## 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.

12:00-1:00pm

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

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

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.