JUNIOR/SENIOR HONORS MATHEMATICS SEMINAR
SPRING '06

Professor József Beck

The seminar (course number 640:492:H1, one credit) will be on Tuesdays, third period (12:00 to 1:20) in Hill Center, room 323, Busch Campus. Students who would like to participate should fill in an application; forms are available in Hill Center, room 303.

The topic of the seminar this semester will be UNIFORMITY VERSUS IRREGULARITY (OFFICIALLY: DISCREPANCY THEORY) which tries to answer the following question:   What is the most uniformly distributed n-element point set in the unit square?   We can test uniformity with respect to

(1) axis-parallel squares, or
(2) axis-parallel rectangles, or
(3) tilted squares, or
(4) tilted rectangles, or
(5) axis-parallel right triangles, or
(6) circles,
or whatever your favorite shape is.

For different shapes we have very different answers! Most of the questions have a remarkably complete answer. There are, however, some old open problems; for example, the case of axis-parallel boxes in the unit cube. How come that the 2-dimensional version is solved, but the 3-dimensional version is wide open? This remains a mystery!!