Below you will find a schedule of speakers for the Spring 2006 seminar. The schedule will be updated every weekend with information about the following week's lecture.

• No talk this week.

• Speaker Sara Blight
• Title Prime Connections
• Time/place Tuesday, 4/11/2006 2:00pm in Hill 124
• Abstract In his paper, "Prime Numbers: From Recreational Mathematics to Practical Applications," Enrico Bombieri discusses several of the questions that primes inspire and the limited answers that have been found. In all cases, it seems that surprising links reveal themselves. I'll discuss a couple of these interesting connections.

• Speaker Nicole Raulf, Princeton University
• Title Asymptotics of class numbers via Hecke operators
• Time/place Tuesday, 4/04/2006 2:00pm in Hill 124
• Abstract I will show how the theory of automorphic forms can be used to prove asymptotic results for class numbers.

• Speaker David Jao
• Title Elliptic Curve Cryptography, Expander Graphs, and the Generalized Riemann Hypothesis
• Time/place Tuesday, 3/28/2006 2:00pm in Hill 124
• Abstract The talk concerns the discrete logarithm problem from elliptic curve cryptography and a family of graphs which originally arose in the context of studying the difficulty of this problem. These graphs can be shown to have a certain property known as expansion, assuming the Generalized Riemann Hypothesis for Hecke L-functions. From this, one can deduce a relationship between the difficulty of the discrete logarithm problems on many of the elliptic curves that are used in cryptographic applications today. The rapid mixing of random walks on these graphs had been previously assumed in a number of recent attacks on elliptic curve cryptosystems. One application of our estimates of the graph eigenvalues is to give useful and explicit bounds for these mixing times. The graphs themselves represent a new (conditional) construction of expander graphs having many other applications and generalizations which we will also discuss during the course of this talk.

• Speaker Roman Holowinsky
• Title Ramanujan-Petersson Conjecture and Symmetric Power $L$-Functions
• Time/place Tuesday, 3/21/2006 2:00pm in Hill 124
• Abstract Connections between Ramanujan-Petersson and Symmetric Power $L$-functions. We remove Ramanujan-Petersson conjecture from a work of Elliott, Moreno, and Shahidi by introducing information about the Symmetric sixth $L$-function.

• Time/place Tuesday, 3/14/2006 2:00pm in Hill 124
• SPRING BREAK-NO SEMINAR

• Speaker Prof. Henryk Iwaniec
• Title Negative Hecke Eigenvalues
• Time/place Tuesday, 3/7/2006 2:00pm in Hill 124
• Abstract Estimates for the first negative Hecke eigenvalue of congruence groups in terms of the level. Joint work with W. Kohnen and J. Sengupta.

• Speaker Samar Jaafar
• Title Elliptic curves and cryptography
• Time/place Tuesday, 2/28/2006 2:00pm in Hill 124
• Abstract A basic introduction to elliptic curve cryptosystems. The main goal is to expose Rene Schoofs algorithm.

• Speaker Roman Holowinsky
• Title Shifted Convolution Sums
• Time/place Tuesday, 2/21/2006 2:00pm in Hill 124
• Abstract We will see how shifted convolution sums arise naturally in the analysis of inner products related to the quantum unique ergodicity conjecture on SL(2,Z)\H. Taking absolute values of the terms in relevant shifted convolution sums, we'll state some results which come from exploiting positivity through an upper bound sieve.

• Speaker Hoi H. Nguyen
• Title Selberg's Sieve
• Time/place Tuesday, 2/14/2006 2:00pm in Hill 124
• Abstract I will talk about Selberg's Sieve, mainly about the upper bound. If time permits, I will cite some applications, too.

• Speaker John Bryk
• Title The ABC Conjecture and its Consequences(Part 2)
• Time/place Tuesday, 2/7/2006 2:00pm in Hill 124
• Abstract We will discuss the ABC conjecture and its far-reaching consequences for many problems in number theory. In addition to giving a simple proof of Fermat's Last Theorem, we will see that the ABC conjecture is equivalent to extensions of some important theorems, including results of Roth, Faltings, and Wiles.

• Speaker John Bryk
• Title The ABC Conjecture and its Consequences
• Time/place Tuesday, 1/31/2006 2:00pm in Hill 124
• Abstract We will discuss the ABC conjecture and its far-reaching consequences for many problems in number theory. In addition to giving a simple proof of Fermat's Last Theorem, we will see that the ABC conjecture is equivalent to extensions of some important theorems, including results of Roth, Faltings, and Wiles.