). The system is a set of four differential equations in these unknowns.

 dx/dt = A - ax - h(y,y')x + bydy/dt = h(y,y')x - dydx'/dt = A'- a'x'- h'(y,y')x'+ b'y'dy'/dt = h'(y,y')x'- d'y'.
A + A' is the totalflow of uninfected hosts into the population; nonlinearizing factors, the forces of infection"  h(y,y')  and  h'(y,y'), are linear combinations of y andy'.

More about John Nash. The enormous success of the movie (A Beautiful Mind wins Four Golden Globes'' - New York Times, January 20 2002) has elicited memoirs from those who knew him when. S. G. Gwynne, a Princeton undergraduate in 1970, contributed The Genius behind the Tree'' to the Times Op-Ed page for December 26 2001. Apparently Nash had been nicknamed Purple Sneakers'' by the undergrads. Daniel Rockmore, an Princeton mathematics major in the early 1980s (and now a math and computer science professor at Dartmouth), wrote Exploiting a Beautiful Mind'' for the January 25 2002 Chronicle of Higher Education. He remembers Nash as a thin, raincoat-clad, umbrella-carrying specter in the bowels of Fine Hall'' and, expecting a faithful biography, sat through the film with a little shock, and much dismay.'' Rockmore goes on to survey Nash's mathematical works in language appropriate for the Chronicle.  Nash's game-theoretic work places the real world of human interaction in the confines of the ideal and Platonic, and his achievements in geometry were of the same flavor, '' although readers who know little advanced mathematics may wonder why it was such an achievement to show that spaces defined only as solutions to families of polynomial equations ... could actually be described in a Euclidean setting.''
Sylvia Nasar, the author of A Beautiful Mind, was interviewedon the National Public Radio programFreshAir, January 24, 2002. Terry Gross was the interviewer.Topics: The great paradox of Nash's life: he lived in a worldof numbers and had deep faith in rationality, yet his illness made his thinking irrational. Examples of Nash's numerological investigations.Blackboard messages from the Phantom of Fine Hall (witty, very eruditebut entirely bizarre'') The originality and usefulnessof Nash's game-theoretic discoveries. Nash's work atRAND. Did the cold war paranoid fears feed into theparanoia that he later developed? They created the content of hisdelusions, in part, but they didn't cause his paranoia, if you seethe distinction.'' The movie left out any reference toNash's non-standard love-life:I think it would have been really weird if the movie had focused on it.''The movie did not mention John and Alicia's divorce: If she's not a wife, I don't know who of us is.'' Their son's travails with schizophrenia: For me ... over the twoand a half years I was working on the book ...thinking about Nash's illness and the factthat it stretched on for three decades never was quite as sad to me aswhen I met Johnnie. ... That's when the full import of Nash's experiencereally came home to me.''Fear, on the part of some members ofthe Nobelcommittee, that because of his illnessNash's getting the prize would tarnish the prize'' andvery specifically, they were worried aboutwhat he might do during the wonderful Nobelceremony.'' What did he do during the ceremony? He was wonderful.''How the prize has changed Nash: The effect of this acceptanceand recognition and appreciation has dramatically altered him asa person. ... . He's gotten a life back. And just getting to see thisre-emergence, that is continuing, it's been an incredible experience.''The interview (20 minutes) is availableonline.

Dynamic Catastrophe Theory. The September 14 2001 Science has an article by David J. Wales (Universal Chemical Laboratories, Cambridge UK) on a new application of catastrophe theory to the study of the kind of potential energy landscapes'' that occur in complicated energy-minimization problems like protein folding. His principal result in this context is a quantitative connection between the potential energy barrier, the path length, and the lowest vibrational frequencies for a steepest-descent path linking a minimum and a transition state. This result may appear counterintuitive, for one might suppose that these quantities are independent.'' In a commentary piece (Flirting with Catastrophe'') in the same issue of Science, Robert Leary (San Diego Supercomputing Center) explains Wales' result in these terms: He shows that neighboring stable states and the reaction paths that connect them can often be described by universal functional forms dictated by catastrophe theory.'' He mentions that The results are validated with large databases of paths for various potentials, with excellent agreement where the minimum lies in close vicinity of the transition point'' and concludes Wales' application of catastrophe theory, an analytical tool not widely familiar to the scientific community, to energy landscapes is an exciting new development.''

-Tony Phillips
Stony Brook