IAS/Park City Mathematics Institute 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. 2002 Course Descriptions: 15 lectures This course will introduce students to expander graphs - graphs that efficiently
propagate information quickly to many nodes along short paths. Recent explicit
constructions of such graphs have created an explosion of interest in their potential
applications to network design, complexity theory, coding theory and cryptography, to name
just a few. We will discuss applications and constructions of expander graphs. The
constructions are algebraic in nature, and provide a beautiful example of how abstract,
seemingly unrelated topics in number theory, group theory and analysis can be elegantly
combined to solve an important real-world problem. Steve Gelbart, The Weizmann Institute of Science, Rehovot, Israel In this course, the basic object of study shall be the Riemann zeta-function
Z. However, we shall also look at Dirichlet L-functions which generalize
Z, and more generally Hecke's L-functions. Ultimately, we are interested in the many
properties of the integers that are reflected in the analytic properties of the
zeta-function. For example, the Euler product representation for the zeta-function is a
reflection of the unique factorization of integers into primes. Topics to be considered
are: 1) the analytic continuation and functional equation of L-functions 2) Dirichlet's theory of L-functions 3) "explicit formulae" of prime number theory 4) zeros of Z and the prime
number theorem 5) Riemann's Hypothesis 6) Hecke's theory of L-functions. questions or concerns should be directed to C. Giesbrecht |