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. There will be many organized activities, with some specifically targeted at students at the introductory level and others at more advanced students. There will also be time for study groups and individual projects guided by advisors, as well as other activities.

**2012 Course Descriptions:**

**Basic Class**

Jennifer Taback, Bowdoin College

Title: Groups, Graphs and Trees

Abstract: This course will be a first introduction to geometric group theory, beginning with the basic question of how one can construct a geometric model of a finitely generated group. Geometric group theory is a field of mathematics which thrives on interesting examples, and interspersed with proving some introductory theorems we will focus in detail on several major families of groups which are studied from this perspective. These include free groups, Baumslag-Solitar groups, lamplighter groups and Thompson's group F.

This course will loosely follow the book Groups, Graphs and Trees by John Meier, and a first course in group theory will be assumed.

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**Advanced Class**

Kevin Wortman, University of Utah

Title: Introduction to arithmetic groups Abstract: A basic example of an arithmetic group is SL(n,Z): the group of n by n invertible matrices with integer entries and determinant equal to 1. Arithmetic groups have a lot to do with the geometry of symetric spaces, which are certain kinds of manifolds that display a lot of symmetries (hyperbolic space is an example). The interplay between the geometry and the algebra of arithmetic groups is very deep, and provides for a good introduction to understanding the interplay between geometry and algebra that one hopes to find in geometric group theory more generally.

In this course, we'll focus on examples of arithmetic groups. By the end of the course, we'll be able to hint at what current research in arithmetic groups is focused on.

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The Coordinators of PCMI's Undergraduate Summer School Program are Aaron Bertram, University of Utah, Steven Cox, Rice Unviersity, and Thomas Garrity, Williams College .

PCMI's Undergraduate Summer School is supported in part by the National Security Agency and in part by the National Science Foundation grant #DMS- 0437137.