Cornell Math - MATH 752, Spring 2000

MATH 752 — Spring 2000
Seminar in Topology

This course — known as the Berstein Seminar — is a geometry/topology seminar in which the students do most of the lecturing and the professor tries to provide guidance without getting in the way.*

I propose two possible topics.

I. Diagrams, Equations Over Groups and Asphericity

(Prerequisite: Math 651)

This course would introduce one of the major themes in geometric group theory.

To every presentation P of a group G there is a naturally associated 2-dimensional CW complex K_P whose fundamental group is isomorphic to G. Maps of the 2-disk D² and the 2-sphere S² give important information about triviality of elements of the group and properties of the group. Such maps induce (upon judicious adjustment) cell structures (''diagrams'' and ''pictures'') on D₂ and S² and the geometry and combinatorics of these structures tells us much about the group.

We will read papers on diagrams and their applications to equations over groups and asphericity of the 2-complexes. These words will be explained and studied. Papers of Gersten, Pride, Bogley, Klyachko, Sieradski and Rothaus are candidates of papers to read (as are many others.)

II. Geometric Topology and Knot Theory

(Prerequisite: Concurrent registration in Math 651)

Topics in geometric and piecewise linear topology will be studied. Possibilities, depending on the interests of the students, are:

1. Classical results on spheres: The Alexander Horned Sphere, the Schoenflies Theorem
2. Piecewise linear topology as a follow up to Math 661 - fall 1999. The h-cobordism theorem, co-dimension 3 unknotting.
3. Classical knot theory (and perhaps some non-classical).

For this option of the course Rolfsen's Knot Theory book would be ideal. It and original papers would be the sources.

Students who may be interested should contact Marshall Cohen as soon as possible for discussion of possibilities which they would like to explore. The decision of which path we will take will be announced on the first day of class, or sooner if there is a clear student preference.

*This form of the seminar was originated in 1974 by Professor Israel Berstein when he could no longer lecture traditionally due to Parkinson's disease. He taught it for the next 16 years. His quickness of mind, lively humor and warm non-judgmental support of the students have left a legacy which subsequent professors take as the model of what we should be doing in this seminar. When it works well it can play a significant role in a student's mathematical growth and empowerment.