The Undergraduate Summer School provides opportunities for talented undergraduate students to enhance their interest in mathematics. This program is open to undergraduates at all levels, from first-year students to those who have just completed their undergraduate education. Based on the backgrounds of the accepted students, they will be divided into two groups - introductory and advanced. There will be several activities organized for these groups, with some specifically intended for either the introductory or advanced group. There will be time for study groups and individual projects guided by advisors, as well as other activities.

2006 Course Descriptions:

1. Introductory Course: Topology of 2-dimensional and 3-dimensional spaces, knot theory and applications of topology to chemistry
Erica Flapan, Pomona College

The course will be an introduction to the topology of 2-dimensional and 3-dimensional  spaces, knot theory and applications of topology to chemistry.  More specifically, the lectures on 2 and 3-dimensional spaces will include manifolds, orientability, quotient spaces, connected sums, products, flat manifolds, and geometries on surfaces.  The lectures on knot theory will include Reidemeister moves, knot and link invariants (including knot polynomials), invertability, amphicheirality, and properties of alternating knots.  Finally, we will show how knot theory and other topological techniques can be used to analyze molecular symmetries.  The only prerequisites for the course are linear algebra and the ability to write proofs.

2. Advanced Course: Hyperbolic Geometry
Francis Bonahon, University of Southern California

Hyperbolic (non-euclidean) geometry plays an unexpectedly important role in low dimensional topology and geometry.  After discussing the basic properties of hyperbolic space, the course will branch out in two directions: kleinian groups and their fractal limit sets; and applications of hyperbolic geometry to topology and knot theory.  Although not a formal prerequisite, prior exposure to real analysis and/or pointset topology would be useful.

Undergraduate Summer School & Research Experience at the University of Utah

June 5- July 15, 2006

For the purpose of providing students in the USS with a more lengthy and richer summer program, some students participating in the PCMI Undergraduate Summer School (USS) will be given the opportunity to supplement the USS with an REU experience at the University of Utah.  The program at the University of Utah is entitled, "The Geometry of Mobius Transformations," and runs from June 5- June 23, 2006.  After the REU, the students will spend the next three weeks at PCMI in the Undergraduate Summer School.  The Park City Mathematics Institute runs from June 25 - July 15, 2005.

In order to apply for this joint program, applicants should apply directly to the PCMI Undergraduate Summer School and indicate at the designated place on the application their interest in the REU.

More information about the REU at the University of Utah can be found at  www.math.utah.edu/vigre/reu/summer06/index.html.

The Undergraduate Summer School is supported in part by the National Security Agency and in part by the National Science Foundation grant no. 0437137.