June 29, 2016
The Fields Institute, Toronto, Canada
Monday, September 30, 2013
7:00 – 9:00 p.m. at the Isabel Badel Theatre, 93 Charles St W, Toronto

The 2013 Fields Medal Symposium is centred on the work of Elon Lindenstrauss (Hebrew University), and its current and potential impact. The goal is to bring Lindenstrauss' general area of study to a broader audience.

The public opening will include remarks by The Honourable Brad Duguid, Minister of Training, Colleges and Universities, the Honourable Reza Moridi, Minister of Research and Innovation, and Paul Young, VP – Research for the University of Toronto. Following the opening remarks, there will be two public lectures, by Lindenstrauss and Peter Sarnak (Institute for Advanced Study).

Welcome and Opening Remarks

Professor Paul Young, PhD, FRSC, Vice President – Research and Innovation, University of Toronto

The Honourable Reza Moridi, Minister of Research and Innovation

The Honourable Brad Duguid, Minister of Training, Colleges, and Universities

Public Lectures

Randomness in Number Theory
Peter Sarnak, Institute for Advanced Study (slides)

By way of concrete examples we discuss an apparent dichotomy in number theory; that the basic phenomena are either highly structured or very random. The models for randomness for different problems can be quite unexpected and understanding, and establishing the randomness is often the key issue.The ergodic theory of certain dynamical systems has proven to be a powerful tool in this regard. Conversely, the fact that certain number-theoretic quantities behave randomly is a powerful source for the construction of much sought-after pseudo-random objects.

Information, Entropy, and Numbers
Elon Lindenstrauss, Hebrew University

Claude Shannon is considered the founding father of information theory, with a profound impact on today's digital world. Perhaps his greatest insight was understanding the theoretical capacity of a channel of communication and the dual problem of how many bits are needed to encode a random source under an optimal encoding. This remarkable piece of applied mathematics was introduced into the pure mathematical field of ergodic theory (whose origins can be traced to attempts to understand the dynamics of naturally occurring systems such as the solar system) by Kolmogorov and Sinai in the late 50's, dramatically changing the subject.

We will discuss Shannon and Kolmogorov-Sinai entropies, which can be thought of as being measured in the units of bits per unit time, as well as a newer notion called mean dimension measuring parameters per unit time.

It turns out that entropy theory can be a powerful tool in analysing the distribution of integer solutions of certain types of equations, i.e. to number theory. For instance, work of the mathematician Linnik on the distribution of integer points on spheres in three dimensions can be recast in these terms. We also (briefly) discuss recent work in this direction.

Thus we see here a story of an applied mathematical idea being used as a decisive tool to study a question in pure mathematics - namely in number theory. It is amusing to note that results in number theory, which were originally thought of as being as theoretical as could be, have found many real-life applications — indeed, to Shannon's favorite topic of digital communication — but this fascinating story will have to be told elsewhere.

Gold Level Sponsor

Silver Level Sponsor

James Stewart, Prof Emeritus, McMaster University; textbook author, donor of the Fields Institute Library
Bronze Level Sponsors

Edward Bierstone, Fields Institute and the University of Toronto
George Elliott, Fields Institute and the Unviersity of Toronto
John R. Gardner
Dan Rosen, R2 Financial Technologies
Philip Siller, BroadRiver Asset Management, L.P.