Number Theory at UBC

SFU/UBC Number Theory Seminar
Winter Semester 2003


Monday, January 6, 2003
UBC Campus, PIMS Seminar Room (West Mall Annex 216)
2:00-2:50
David Savitt (McGill/IHES)
Strongly divisible lattices and (almost) explicit Galois deformation rings

Abstract: In this talk, I will explain how C. Breuil's work on integral p-adic Hodge theory (strongly divisible lattices) may be used to compute certain deformation rings of potentially semistable Galois representations, and I will describe some expected consequences for the modularity of such representations.


Thursday, January 30, 2003
UBC Campus, MATX 1102 (Math Annex)
4:15-5:05
Nike Vatsal (UBC)
Ostrowski's Theorem

Abstract: Ostrowski proved in 1917 that any “reasonable” norm on the rational numbers is equivalent either to the usual absolute value or to one of the p-adic norms.  In this talk we give an elementary exposition of the proof of this result.


Thursday, February 13, 2003
SFU Campus, Room K9509 (Shrum Science Building)
4:00-4:50
Sheldon Yang (Canadian College for Chinese Studies, Victoria)
Transformation of Four Titchmarsh Infinite Integrals

Abstract: This talk relates succintly how four Titchmarsh-type infinite integrals involving the Riemann Ξ-function can be transformed correspondingly into four infinite integrals involving Hardy's Z-function all resulting with a unified common factor π5/4.
4:50-5:10
tea break
5:10-6:00
Michael Bennett (UBC)
Can the product of consecutive terms in an arithmetic progression be a perfect power?

Abstract: An old question of Erdős, restated by Darmon and Granville, concerns the possible occurences of powers as products of terms in arithmetic progression. We will discuss this problem and see what can be said about it in the post-Fermat era.


Thursday, February 27, 2003
UBC Campus, MATX 1102 (Math Annex)
4:00-4:50
Greg Martin (UBC)
Prime number races, part I
4:50-5:10
tea break
5:10-6:00
Greg Martin (UBC)
Prime number races, part II

Abstract: The prime number theorem for arithmetic progressions says that all of the “reasonable” arithmetic progressions modulo q (where “reasonable” means relatively prime to q) contain asymptotically the same number of primes.  For example, the ratio between ψ(x;4,1), the number of primes less than x that are congruent to 1 mod 4, and ψ(x;4,3), the number of primes less than x that are congruent to 1 mod 4, approaches 1 as x tends to infinity.  However, try finding an x for which ψ(x;4,1) > ψ(x;4,3), and you'll see there's more to the story!  In general, a given difference ψ(x;q,a)–ψ(x;q,b) is often biased towards positive values or towards negative values.  Recently, we have gained a great deal of understanding (at least modulo some high-powered conjectures in analytic number theory) regarding when these biases occur and how strong they are in some suitable quantitative sense.  There are also related phenomena involving three or more of the counting functions ψ(x;q,a) at the same time.  In part I of the talk, we give an exposition of these phenomena and discuss some of the recent discoveries; part II of the talk will be more technical and detailed, as the methods of proof of these recent results are described.


Thursday, March 13, 2003
SFU Campus, Room K9509 (Shrum Science Building)
4:00-4:50
Peter Borwein (SFU)
The Mahler measure of polynomials with odd coefficients

Abstract: We prove that the minimum value of the Mahler measure of a monic nonreciprocal polynomial with all odd coefficients is the golden ratio.  We also determine the smallest measures among reciprocal polynomials with coefficients ±1 and degree at most 72.
4:50-5:10
tea break
5:10-6:00
Stephen Choi (SFU)
On Planar Functions

Abstract: A function f : ZN → ZN is called planar if f(k+t) - f(t) is a bijection in t for any k = 1, 2, ..., N-1.  It is conjectured that a planar function from ZN into ZN exists if and only if N is a prime.  Some partial and conditional results will be discussed in the talk.


Thursday, March 27, 2003
UBC Campus, MATX 1102 (Math Annex)
4:00-4:50
Hiro Goto (Hokkaido University of Education, Hakodate)
K3 surfaces in positive characteristic

Abstract: There are various problems about K3 surfaces existing particularly over the fields of positive characteristic. For instance, we find supersingular K3 surfaces and unirational K3 surfaces which do not exist in characteristic zero. We discuss a conjecture (namely, the Artin-Shioda conjecture) about these K3 surfaces and give some numerical questions related to their geometric invariants.
4:50-5:10
tea break
5:10-6:00
Kevin Hare (University of California, Berkeley)
Generalized Gorshkov-Wirsing polynomials and the Integer Chebyshev Problem

Abstract: The Integer Chebyshev Problem is the problem of finding an integer polynomial of degree n such that the supremum norm on [0,1] is minimized.  Current techniques used to determine possible upper bounds for the limiting case involves heavy computation with algorithms such as LLL and the Simplex method.  The only known technique to determine a lower bound for the limiting case is by use of a sequence of polynomials called the Gorshkov-Wirsing polynomials.

This talk will discuss some of the property of the Gorshkov-Wirsing polynomials.  We will show how to construct generalized Gorshkov-Wirsing polynomials on any interval [a,b].  Extensive searches have been done using these Generalized Gorshkov-Wirsing polynomials for various intervals.  The results of these searches will be discussed.