The Elementary Particles of Arithmetic

  • Date: 11/06/2008

Dr. Clifton Cunningham, PIMS Calgary Site Director, Mathematics & Statistics, University of Calgary


Calgary Place Tower (Shell)


The Large Hadron Collider is gathering steam at CERN and
beginning its search for Higgs boson ---
one of the most important particles in the Standard Model. Although the scale
of the experimental apparatus is enormous, physicists are actually testing a
very precise prediction, which, if confirmed, will appear as specific
resonances in a newly constructed linear accelerator. Where did these
predictions come from? How did it come to be that physicists have so much
information about something they have not observed?



The answer goes back to an idea developed in the late 1950s and early 1960s by
Sakata, Gell-Mann, NĂ©eman, Zweig and others. In its ultimate expression, the
idea was to use a branch of mathematics known as Group Representation Theory to
find some order in the zoo of elementary particles already detected at that
time. The result was the famous quark theory of hadrons, known affectionately
as the Eightfold Way of Gell-Mann. In a spectacular confirmation of this idea,
Group Representation Theory also predicted the existence (and masses and other
properties) of other elementary particles which were later confirmed. Since
this time, the tools used to classify and predict the existence and properties
of elementary particles have developed significantly, but Group Representation
Theory remains a vital mathematical tool in this endeavour.



At the same time, number theorists were grappling with a seemingly unrelated
problem --- that of understanding a class of functions known as L-functions.
The simplest L-function is the celebrated Riemann Zeta function. In 1968 a
Canadian mathematician named Robert Langlands proposed using Group
Representation Theory to find some order in the zoo of L-functions, and the
result is a theory known as the Langlands Programme. At the heart of the
Langlands Programme is the notion of automorphic representations, which may
also be viewed as a generalization of the notion of modular forms. Forty years
later, automorphic representations, and their associated L-functions, are now
central objects in number theory. Indeed, the proof of Fermat's Last Theorem,
which made front-page news in the New York Times in 1990s, is best understood
from the perspective of the Langlands Programme. L-functions also play a role
in modern cryptography.



In this talk I will explain why I think of automorphic representations as the elementary
particles of number theory, and even suggest that the analogy may have something
important to tell us about the nature of our universe.


Calgary Place Tower I (330 5th Avenue SW), Room 1104/1106.

Everyone is welcome to attend. A light lunch will be provided

Other Information: 

The Pacific Institute for the Mathematical Sciences is grateful for the support of Shell Canada Limited, Alberta Advanced Education and Technology, and the University of Calgary for their support of this series of lectures.