Math in the Media 0304
**March 2004** **Google salutes Julia.** February 3 is the birthday of Gaston Julia (1893-1978), one of the fathers of complex dynamics. The web search engine Google celebrated by posting a special Holiday Logo. The standard logo is embellished with fractal trimmings; the formula `z`_{n+1} = z_{n}^{2} + e appears in the background. A look into hyperbolic dodecahedral space. *Nature* used this image, reproduced here courtesy of Charles Gunn & Stuart Levy, © Geometry Center, University of Minnesota, to show one of the eight basic geometries entering into Thurston's geometrization conjecture. | Grisha at the blackboard, during one one of his lectures last year. Photo Celene Chang/The Daily Princetonian | **Perelman in ***Nature*. The January 29 2004 issue contains a piece by Emily Singer entitled "The reluctant celebrity," about Gregory Perelman and his attack on the Poincaré conjecture. Singer gives a sketch of the problem, including a correct intuitive picture of the 3-sphere. Unfortunately one might get the impression that Poincaré was not able to prove that the 3-sphere is simply connected, but let's not quibble. The roles of Thurston and Hamilton in beginning and continuing work on the Geometrization Conjecture are well described, as is Hamilton's Ricci flow program ("a systematic procedure that smooths an object's surface into a simpler ... shape by spreading its curvature") and the singularities that obstructed it ("Some parts of the surface may transform faster than others, resulting in a "lumpy' shape"). There are nice quotes from mathematicians who knew Perelman before he embarked on his eight-year quest to iron out Hamilton's singularities. Jeff Cheeger: "He was already considered extremely brilliant; this was apparent in conversation and on the basis of his work." But the main focus of the article is the reluctance mentioned in the title. That Perelman does not want to bask in the limelight or accept one of the opulent offers dangled before him by american universities is apparently almost as unfathomable as the mental processes that led to his discoveries. | **Groupoids and the leech heart.** The heart of the leech *Hirudo medicinalis* beats to a rythm of its own. There are two disjoint tubes, running most of the length of the creature's body, each partitioned into 16 connecting chambers. In one of the tubes, all the chambers contract synchronously; in the other the contractions are staggered so as to produce traveling waves. Every 50 beats or so (about every 25 minutes), the two tubes trade roles. "This curious pattern is coordinated by a small nework of neurons at the rear of the animal," we learn from Ian Stewart in a piece ("Networking Opportunity") he contributed to the February 12 2004 *Nature*. The problem in *network dynamics* is to understand "how the network architecture produces the distinct coordinated states and switches between them." And one of the mathematical tools that has proved useful in problems like this is the theory of groupoids, "the ideal tool for describing symmetries that apply only to parts of systems." For example, network **c** in the figure is completely symmetrical, and is consequently acted on by the group of cyclic permutations of the nodes. Network **b** is not symmetrical. But if we examine the "input sets -- all nodes that emit an arrow pointing to a given node" then there is a symmetry: each node has a one-element input set, and the nodes can be partitioned into equivalence classes. For example **1** is equivalent to **4**, since they both have a single input from **3**. Then **2** and **5** are equivalent, since they each have an unput from the {**1**,**4**} set. Likewise **3** and **6** are equivalent; finally **7** must fall in the equivalence class of **1** and **4**. This diagram is adapted from Stewart's article. Network **a** is unsymmetrical: node **1** is functionally different from the others. The modified network **b** is still geometrically unsymmetrical, but admits a projection onto a *quotient network* **c** with three symmetrical states. | This partition into equivalence classes can be called a *synchrony*; in fact the equations of network **b** admit a solution in which equivalent nodes are synchronized. Groupoids come into the picture, Stewart tells us, when we try to work out "all robust patterns of synchrony in a network." The synchrony in network **b** gives rise to a traveling wave of excitations, propagating down the tail of the network. If the network were modified again so that the feedback arrow led from node **1** to itself, then all the nodes would be in a single equivalence class, and the solution to the network equation would have all nodes acting synchronously. "Qualitatively these two patterns resemble those observed in the leech heartbeat ..." **Love Model Equations.** The AAAS annual meeting was in Seattle last month, and the February 13 *Seattle Times* reported on some of proceedings. A local team of psychologists and applied mathematicians presented no less than a "mathematical formula for marital bliss." Unfortunately this formula, derived by John Gottman, James Murray, Kristin Swanson and their collaborators, is not an algorithm for achieving bliss. Rather it is a mathematical model of a relationship, based on the analysis of how a couple interacts when arguing, that can predict "with 94 percent accuracy which marriages will last and which will end in divorce." The model is a set of "coupled" first-order ordinary differential equations. In LoveModelEquations-2.pdf (available from the online *Seattle Times* article) Swanson spells them out: Rate of change of husband's score | | Husband's emotional inertia | Deviation from husband's uninfluenced steady state | | Influence of wife on husband's score | *dx* --- dt | = | *q*_{1} | (*x*_{0} - *x*) | + | *I*_{1}(*y*) | Rate of change of wife's score | | Wife's emotional inertia | Deviation from wife's uninfluenced steady state | | Influence of husband on wife's score | *dy* --- dt | = | *q*_{2} | (*y*_{0} - *y*) | + | *I*_{2}(*x*) | Here *I*_{1} and *I*_{2} are piecewise linear functions (two different positive slopes, changing at 0) which encode the couple's argument-interaction behavior. Geometrically speaking, the health of the relationship can be read off from the convexity of *I*_{1} and *I*_{2}. Both close to straight lines gives a "validating style of interaction." Both are very convex downward in conflict-avoiding couples, very convex upward in volatile couples. We are not told the prognosis for a mixed marriage. -*Tony Phillips* Stony Brook Math in the Media Archive |