Emerging Applications of Number Theory

IMA Program for July 1996

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:

• (a) wave phenomena in quantum mechanics (so-called quantum chaos);
  

• (b) graph theory (especially expander graphs).

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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