AMS Centennial Fellowship

Named to commemorate the Centennial of the American Mathematical Society in 1988, the Centennial Fellowship is awarded annually to one or more researchers who have held the Ph.D. for between three and twelve years. The primary selection criterion for the Centennial Fellowship is excellence in research. Christopher Hoffman was the sole recipient in 2008.

Christopher Hoffman: AMS Centennial Fellow

Christopher Hoffman received his Ph.D. from Stanford University in 1996 under the supervision of Don Ornstein. After postdoctoral positions at the Hebrew University of Jerusalem and the University of Maryland, in 1999 Hoffman joined the faculty at the University of Washington, where he was promoted to full professor in 2008.

Hoffman’s work has been in ergodic theory, probability and combinatorics. One theme that is common through much of his work recently is that of phase transitions, an important phenomenon in physics. The most familiar phase transition is water changing to ice when the temperature falls below 32 degrees. A different phase transition in magnets has been intensively studied by mathematical models. At room temperature the magnetic poles of most of a magnet’s atoms are lined up in the same direction. If you heat up the magnet a little bit, most of the poles will remain lined up in the same direction. But near a certain temperature, called the critical temperature, the fraction of atoms whose poles line up in the same direction decreases rapidly. When the temperature of the magnet gets above the critical temperature, the magnetic poles of the atoms lie in all different directions and there is very little correlation between the directions of the magnetic poles of any two different atoms.

Many models in mathematical physics display a similar behavior to the behavior of a magnet described above. For the system there are a variety of different states indexed by the value of one parameter, such as temperature in the examples above. We say the system exhibits a phase transition if the behavior of the system is very different below the critical value than it is above the critical value. (Often the system will exhibit a third behavior at the critical value that is quite distinct from the behavior either above or below the critical value.)

Hoffman has worked on phase transitions in many different contexts. An example of his work in this area are the papers that deal with stable matchings. Stable matchings have been studied in a wide variety of contexts. The original motivation for their study came from the following scenario.

Imagine a high school preparing for prom where there are an equal number of males and females (and everyone is heterosexual). We consider all possible matchings of the boys and the girls so that every boy gets matched with exactly one girl (and vice versa). Any such assignment of dates is likely to make some people unhappy. In particular, we say that the matching is unstable if there are a boy and a girl who would both rather go to the prom with each other than with the partners they are actually paired with, and stable if there is no such boy-girl pair. This problem was first studied by Gale and Shapley, who asked if there always exists a stable matching no matter what the preferences of the boys and girls are. They showed that the answer is yes by devising two algorithms to find a stable matching. (The two algorithms usually don’t produce the same result. One always produces the matching that makes the boys the happiest, while the other does the same for the girls.) Such algorithms have widespread applications. For instance, the assignment of new doctors to their internships is done by one of the Gale-Shapley algorithms.

Together with Ander Holroyd and Yuval Peres, Hoffman has studied a stable matching problem between two measures in the plane. For each positive value of the parameter a, the matching problem consists of choosing a random set in the plane and assigning to each point in the set a region of area at most a in a stable manner, where points “prefer” regions that are located nearby and vice versa. To see how this system behaves we include pictures of matchings with four different values of a.

Figure 1. a = .02	Figure 2. a = .08	Figure 3. a = 1	Figure 4. a = 2

When a is less than 1 (Figures 1 and 2), the image of every point in the random set has area a, but some points (the white area) are not assigned to any point in the random set. The picture looks very different if a is greater than one (Figure 4): the image regions completely fill the plane, and some of them have area smaller than a. The critical point occurs when a = 1 (Figure 3). In this case the image regions fill the plane (as when a > 1), and they all have area a (as when a < 1).

Hoffman plans to use his Centennial Fellowship to continue his work on stochastic growth processes by attending the program on discrete probability at the Institut Mittag-Leffler in Djursholm, Sweden as well as the program on probabilistic methods in mathematical physics at the Centre de Recherches Mathématiques in Montreal.