Department Of Mathematics, Kansas State University
| September 2008 : | David Auckly wins Lester R Ford Award from the Mathematical Association of America |
| In a discussion that sets off to explain
a procedure for solving quartic polynomial equations, Auckly leads
a tour of some of the most mathematically rich and variegated native
lands of the Algebraic Geometers. The main attractions along his
tour are pencils of curves, which he ultimately employs as the chief
tools for illustrating how one solves the general quartic,
exemplified here by the consideration of the equation
x^4 - 7x^2 + 6x = 0.
This quartic is turned into the pencil of conics
y^2 - 7y + 6x + \lambda(y - x^2) = 0, whose four base points provide the solutions to the original quartic. Along the way to the solution, the author, like the best tour guides, takes us along pathways that lead deep into the mathematical countryside, hinting at what we can learn if we choose to dally longer. At a number of points along the way, he stops and opens the doors of the bus to allow his passengers to debark and probe a bit further with the call, "Talk about this with the mathematically adept or. . . read all about it at your library." From the safety of the motorcoach our host points to how the landscape has been developed by algebraic geometers over the centuries via the construction of rather sophisticated machinery: algebraic closure and the complexification of curves followed by projectivization, linear systems and pencils. Even more exotic notions follow on later in the tour: rational elliptic surfaces, birational equivalence, symplectic manifolds, Lefschetz fibrations. Auckly's travelogue engages readers with a familiar problem and keeps them interested through an exposition of some deep and significant notions in algebraic geometry. After competing his tour, we realize the need to keep our pencils sharpened. |