ALBERTA NUMBER THEORY DAY


UNIVERSITY OF CALGARY --- 30 APRIL, 2009


A one-day regional conference featuring five invited lectures with ample opportunity for interaction, discussion and socialization between participants.


Graduate students are especially encouraged to attend. Funding is available!


Poster: pdf


Organizers: Matthew Greenberg, Renate Scheidler, Andrew Shallue


Contact (registration or information): mgreenbe@ucalgary.ca or (403) 220 3952


Registration Fee: $20 (includes coffee, lunch and banquet)


Invited speakers:

Paul Buckingham (Alberta)

Laurent Imbert (Calgary, CNRS)

Nathan Ng (Lethbridge)

Lior Silberman (British Columbia)

Kaneenika Sinha (Alberta, PIMS)




Program:

Time

Activity

9:00-9:30

Registration

9:30-10:30

Invited lecture: Nathan Ng

Title: Moments of the Riemann zeta function

In this talk I will discuss the problem of evaluating integral and discrete moments of the Riemann zeta function. In 1918 Hardy and Littlewood introduced the integral moments in order to study the Lindelof hypothesis. In the 1980's Gonek and Hejhal independently introduced discrete moments in which the zeta function is averaged over its zeros. These discrete variants are important for various number theoretic applications such as finding simple zeros of the zeta function or finding small gaps between the zeros of the zeta function. I will discuss some of the different techniques for evaluating these moments and the challenges we face in evaluating high moments in both cases.

10:30-10:50

Coffee

10:50-11:50

Invited lecture: Laurent Imbert

Title: Strictly chained (p,q)-ary partitions

A partition of an integer n is a non-increasing sequence of positive integers, called parts, summing up to n, possibly subject to one or more constraints. For instance, one may want the parts to be distinct, odd, prime, powers of some integer, etc. Strictly chained partitions are finite sequences of integers that decrease for the divisibility order. In other words, partitions of the form n = a1 + a2 + ... + ak into distinct positive integers a1, ..., ak such that ak | ak-1 | ... | a1. If p(n) denotes the number of strictly chained partitions of n, Erdös and Loxton proved that the sum function P(x) = ∑1n≤xp(n) behaves like cxρ, where c is an unknown constant and ρ is the unique root of ζ(ρ) = 2, where ζ is the Riemann zeta function.

In this talk, we discuss a special case of this type of partitions. We consider strictly chained (p,q)-ary partitions, i.e., partitions with distinct parts of the form paqb, with the further constraint that each part is a multiple of the following one. For sake of simplicity, we further assume that p and q are relatively prime integers greater than 1. This work arose from recent developments on so-called double-base chains, in particular their use in speeding-up exponentiation in various groups. We investigate several problems including those of generating, encoding and counting these partitions. We also give some results on the length (the number of parts) of the shortest partitions of that type.  The special case min(p,q)=2 will be given a special attention, in particular the case (p,q)=(2,3).

11:50-13:00

Conference lunch

13:00-14:00

Invited lecture: Lior Silberman

Title: Equidistribution of eigenfunctions on locally symmetric spaces

The Arithemtic Quantum Unique Ergodicity Conjecture is a statement about the analytic behaviour of Hecke eigenforms on the modular surface and its covers.  It is now known for both Hecke-Maass forms (in the eigenvalue aspects) and holomorphic newform (in the weight aspect).

I explain the equidistribution problem of automorphic forms, and discuss higher-dimensional cases.  Some progress can be made using analysis and ergodic theory alone, while some results require using the Hecke operators.

14:00-14:10

Break

14:10-15:10

Invited lecture: Paul Buckingham

Title: On the Fitting ideals of class-groups of global function fields

Global function fields are geometric analogues of number fields, and can often be studied in the same contexts as number fields. In particular, each function field has a class-group, built from the points of a curve canonically associated to the field. We describe a construction which gives algebraic information about the class-group in analytic terms, along the lines of an analytic class number formula but taking into account the action of a Galois group as well.

15:10-15:30

Coffee

15:30-16:30

Invited lecture: Kaneenika Sinha

Title: Fourier coefficients of certain cusp forms

Modular cusp forms are complex analytic functions on the upper half plane with rich inner symmetry and certain growth conditions. Fourier coefficients of some appropriately chosen modular forms can be interpreted as eigenvalues of some special linear operators defined by Hecke.  We will discuss an important formula that calculates the traces of these operators and use it to study the distribution of Fourier coefficients of newforms.  This has important applications related to arithmetic of elliptic and modular curves.

18:00-20:30

Conference banquet!! (Thai restaurant)


Accomodation:

Quality Inn University

2259 Banff Trail NW, Calgary, AB, T2M 4L2

(403) 289 1973 or (800) 289 1973

Group ID: Alberta Number Theory Day


Participants:

Paul Buckingham (U. Alberta)

Mark Bauer (U. Calgary)

Perlas Caranay (U. Calgary)

Amy Cheung (U. Calgary)

Sarah Chisholm (U. Calgary)

Clifton Cunningham (U. Calgary)

Chuck Doran (U. Alberta)

Hanafi Farahat (U. Calgary)

Brandon Fodden (U. Lethbridge)

Felix Fontein (U. Calgary)

Matthew Greenberg (U. Calgary)

Razieh Hosseini (U. Calgary)

Laurent Imbert (U. Calgary, CNRS)

Thomas Izard (Montpellier)

Michael Jacobson (U. Calgary)

Habiba Kadiri (U. Lethbridge)

Michael Lamoureux (U. Calgary)

Eric Landquist (U. Calgary)

Ahmad Lavasani (U. Calgary)

Greg Martin (UBC)

Matt Musson (U. Calgary)

Nathan Ng (U. Lethbridge)

Jason Nicholson (U. Calgary)

Pieter Rozenhart (U. Calgary)

Renate Scheidler (U. Calgary)

Andrew Shallue (U. Calgary)

Lior Silberman (UBC)

Kaneenika Sinha (U. Alberta)

Arthur Schmidt (U. Calgary)

Adrian Tang (U. Calgary)

Jonathan Webster (U. Calgary)

Colin Weir (U. Calgary)

Ursula Whitcher (U. Washington)

Hugh Williams (U. Calgary)

Kjell Wooding (U. Calgary)


Thanks to our sponsors:

PIMS

iCore

iCore Chair in Algorithmic Number Theory and Cryptography

University of Calgary, Faculty of Science

University of Calgary, Department of Mathematics and Statistics