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Low Dimensional Topology and Geometry:

A Conference for Undergraduate Mathematics Majors

Rice University, November 20-22, 1998

The general goal of this conference is to expose undergraduates from many universities to, and stimulate interest in, current research in mathematics.  Several research mathematicians from Rice University will discuss classical and current results and open questions at a level accessible to senior mathematics majors. There will also be some interactive sessions Saturday afternoon. 


Tim Cochran
Zhiyong Gao
John Hempel
David Metzler
Jennifer Slimowitz
Richard Stong
Michael Wolf

Conference Schedule

Friday, November 20:

6:30 - 7:30 pm: Complimentary Buffet, HB (Herman Brown Building) 438

 8:00 pm (HB 227): Tim Cochran
Title: Topology and Geometry in Low Dimensions

Abstract:  I will attempt to give an overview of "topology and geometry in low dimensions".  I will discuss what this means and why it is a separate, important, appealing and difficult field of study.  I will discuss several specific subfields within this area, their major questions and major techniques.

Saturday, November 21:

9:00 am (HB438) Coffee, donuts, fruit

9:30 am (HB227): Richard Stong
Title: Four Manifold Topology

Abstract: The key to understanding a four dimensional manifold is the ability to find embedded disks inside it. Unfortunately finding such disks is very difficult. We will discuss why finding disks is hard and outline how one can sometimes produce them.

11:00 am (HB227): Jennifer Slimowitz
Title: What is a symplectic matrix and why can't it squeeze?

Abstract: Non-squeezing, one of the hot topics in symplectic geometry and topology today, can be described using linear algebra.  We will define the group of symplectic matrices, discover some properties, and eventually prove that a symplectic linear transformation can't squeeze.  This result is a special case of some recently proven deep theorems about general symplectic diffeomorphisms.

12:00 noon: Picnic Lunch with Rice Undergraduates

1:30 pm (HB227): Michael Wolf
Title: Classification of Surface Diffeomorphisms

Abstract: What are the possibilities for the shapes of a homeomorphisms of a surface?  At a time (1976) when it was not even clear how to begin a classification, Bill Thurston gave a complete answer.  We will describe his solution.
3:00 pm: Three concurrent "lab" activities: Tim Cochran, David Metzler, Richard Stong

I. Tim Cochran (HB423):  Problems with Knots and Links

Abstract:  A (mathematical) knot is a circle embedded in 3-dimensional space. One can make a model of a knot with a knotted piece of rope by glueing the two "free ends" together. Despite their concrete nature, knots are very complicated "non-commutative" objects. the tools used to study knots have become qutie sophisticated, and knot theory is a major research area. For about 30 minutes we will discuss knots, "colorings of knots" and ways of associating non-abelian groups to knots. Then the participants will break up into small groups and experiment. We will try to discover, among other things, a proof that there are an infinite number of distinct knots. Different groups will work on different problems, depending on their mathematical background.

II. David Metzler (HB453): How to Hold a Four Dimensional Manifold in Your Hands.

Abstract:  We will construct and play with some "floppy polygons", polygons in 3-space with flexible joints, out of dowels and rubber tubing. The configuration spaces of these objects turn out to be fascinating examples of symplectic manifolds called "toric varieties." We will extract as much information about their geometry and topology as we can by manipulating themby hand.

III. Richard Stong (HB427): How to Play and Win Combinatorial Games.

Abstract: There are a large number of games which are easy to describe and which can be analyzed mathematically. We will describe a few of these games and winning strategies for them.

6:00 pm: Pizza party with Rice graduate students.

Sunday, November 22:

9:00 am (HB438): Coffee, donuts, fruit

9:30 am (HB227): John Hempel
Title: Names and aliases for 3-dimensional manifolds

Abstract: We describe a method for attaching an identifying notation (name) to each compact, oriented 3-manifold which contains a set of instructions for reconstructing the manifold. We discuss how different names for the same 3-manifold are related. The main point is to try to explain why the (yet to be solved) problem of classifying 3-manifolds is difficult.

11:00 am (HB227): Zhiyong Gao
Title: Geometry in Low Dimensions

Abstract: The hyperbolic plane H^2 (with the complete metric of constant curvature -1) admits no isometric C^2-immersion into three dimensional Euclidean space by a famous theorem of Hilbert. We will study how to construct isometric C^2-immersion and embedding of hyperbolic plane H^2 into five and six dimensional Euclidean spaces, and etc.

12:00 noon: Lunch (in Rice Village ?)

Afternoon: games (frisbie, soccer, chess) 


Most activities will be held in the Herman Brown Building on the Rice University campus. The Herman Brown Building is located near entrance 14, on Rice Boulevard on the north side of campus. Near this entrance there should be adequate visitor parking available. If not, proceed west to the stadium lot.

For a map and further information see "Travel Information" in the Rice Mathematics Department homepage (http://math.rice.edu).


Please submit--by November15th--the following information by e-mail (stong@math.rice.edu) or fax (713-285-5231) or snail-mail (Math. Dept., Rice University, PO Box1892, Houston, TX 77251).

Phone Number:
Class (junior, senior, etc.):
Arrival date and time:
Departure date and time:
Do you plan to attend the complimentary buffet on Friday?
Do you plan to attend the Saturday evening party?