This month's topics:

• E₈ in the New York Times
• E₈ in the media
• Intel silver and bronze for math projects
• Cooking Gaussian curvature
• "Journeys to the Distant Fields of Prime"

The Times printed this image, drawn by Peter McMullen in 1964, giving a glimpse of the size and complexity of the Lie group E₈. The configuration (projected here into 2 dimensions) shows part of the arrangement of closest packed balls in 8-dimensional space; the vertices represent a ball's 240 nearest neighbors in 8-space, with bonds drawn between nearest neighbors among the neighbors. E₈ contains a discrete subgroup mapping 256-to-one onto the 696,729,000-element symmetry group of this configuration. A larger, color image, on the Atlas of Lie Groups website, was made by John Stembridge, who explains it here.

The most straightforward Lie groups are groups of n by n matrices characterized by some linear algebraic condition preserved in products, e.g. determinant nonzero, determinant = 1. The product of two matrices is a matrix whose entries are analytic functions (actually sums of products) of the entries in the factors. That's all it takes to make a Lie group. The building blocks of Lie theory, the simple Lie groups, fall into four infinite families of larger and larger matrices, plus five exceptional groups F₄, G₂, E₆, E₇, E₈. The last, largest (248-dimensional) and gnarliest of the exceptionals, E₈, has been in the news recently. Kenneth Chang reported, in the March 20 2007 New York Times, the culmination of a four-year effort by a team of 18 mathematicians, led by Jeffrey Adams (Maryland), to work out the details of its algebraic structure. His description of exactly what they were calculating is very vague, perhaps inevitably, but he clearly conveys the message that the task was enormous. "To understand using E₈ in all its possibilities requires calculation of 200 billion numbers," Chang tells us. "Possibilities" presumably refers to the set of unitary representations of E₈: the main way a group can be analyzed is through representations (projections which preserve multiplication) onto finite or infinite-dimensional matrix groups. The many episodes of the huge computation are laid out in David Vogan's narrative, a good story well told.

"Jeffrey D. Adams and a Lie group," as seen in the Times. Photo by Mark Tilmes, used with permission.

Second and third place in this year's Intel Science Talent Search went to mathematics projects, as reported by Aimee Cunningham in Science Online for March 17, 2007. "Second place and a $75,000 scholarship went to John Vincent Pardon, a 17-year-old from Durham Academy in Chapel Hill, N.C. In his mathematical project, Pardon proved that a closed curve can be made convex without permitting any two points on the curve to get closer to one another.
    Mathematics research also won the third-place prize, which comes with a $50,000 scholarship. Eighteen-year-old Dmitry Vaintrob of South Eugene High School in Eugene, Ore., found a connection between different descriptions of certain mathematical shapes."
    [Vaintrob's project was reported on the Intel site in more detail: the award was "for his sophisticated investigation of ways to associate algebraic structures to topological spaces. Dmitry proved that loop homology and Hochschild cohomology coincide for an important class of spaces." Pardon's Intel citation also mentioned that his project had "solved a classical open problem in differential geometry."]
    Pardon and Vaintrob's scholarship awards were also reported in the March 14 2007 New York Times.

Gaussian cuisine. Low-concentration solution (A) and high-concentration solution (B) of N-isopropylacrylamide (NIPA) are mixed (C) in continuously varying proportion and extruded centrally between parallel plates (D) to form a gelatinous disc (E) with radially varying NIPA concentration, which is placed (F) in a hot bath; the heat makes the low-concentration areas shrink faster than the high, resulting in a non-Euclidean metric. Adapted from Science 315 1117.

Anyone who has considered a potato chip mathematically has seen how Gaussian curvature can be produced by cooking. A team at the Hebrew University have found a way to control this process so as to produce (within a certain range) discs whose Gaussian curvature is a prescribed function of the radial coordinate. Their report, in the February 23 2007 Science, is entitled: "Shaping of Elastic Sheets by Prescription of Non-Euclidean Metrics." The authors (Yael Klein, Efi Efrati and Eran Sharon) present their project as a "novel shaping mechanism" for 2-dimensional objects. "Rather than aiming at a specific embedding, one prescribes on the sheet only a 2D metric, the 'target metric' g_tar ... . The free sheet will settle to a 3D configuration that minimizes its elastic energy. In this mechanism, the selected configuration is set by the competition between bending and stretching energies, and its metric will be close to (but different from) g_tar." Bending energy comes into the picture because the gel is not a 2-dimensional object: it has a finite thickness and resists bending. Nevetheless, "We show that the construction of elastic sheets with various target metrics is possible and results in spontaneous formation of 3D structures." The authors spend some time discussing the difference between the positive curvature case ("The surfaces of K_tar > 0 preserve the radial symmetry of g_tar, generating surfaces of revolution") and the negative ("The surfaces of K_tar < 0 break this symmetry, forming wavy structures"). They report: "A more surprising observation is the asymmetric distribution of the Gaussian curvature. Instead of the negative, rotationally symmetric K_tar, K(ρ,&theta) varies periodically in θ, attaining positive and negative values." [It looks to me like they are measuring normal curvature here. -TP]

Kenneth Chang's article took up the top of the first page in the New York Times Science section for March 13, 2007. It is a "Scientist at Work" profile of Terence Tao (UCLA), one of this year's Fields Medal winners. Don't be put off by the absurd title; Chang gives us a balanced and sympathetic look at this mathematical star. He takes us to Tao's public lecture on prime numbers (slides available here, video here), but then focuses on a "real-world" area of Tao's research, his work on compressed sensing. In a digital camera millions of sensors record an image which then gets compressed. Tao: "Compressed sensing is a different strategy. You also compress the data, but you try to do it in a very dumb way, one that doesn't require much computer power at the sensor end." In fact, Chang tells us, Tao and Caltech professor Emmanuel Candè have shown that "even if most of the information were immediately discarded, the use of powerful algorithms could still reconstruct the original image." Chang also interviewed Billy Tao, Terence's father, who had the insight to consult with experts on educating very gifted children: "To get a degree at a young age, to be a record-breaker, means nothing. I had a pyramid model of knowledge, that is, a very broad base and then the pyramid can go higher. If you just very quickly move up like a column, then you're more likely to wobble at the top and then collapse." A nice quote from Charles Fefferman: "Terry has a style that very few have. When he solves the problem, you think to yourself, 'This is so obvious and why didn't I see it? Why didn't the 100 distinguished people who thought about this before not think of it?'" And he has an elegant blog.
   The next day the Times printed this correction, surely one for the books: "A profile of Terence Tao, a world-renowned mathematician, in Science Times yesterday referred incorrectly to work he did with another mathematician on prime numbers. They proved that it is always possible to find, somewhere in the infinity of integers, a progression of any length of equally spaced prime numbers -- not a progression of prime numbers of any spacing and any length."