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IAS/Park City Mathematics Institute The Undergraduate Program receives major funding from the National Security Agency. For 2004 Application, please click here. The Undergraduate Program 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. 2004 Course Descriptions: 1.
Introductory Course: From
polytopes to enumeration Polytopes are n-dimensional analogs of polygons and polyhedra. They arise in many contexts, they have many applications, and they are a rich source of problems, which call on techniques from geometry, combinatorics and algebra. We will begin with the basic definitions, geometry, and constructions of convex polytopes. This will be followed by an examination of their enumerative properties such as their Euler characteristic, shellability, and Dehn-Sommerville relations. We will consider the close relationship between zonotopes, a special class of polytopes, and real hyperplane arrangements. In our attempt to count the number of regions in the complement of a hyperplane arrangement, we will be led to one of the most powerful tools in combinatorics, the Mobius function of a partially ordered set. Prerequisites: While not essential, an understanding of elementary concepts of linear algebra such as linear independence and dimension will be useful, particularly in the latter course. 2.
Advanced Course: Groebner
bases and polytopes We begin with a crash course in Groebner bases, a theory that was developed by algebraists in the last thirty years. These bases have numerous applications, and in particular, can be used to study convex polytopes and simplicial complexes. We will examine the basic connections in detail, and investigate some of the resulting applications to optimization, polynomial systems, counting lattice points in polytopes, and the study of combinatorial and topological invariants of simplicial complexes. Prerequisites: A background in the theory of polynomial rings at the undergraduate level is recommended. Computer experiments will be an integral part of this course. Send questions and comments to Carleen Inderieden
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