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Combinatorial Aspects of Geometry and Topology:

A Conference for Undergraduate Mathematics Majors

Rice University, April 6-8, 2001

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 junior mathematics majors. There will also be some interactive sessions Saturday afternoon. 


Tim Cochran
Stanley Chang
Richard Evans
Robin Forman
Nate Dean
Diane Hoffoss
Richard Stong
Michael Wolf

Conference Schedule

Friday, April 6:

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

8:00 pm (HB 227): Robin Forman
Title: From Euclid to Euler to Gauss and Beyond

Abstract: All of modern geometry and topology has grown out of some wonderful geometric formulas involving angles, vertices, and edges.

Saturday, April 7:

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

9:30 am (HB227): Diane Hoffoss
Title:  Surgery, Mutants, and the Jones Polynomial

Abstract:  We will discuss why knots are interesting to mathematicians, ways to make mutant knots, how to compute the Jones
Polynomial for a knot, and the curious fact that the Jones Polynomial doesn't distinguish between a knot and any of its mutants.

11:00 am (HB227): Nate Dean
Title: Which unit-length graphs are planar?

Abstract:  A connected graph is roughly a collection of vertices and edges such that any two distinct vertices either belong to the same edge or can be joined by a string of edges.(or both).  Which graphs can be realized in the plane if the edges havelength one (and possibly cross)?  This problem has some interestingapplications, but search algorithms lead to big computations even with arelatively small number of edges.

12:00 noon: Lunch with Rice Undergraduates at Jones College

1:30 pm (HB227) Tim Cochran (HB227) Noncommutative Topology of Links and 3-Manifolds

Abstract:  The advent of quantum mechanics emphasized that "non-commutative mathematics" is necessary to model our universe.  For example, if A and B are matrices, then usually AB is not equal to BA.  Yet, until recently non-commutative mathematics has not had a strong impact in topology.  A link is a collection of circles embedded disjointly in R3.  Links can be intertangled in extremely complicated ways.  Thus it is not surprising that non-commutative groups and rings are useful in reflecting the full mathematical structure of links.  This talk will dscuss the non-commutative nature of links and 3-dimensional manifolds and some invariants derived from non-commuattive algebra which can distinguish among them.

3:00 pm: Three concurrent"lab" activities: Stanley Chang, Richard Evans, Tim Cochran

I. Stanley Chang (HB423):  The Shapes of Soap Bubbles and SoapFilms

Abstract:  Shapes, structures, and models for soap bubbles and soap films have fascinated mathematicians for centuries.  
We will experiment with a variety of topologically and geometrically complex real soap films.

II. Richard Evans (HB453): Playing and calculating with polyhedral surfaces.

Abstract:  We will construct and play with some "floppypolyhedra" "floppypolygons" and, by manipulating them by hand, try to find some interesting formulas and relations between geometry, topology, and combinatorics.

III. Tim Cochran (HB427): Knots.

Abstract:  A (mathematical) knot is a circle embedded in 3-dimensional space.  One can make a model of a knot witha 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 studyknots have become quite 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 theparticipants will break up into small groups and experiment.  We will tryto discover, among other things, a proof that there are an infinite numberof distinct knots.  Different groups will work on different problems, depending on their mathematical background.

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

Sunday, April 8:

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

9:30 am (HB227): Richard Stong

Title: Discrete approximations to rotations

Abstract:  If we are working with continuous maps, a rotation of the plane is a very natural map to consider. Suppose however that we wish to work discretely, say with the lattice  Z2  in the plane. One might ask if there are one-to-one maps which well approximate rotations. (For example, a finite analog of this question arises when one wants to rotate a digital image.) Such approximations can be built in a number of ways and exhibit a great deal of interesting structure.

11:00 am (HB227): Michael Wolf
Title: Trees in Geometry and Topology

Abstract: When we think iftrees in mathematics, we think of Z-trees:  edges come into vertices, andat vertices, a finite number of edges branch off, each edge ending at a newvertex.  These have certainly been useful combinatorial objects in topology.  Recently, though, a generalization of these objects, called R-trees, have emerged as important objects.  When we construct these trees, we allow arbitrarybranching at the vertices (even an uncountable number of edges incident toa vertex), and we no longer require the vertex set to be discrete in the tree.  These are particularily important when we consider limits ofmetricspaces, say under rescalings.

12:00 noon: Lunch (in RiceVillage )

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 RiceBoulevard on the north side of campus. Near this entrance there shouldbe 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--as soon as possible--the following information by e-mail (hardt@math.rice.edu) or fax (713-348-5231) or snail-mail (Math. Dept., Rice University, PO Box1892, Houston, TX 77251).

Phone Number:
Class (sophomore, 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?