## Emerging
Applications of Number Theory

## IMA
Program for July 1996

**Organizers:**

*Name* *Present Institution*
Dennis Hejhal University of Minnesota
Joel Friedmann University of British Columbia
Martin Gutzwiller IBM Yorktown Heights
Andrew Odlyzko AT&T Bell Laboratories

Until
comparatively recently, most people tended to view number theory
as the very paradigm of pure mathematics. With the advent of
computers, however, number theory has been finding an increasing
number of applications in practical settings, such as in cryptography,
random number generation, and coding theory. Yet other applications
are still emerging, providing number theorists with some major
new areas of opportunity.

This
workshop will try to foster the further development of some
of these new applications of number theory, focusing in particular
on recent links with:

Though
the central questions of number theory continue to refer to
the most basic of mathematical objects, *viz.* the integers,
the techniques that are used nowadays to attack these questions
frequently come from diverse areas of mathematics as well as
physics and computer science. It is therefore desirable to bring
together number theorists and researchers in these other areas
who have relevant expertise.

This
Summer Program will provide a congenial setting where interested
workers from the respective pure and applied groups can come
together for two weeks to listen to a variety of state-of-the-art
expository lectures, participate in discussions concerning current
and emerging applications of number theory, and make new scientific
contacts of a cross-disciplinary nature.

During
the first week, the program will focus mainly on **quantum
mechanics.** Whereas classical physics was based primarily
on differential equations -- ordinary as well as partial --
quantum physics has drawn from its very beginning on mathematics
of a more discrete type, in particular group theory and algebra.
The relevance of number theory in such a setting is thus not
entirely unexpected. As the phenomenon of chaos in classical
mechanics became better understood, the spirit of number theory
was found to enter physics even at the classical level, *e.g.,*
in the study of critical resonances and the use of symbolic
dynamics. Manifestations of classical chaos are visible not
only in quantum mechanics, but also in a variety of other wave
theories such as optics, electromagnetism, and acoustics.

The
distribution of energy levels and eigenfrequencies, as well
as their relation with the classical periodic orbits (through
the trace formula), has led to new methods in spectroscopy which
are at least partially based on number theory. Prime examples
are particles moving either in a cavity or on a Riemann surface
of negative curvature, and the resultant description using zeta
functions. Applications of these concepts have recently been
made to small switching elements, quantum computers, polymers,
* etc.*

The
second week will see the focus gradually shift over to one of
** graph theory.** Application-wise, graphs give a very good
model for communication networks, both among people and processors.
In many types of analyses, the optimal network often turns out
to be one having properties similar to random networks. One
good example of such a property is "expansion", which guarantees
an absence of "hot spots" in the network. It turns out that
random graphs have a great deal of expansion, but it is a famous
open problem to give explicit constructions of graphs with as
much expansion. A lot of excitement has been generated in recent
years with the discovery of explicit constructions having quite
good expansion; these constructions require number theory and
the spectral theory of graphs to prove that they do, in fact,
have the necessary amount of expansion. These developments also
have ties to the spectral theory of differential operators,
algebraic geometry, and representation theory.

The
graph-theoretic part of the workshop will concentrate on the
use of number theory to construct graphs having desirable features
such as the good expansion property mentioned above. There are
a fair number of ways number theory can be used to produce a
large class of such graphs. To show that these graphs have desirable
properties, one combines the spectral theory of graphs with
number theory to prove that these graphs' eigenvalues are small,
and then checks that any graph with small eigenvalues necessarily
has good behavior. There are a number of relationships between
eigenvalues and graph properties that are not yet well understood,
and some of the lectures will touch on this. In addition, there
are a number of new applications of expanders, including to
coding theory, which will be addressed. Finally, since the properties
of number-theoretical graphs are often mimicked by random graphs,
there will be some presentations devoted to random graphs as
well.

To:
*MATHEMATICAL MODELING FOR INSTRUCTORS AND GRADUATE STUDENTS*:

July 29 - August 16, 1996

To: *APPLICATIONS AND
THEORY OF RANDOM SETS*:

August 22-24, 1996

To:
*MATHEMATICS OF HIGH PERFORMANCE COMPUTING*

September 1996 - June 1997

To:
*EMERGING APPLICATIONS OF DYNAMICAL SYSTEMS*

September 1997 - June 1998

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