In a special "Current Events" session at the Joint Meetings, four outstanding expositors presented talks on exciting topics at the forefront of research.
Margaret Wright of the Courant Institute of Mathematical Sciences began with a history of the interior point method from the 1960's to the present. She emphasized the "revolution" that began with Narendra Karmarkar's 1984 announcement of a polynomial-time LP method, which he claimed ran 50 times faster than the simplex method on some problems. Recognition of an equivalence between Karmarkar's method and "classical" barrier methods has led to improvements in, and reliance upon, both interior point methods and the simplex method. As a result, there have been major advances in such areas as complexity theory and linear algebra. Currently benefiting from the use of interior methods are system and control theory as well as approximation techniques for NP-hard combinatorial problems.
Next, Thomas Hales of the University of Pittsburgh discussed the new concept of "motivic measure," for which the objects being measured are formulas, rather than solution sets of those formulas. This new theory, which was sparked by a 1995 lecture by Maxim Kontsevich, has found applications to such topics as the geometry of varieties, p-adic integrals, and generating functions.
In the third lecture, Andrew Granville of the Université de Montréal discussed the recently published "polynomial time primality test" of Manindra Agrawal, Neeraj Kayal and Nitin Saxena. Their algorithm--referred to as AKS--decreased the amount of time to determine the primality of a number nfrom d^(log log d) to approximately (log n)^7.5 or d^7.5 steps, where d is roughly (log n)/(log 2). Granville provided both the proof of this test and an historical context for this discovery by building on ideas likely to be familiar to anyone having taken a course in undergraduate number theory. In this way, he held one's attention all the way to his "stop the presses" punch-line revelation of an improved result by Lenstra and Pomerance: a new algorithm with an even faster running time of (log n)^6.
The final lecture was by John Morgan of Columbia University, who spoke on the work of Grigory Perelman on the Poincaré Conjecture. Morgan provided an accessible introduction to Richard Hamilton's ideas about the Ricci flow, which is the basis for Perelman's work, and then went on to discuss the new methods Perelman has introduced to handle the singularities that arise in the flow. At the time the talk was given, three papers by Perelman had been made public, and these three purport to prove the Poincaré Conjecture. A fourth paper, which is supposed to complete a proof of the Geometrization Conjecture, remained unfinished. A member of the audience asked what it would take to convince Morgan that the first three papers really constitute a proof of Poincaré. Morgan replied that the papers present a long and complicated argument, and not all of the steps are written down in full detail. If this were just a normal research paper, and not one claiming to prove one of the biggest outstanding questions in mathematics for the past century, he said, the papers could be accepted as correct. But, since the Poincaré Conjecture is at stake, the papers have to be held to a higher standard.
-- Allyn Jackson, Deputy Editor of The Notices, and Claudia Clark, AMS-AAAS Media Fellow