|Monday, September 30, 2013
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
The Honourable Brad Duguid, Minister of Training,
Colleges, and Universities
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.