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— Application DEADLINE January 28, 2009
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.
2009 Course Descriptions:
1. Elliptic Curves, Modular Forms, and L-functions: Alvaro Lozano-Robledo, University of Connecticut
This course will be an introduction to elliptic curves and modular forms, with an emphasis on examples. We will begin with some motivating problems, such as the congruent number problem, and the definitions, and then explain how a link between elliptic curves and modular forms is suggested through L-functions. Students will learn how to manipulate elliptic curves, modular forms and L-functions to extract interesting arithmetic information such as rational points, the rank, or congruences (using the free software SAGE). We will discuss some of the big theorems and conjectures - such as the Mordell-Weil theorem and Birch and Swinnerton-Dyer conjecture -, and their consequences. For example, we will sketch how the modularity of elliptic curves is used to prove Fermat's Last Theorem.
The prerequisites for this course are elementary number theory, linear algebra and group theory.
Students should CLICK HERE for full information for this course.
2. Dirichlet L-functions, Generalizations, and Applications: Keith Conrad, University of Connecticut
The first use of L-functions, by Dirichlet (1837), was in his proof that there are infinitely many primes in any arithmetic progression a, a + m, a + 2m,... where a and m are relatively prime. This course will begin by proving Dirichlet's theorem and developing some basic properties of Dirichlet L-functions: analytic continuation, functional equation, and special values.
Analogies between Z and F[T], where F is a finite field, are an important theme in number theory, and we will see the analogue of Dirichlet's theorem and Dirichlet L-functions for F[T]. Moreover, the Riemann hypothesis can be proved in this setting, which leads to bounds on character sums of classical interest (with applications to counting points mod p on elliptic curves).
Returning to Z, we will develop algebraic number theory for cyclotomic fields in order to rewrite Dirichlet L-functions in terms of characters on abelian Galois groups. Then we will see how to do this for the analogous L-functions on F[T]. At the end we will introduce Artin L-functions, which are a generalization of Dirichlet L-functions that are associated to possibly non-abelian Galois groups.
The background required for this course is complex analysis, Galois theory, and familiarity with elementary number theory up through the quadratic reciprocity law.
Students should CLICK HERE for full information for this course.
The Coordinators of PCMI's Undergraduate Summer School Program are Andrew Bernoff, Harvey Mudd College, and Aaron Bertram, University of Utah.
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.