#
JUNIOR/SENIOR HONORS MATHEMATICS SEMINAR

SPRING '06

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** 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!!