# The many aspects of Mahler's measure (03w5035)

Arriving in Banff, Alberta Saturday, April 26 and departing Thursday May 1, 2003

## Organizers

David Boyd (University of British Columbia)

Christopher Deninger (University of Muenster)

Douglas Lind (University of Washington)

Fernando Rodriguez Villegas (University of Texas at Austin)

## Objectives

The purpose of the workshop will be to explore the many apparently
different ways in which Mahler's measure appears in different
areas of Mathematics. For this purpose, we are bringing
together experts specializing in many different fields:
dynamical systems, K-theory, number theory and topology,
with the hope that common threads will emerge from the
interaction between the participants.

The logarithm of Mahler's measure for one variable polynomials is a quantity that occurs naturally in many problems as an entropy or grown rate. For example, it occured in Lehmer's investigation of certain cyclotomic functions and led him to ask his famous question now known as Lehmer's conjecture. In spite of fundamental work by Smyth, Dobrowolski and others on this conjecture, it has not yet been proved.

Mahler's measure for polynomials in many variables was introduced by him as a device to provide a simple proof of Gelfond's inequality for the product of polynomials in many variables. It has turned out to have a much more fundamental signifigance. The starting point was the proof in the late 1970's that the Mahler measure of a several variable polynomial is the limit of the Mahler measure of one variable polynomials.

The limit theorem was used by Lind, Schmidt and Ward in their proof that the logarithmic Mahler measure is the entropy of a Z^d action. Recently Deninger has shown that the Mahler measure of a many variable polynomial is related to Beilinson's higher regulators. Since then Boyd, Rodriguez-Villegas and others have explored this connection with K-theory. Some of these results can be regarded as verifications of conjectures of Kontsevich and Zagier in their general theory of periods.

In another direction, connections have been found between the Mahler measure of certain two variable polynomials called A-polynomials and invariants of hyperbolic 3-manifolds such as the volume and Borel regulator. There is an intriguing conjecture of Chinburg about realizing special values of Dirichlet L-functions as logarithmic Mahler measures that seems to have a close connection to the study of the A-polynomials of arithmetic hyperbolic manifolds.

We hope that most participants will give a lecture on recent work. Since the main purpose of the workshop is the interaction between individuals in different fields, we hope that the lecturers will not assume too much specialized background but will pitch their lectures to a general educated mathematical audience. In order to facilitate communication, there will be ample time for discussion which we hope will be enhanced by the beautiful setting of the workshop.

The logarithm of Mahler's measure for one variable polynomials is a quantity that occurs naturally in many problems as an entropy or grown rate. For example, it occured in Lehmer's investigation of certain cyclotomic functions and led him to ask his famous question now known as Lehmer's conjecture. In spite of fundamental work by Smyth, Dobrowolski and others on this conjecture, it has not yet been proved.

Mahler's measure for polynomials in many variables was introduced by him as a device to provide a simple proof of Gelfond's inequality for the product of polynomials in many variables. It has turned out to have a much more fundamental signifigance. The starting point was the proof in the late 1970's that the Mahler measure of a several variable polynomial is the limit of the Mahler measure of one variable polynomials.

The limit theorem was used by Lind, Schmidt and Ward in their proof that the logarithmic Mahler measure is the entropy of a Z^d action. Recently Deninger has shown that the Mahler measure of a many variable polynomial is related to Beilinson's higher regulators. Since then Boyd, Rodriguez-Villegas and others have explored this connection with K-theory. Some of these results can be regarded as verifications of conjectures of Kontsevich and Zagier in their general theory of periods.

In another direction, connections have been found between the Mahler measure of certain two variable polynomials called A-polynomials and invariants of hyperbolic 3-manifolds such as the volume and Borel regulator. There is an intriguing conjecture of Chinburg about realizing special values of Dirichlet L-functions as logarithmic Mahler measures that seems to have a close connection to the study of the A-polynomials of arithmetic hyperbolic manifolds.

We hope that most participants will give a lecture on recent work. Since the main purpose of the workshop is the interaction between individuals in different fields, we hope that the lecturers will not assume too much specialized background but will pitch their lectures to a general educated mathematical audience. In order to facilitate communication, there will be ample time for discussion which we hope will be enhanced by the beautiful setting of the workshop.