KP equation and plabic graphs

Welcome to the Department of Mathematics and Statistics. We offer a wide variety of undergraduate and graduate degree programs designed for students with diverse career or higher educational goals. Our faculty members maintain active research programs in the fields of combinatorics, algebra, analysis, applied mathematics and applied statistics. In a nod toward the unity of mathematics, we offer the following question—whose answer requires several of the above fields, as well as geometry:

A collection of small waves are travelling through shallow water and happen to collide. What happens next?

The first half of the above sentence is governed by the famous KdV equations. (Jerry Bona, UIC, spoke at our colloquium about these waves not long ago.)

The second half of the above sentence is governed by cells in the totally positive part of the Grassmannian and plabic graphs. (Dr. Lauve can tell you more about this aspect of the theory of totally positive matrices.)


See our Faculty Research page for a list of local people to ask for more details, or consult the original sources:

Y. Kodama, L. Williams, KP solitons, total positivity, and cluster algebrasProceedings of the National Academy of Science, v.108, n.22 (2011).

M. Ablowtiz, D. Baldwin, Nonlinear shallow ocean wave soliton interactions on flat beachesPhysical Review E, v.86, n.3 (2012).

Photo Credit: Mark Ablowitz, Colorado. Taken in Nuevo Vallarta, Mexico.

Figure Credits: Lauren Williams, Berkeley. [slides]