This research group, with the aid of funding from the Division of Mathematical Sciences at the National Science Foundation, will conduct research on quasirandom processes, with a view towards laying the foundations of a general theory of quasirandom analogues of discrete random systems, and with the hope of developing real-world applications of some of the fundamental mechanisms of discrete quasirandomness, such as chip-firing and rotor-routing. (See my blurb on quasirandomness for a brief introduction to the notion.)
Undergraduates, graduate students, postdocs, and faculty members are all welcome to participate, from mathematics, computer science, statistics, and other disciplines. The current participants in this project (Spring 2007) are UML undergraduates Stephen Dalton, William D'Angelo, Stephen Giardini, and Anthony Tempesta.
I taught a graduate/undergraduate course (92.584) in Fall 2007 that will lay the foundations for much of this work. (Students in 92.584 need not participate in EQL, but participants in EQL are strongly encouraged to take 92.584.) This course will probably be offered agian in Fall 2008.
Quasirandom approaches to inherently continuous problems (and in particular, the use of quasirandom number generators) are well-studied; see e.g. the results of a Google search on quasirandom+Niederreiter (Harald Niederreiter being one of the pioneers of the field). Although my work on discrete random processes (random walk and random aggregation) takes a different starting point, it has many points of contact with the theory of quasirandom sequences of numbers or points; for instance, the van der Corput sequence (a staple of classical quasirandomness) arises almost automatically from quasirandom walk. More generally, notions of discrepancy, which have been fundamental to the advance of classical quasirandomness, play a basic role in discrete quasirandomness as well.
A good source of information on discrepancy theory is the book The Discrepancy Method: Randomness and Complexity (also available as a printed book) by Bernard Chazelle. One deep question that the work of EQL may shed some light on is, to what extent can probability theory be viewed as a subset of discrepancy theory?
Here is software that demonstrates how rotor-routers work.
For more on discrete quasirandomness, see my discrete quasirandomness web-page. This site includes a link to the grant proposal which was funded by NSF in 2006 ($100,000 for the period 2007-2009).
Other research programs that I have run at other schools are the Tilings Research Group (MIT), the Spatial Systems Laboratory (University of Wisconsin), and Research Experiences in Algebraic Combinatorics at Harvard (Harvard).
If you are interested in finding out more about EQL or getting involved, contact me.
This page was last modified April 3, 2007 by JamesPropp at gmail dot com.