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Number Theory at UBC

## SFU/UBC Number Theory Seminar Winter Semester 2005

 Thursday, January 13, 2005SFU Campus, Room K9509 3:00-3:50 Mike Bennett (UBC) A tale of two surfaces Abstract: In this talk, we discuss two families of surfaces and techniques for finding the integral points upon them, arising from Diophantine approximation. These problems have their roots in a 19th century question about triangular numbers. 3:50-4:10 tea break 4:10-5:00 Greg Fee (SFU) Finding a solution to the sum of squares problem Abstract: The “sum of squares problem” is to find all integer solutions of the equation x[1]2 + x[2]2 + ... + x[k]2 = n, for n a given input integer and k a given positive integer. We are only interested in finding a single solution. A simple example for k=2 is x2 + y2 = 113, and one solution is {x=8, y=7}. For the case when k=4, Euler (almost), Lagrange, and Jacobi proved that a solution always exists if n≥0, but the proof leads to an exponential time algorithm. We would like to improve this to find a polynomial time algorithm. Thursday, January 27, 2005UBC Campus, Room WMAX 216 3:00-3:30 Peter Borwein (SFU) A theorem of Duffin and Shaeffer and related stuff Abstract: A function a0 + a1z + ... + anzn + ... , where the an take only a finite number of different values, is a rational function if it is bounded in a sector of the unit circle. 3:30-3:50 Peter Borwein (SFU) Two problems of Smale that imply a version of P not equal to NP Abstract: These are from a paper of Smale entitled "Mathematical problems for the next century." They are problem 4. One concerns the number of distinct integer zeros of polynomials. The other concerns computing factorials. 3:50-4:10 tea break 4:10-5:00 Patrick Ingram (UBC) Integral points on elliptic curves Abstract: It is a well-known result of Siegel that any elliptic Diophantine equation has at most finitely many solutions in integers. We will examine techniques for uniformly bounding the number of such solutions across certain families of equations, as well as exploring the related question of the divisibility properties of denominators of rational points on elliptic curves. Thursday, February 10, 2005SFU Campus, Room K9509 3:00-3:50 Ron Ferguson (PIMS) The merit factor problem for binary sequences Abstract: Two problem which have received considerable attention from both mathematicians and communications engineers regarding binary sequences are: Do there exist Barker sequences of length > 13? Is the merit factor bounded? I will discuss both theoretical and algorithmic approaches to these problems, including our latest results obtained using methods of stochastic optimization. (joint work with Peter Borwein and Josh Knauer) 3:50-4:10 tea break 4:10-5:00 Nike Vatsal (UBC) Elliptic curves and modular forms This is an expository talk. Thursday, March 3, 2005UBC Campus, Room WMAX 110 (please note alternate room) 3:00-3:50 Brian Conrad (University of Michigan) Irreducible specialization in characteristic 2 Abstract: A few years ago, in joint work with K. Conrad and R. Gross, it was shown that the function-field analogue of classical heuristics on prime specialization of irreducible polynomials over Z are false, due to a new phenomenon ("Mobius bias") unlike anything known in characteristic 0; it was also shown that this Mobius bias permits a plausible correction for the conjecture in the function field case. Though those results (and the associated conjecture) have been extended to higher genus at the expense of using much heavier amounts of algebraic and rigid geometry, in the case of characteristic 2 there remain some basic vexing questions that can be illustrated by very concrete examples. In this talk, we present such concrete examples and discuss some related theorems (proved jointly with K. Conrad and R. Gross), thereby motivating the formulation of some open problems that the speaker has no clue how to solve. 3:50-4:10 tea break 4:10-5:00 Bill Casselman (UBC) How (not) to teach an introductory course in automorphic forms Thursday, March 17, 2005SFU Campus, Room K9509 3:00-3:50 Michael Rubinstein (University of Waterloo) Elliptic curves and random matrix theory Abstract: I'll discuss, in the context of random matrix theory, the value distribution of L-functions associated to the quadratic twists of an elliptic curve. 3:50-4:10 tea break 4:10-5:00 Chandrashekhar Khare (University of Utah) Serre's conjectures on mod p Galois representations Abstract: I will report on recent progress on Serre's modularity conjecture for 2-dimensional mod p Galois representations. The work reported on is joint work with J-P. Wintenberger, and subsequent work of myself. Wednesday, March 23, 2005UBC Campus, Room WMAX 110 (please note alternate room) 3:00-3:50 Carl Pomerance (Dartmouth College) Periods of pseudorandom number generators Abstract: This talk will consider two common pseudorandom number generators based in number theory. The first, due to D.H. Lehmer, is the linear congruential generator, where the (n+1)st iterate x(n+1) is ax(n)+b (mod m) (where a, b, m, and an initial seed x(0) are given). This generator is commonly used in numerical analysis for Monte Carlo simulations. The other is the power generator x(n+1) = x(n)a (mod m) where a, m, and x(0) are given. This generator has cryptographic applications. Among other results, we have that for any nontrivial choice of parameters a, b, and x(0), the linear congruential generator has period m/exp{(1+o(1))loglog m logloglog m} for almost all m, while the power generator has period m/exp{(1+o(1))(loglog m)2 logloglog m} for almost all m. These results, which are conditional on the Generalized Riemann Hypothesis, are joint with Par Kurlberg, Shuguang Li, and Greg Martin. 3:50-4:10 tea break 4:10-5:00 Alexa van der Waall (University of Sydney) Factorisation of differential operators over Laurent-series rings Abstract: Linear differential operators over a rational function field can be factored, just like polynomials over number fields. The main differences between these two, however, is that the multiplication of differential operators is non-commutative and that there might be infinitely many factorisations. There are a few algorithms known for determining a class of factorisations. The one that currently is being implemented in MAGMA uses factorisations over Laurent series rings. These can be used later for the factorisation over rational fields. In this talk I plan to talk about the series implementations and the ideas behind them, and mention some similarities with the number theory case. Thursday, April 7, 2005SFU Campus, Room K9509 3:00-3:50 A. Speaker to be announced Abstract: to be announced 3:50-4:10 tea break 4:10-5:00 A. Noether-Speaker to be announced Abstract: to be announced