When we look out on a clear night, the universe seems infinite. Yet this infinity might be an illusion.  During the first half of the presentation, computer games will introduce the concept of a “multiconnected universe”.  Interactive 3D graphics will then take the viewer on a tour of several possible shapes for space. Finally, we'll see how recent satellite data provide tantalizing clues to the true shape of our universe.

The only prerequisites for this talk are curiosity and imagination. For middle school and high school students, people interested in astronomy, and all members of the Seattle University and surrounding communities.

Physical models and software simulations will introduce participants first to curved surfaces and then to curved 3-dimensional space. The contrasting properties of spherical and hyperbolic geometry will be savored.

The discovery of unexpected examples forced a reexamination of the concept of dimension in the early twentieth century.  Several competing definitions of dimension emerged.  They do not all give the same answer when applied to a subset of Euclidean space and do not even necessarily yield integers as answers.  (Spaces whose dimension varies, depending on which definition is used, are called fractals.)

In this talk I will take a quick look at the examples mentioned above and review the definitions of topological dimension and Hausdorff dimension from an elementary point of view.  Every compact metric space contains a Cantor set whose Hausdorff dimension equals that of the given space, so it is natural to focus on examples of Cantor sets.  There is a simple construction that yields Cantor sets in Rⁿ of dimension s for every real number s in the range 0 ≤ s ≤ n.  Antoine's necklace is a classic example of a wild Cantor set in R³.  I will explain why the Hausdorff dimension of an Antoine's necklace Cantor set must always be at least 1 and how to construct an Antoine's necklace of Hausdorff dimension s for every s in the range 1 ≤  s ≤ 3.  This is a special case of a much more general theorem that relates Hausdorff dimension to embedding dimension and implies that every wild Cantor set is a fractal.

How do you get bright students hooked on mathematics? How do you keep teachers intellectually engaged and pedagogically innovative? A proven way is to involve them both in mathematics competitions with great problems that span the curriculum. The Mathematical Association of America has continuously sponsored nationwide high-school level math contests since 1952. The sequence of contests now has 5 different contests at increasing levels of mathematical sophistication. Students who succeed at the top level on these contests become the team representing the U.S. at the annual International Mathematical Olympiad. I'll showcase some interesting, easy and hard contest problems, and a little bit of history. Along the way, I'll comment about the intersection of these contests with the school mathematics curriculum.

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