Stanley Chang

Richard Evans

Robin Forman

Nate Dean

Diane Hoffoss

Richard Stong

Michael Wolf

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.

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 R^{3}.
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.

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 Z^{2}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)

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

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