IMA Outcomes

A mathematical result may affect—for now—only an elite cadre of theorists, or may subtly alter the day-to-day existence of millions, many of whom profess a profound antipathy towards mathematics. A new concept, construction, or theorem can reshape careers, or even save lives.

Getting emergency calls through.

Damn... no dial tone?! In 2001 customers in high growth local service areas were finding their telephone lines dead due to circuit congestion of unknown origin. Annoying, certainly, but it can be much worse than that. AT&T engineers estimated that emergency calls in life-threatening situations could be affected as many as 90 times per day (imagine picking up the receiver to report a heart attack to 911 and hearing nothing). AT&T Research Labs scientist V. Ramaswami led a group to study this problem using stochastic modeling tools based on a class of probability distributions known as phase type. Using sophisticated methods of modeling and analysis, their research pinpointed the causes of the congestion, particularly the role of very long calls associated with internet dial-ups, and developed remedies that could be applied in both the short and long term.

The modeling not only saved lives, but also identified opportunities for major cost savings, estimated at $15M per year, to AT&T and has resulted in several innovations forming the subject of five U.S. patent filings of which two have been granted. The paper “Assuring Emergency Services Access: Providing Dial Tone in the Presence of Long Holding Time Internet Dial-Up Calls”, was an INFORMS Wagner prize Finalist and will appear in the journal Interfaces in 2005. In this case, as in many important applications of probabilistic models, innovative algorithmic approaches were essential for obtaining a detailed analysis. For example, the collaborative research of Guy Latouche and Ramaswami, part of which was accomplished at the IMA, provided some of the key methodologies for this work.

Ramaswami returned to the IMA in 2004 to join one of his coauthors on the dial tone analysis, statistician Soohan Ahn, a long term visitor at the IMA. Together they developed a new class of probability distributions extending the phase type distributions, whose properties make them very valuable for stochastic modeling of telecommunication systems; for this new class, they forsee a similarly far reaching impact. Their paper “Bilateral Phase Type Distributions” will appear in Stochastic Models in 2005. In addition to this, the researchers also made significant progress in extending the algorithmic work on discrete state space processes described above to efficient algorithms for fluid flow models that are widely used in high speed communication network performance, insurance risk theory, the theory of queues and dams, etc. Their paper on this, to appear in the Journal of Applied Probability in 2005, will provide a powerful generic tool for stochastic fluid flow models.

Ramaswami credits their success in part to the IMA's “quiet but highly charged atmosphere” which enables “serious work that requires collaboration and uninterrupted concentration.”

Conquering the critical case.

In quantum mechanics a particle's position is described by a probability distribution, called a wave function, rather than by precise coordinates. In the 1920s, Bose and Einstein predicted a new state of matter in which many particles at very low temperatures share a similar wave function, a prediction which was verified experimentally in the 1990s. The simplest mathematical model of such a Bose–Einstein condensate uses the nonlinear Schrödinger equation, in which a nonlinear interaction energy between the many particles in the condensate is added to the kinetic energy term of the classical Schrödinger equation. The nonlinearity poses a great challenge to both physicists and mathematicians, leading to interesting and often unpredictable behavior. For example, strong attractive interaction may lead to collapse, infinite density, and model breakdown. The very challenging question of whether solutions remain as smooth as they started—and whether they approach simpler, linear waves at large times—has attracted the attention of many top mathematicians since the early 1970s.

Vortex structure in a
Bose-Einstein condensate.

Mathematicians characterize the relative importance of the kinetic and interaction energies via their scaling properties. In the subcritical case, when the interaction energy becomes small for small densities, the smoothness question was answered in the affirmative in 1985. The more difficult critical case, in which both energy terms are equally important even at low densities, remained open for another 20 years, but finally fell this year. The successful team, consisting of five mathematicians from across the globe—Jim Colliander, Markus Keel, Gigiola Staffilani, Hideo Takaoka, and Terence Tao—came together in summer 2002 at the IMA to attack this singular problem. After two weeks of intensive collaboration at the IMA they devised an audacious plan of attack. It took another year and a half of work to pull it off, but by early 2004 they had arrived at a comprehensive description of the long-time behavior of the nonlinear Schrödinger equation in the critical defocusing case, corresponding to strong repulsive interaction of a Bose–Einstein condensate. They show that collapse does not occur even for large densities, and, moreover, after an initial transient, the density evolves according to the linear Schrödinger equation: although particles interact strongly, the evolution of the density can be described by completely neglecting the interaction! The massive proof, a milestone in the analysis of partial differential equations, first appeared in the IMA preprint series in February 2004 and will be published in the Annals of Mathematics in 2005.