David W. Boyd
 Position: Professor Emeritus
 Background: B.Sc., Carleton University (1963)
 M.A., University of Toronto (1964)
 Ph.D., University of Toronto (1966)
 Fellow of the Royal Society of Canada
 Fellow of the American Mathematical Society
 Email Address: boyd at math dot ubc dot ca
 Department Telephone: (604) 8222666; Fax: (604) 8226074
Research Interests
Analysis, Number Theory, Hyperbolic Geometry, Mathematical Computation
Research in Progress
My research has concentrated on applying the methods of
classical analysis to the study of discrete phenomena, particularly in number theory
and geometry. The computer has always played an important role in my research.
Recently I have been studying the question of explicit formulas for
Mahler's measure of many variable polynomials. Mahler's measure m(P)
for a polynomial P in n variables
is the average of the logarithm of the absolute value of P over the product
of n unit circles. This quantity measures the entropy of certain
dynamical systems and occurs naturally in many situations. Recently
experiment has discovered many cases in which m(P) is apparently expressible
as a rational multiple of special values of certain Lfunctions, either
Dirichlet Lfunctions or Lfunctions of elliptic curves. Some of these
formulas can be proved but most have only been numerically verified to
many (e.g. 50) decimal places. The phenomenon is connected with the
conjectures of Bloch and Beilinson from Ktheory.
Recently, investigations have centered on certain polynomials called
Apolynomials that can be computed from the ideal triangulation of a
hyperbolic manifold. The Mahler measure of these polynomials can
always be written as a sum of dilogarithms of algebraic numbers related
to pseudotriangulations of the manifold (by tetrahedra that may have
negative or zero volume). In certain cases this gives a relation
between the Mahler measure and the volume of the manifold.
In these cases the Mahler measure can be expressed in terms of the
value at s = 2 of the zeta function of a number field called the
invariant trace field of the manifold.
Another project which combines number theory and computation is the
search from polynomials whose coefficients are all either +1 or 1
and which vanish to high order at the point x = 1. Such polynomials
are used in the design of notch filters in antenna theory. We have
determined the smallest degree possible for each order of vanishing
up to m = 7 and lower bounds on the degree in terms of m for larger
values of m. Recently Berend and Golan have extended this to m = 8.
Selected Publications on Mahler's Measure
 Speculations concerning the range of Mahler's measure,
(CoxeterJames Prize lecture)
Canadian Mathematical Bulletin 24 (1981), 453469.

Mahler's measure and special values of Lfunctions ,
Experimental Mathematics, 3 (1998), 3782.
 Mahler's measure and
the dilogarithm (I), with F. Rodriguez Villegas,
Canadian Journal of
Mathematics 54 (2002), 468492.

Mahler's measures and the dilogarithm (II), with F. Rodriguez Villegas
and N. M. Dunfield,
posted to the Mathematics ArXiv, August 5, 2003,
 Mahler's measure and invariants of hyperbolic
manifolds, in Number Theory for the Millennium
(M.A. Bennett et al., eds.), A K Peters, Boston, 2002, 127143.

Mahler's measure, hyperbolic manifolds and the
dilogarithm , (JefferyWilliams Prize lecture)
Canadian Mathematical Society Notes, 34.2 (2002) 34 & 2628.
(Note: the typesetting of this paper introduced typos into most
of the formulas starting at displayed
formula (15).
Most two digit numbers should be read as fractions! e.g.
in 16d_{19} + 16d_3, the 16's should be read as 1/6.)

Small Limit Points of Mahler's Measure , with M. J. Mossinghoff,
Experimental Mathematics, 14 (2005), no.4, 403414.

Explicit formulas for Mahler's measure , Bulletin CRM, Autumn
2005, 1415 (CRMFields Prize lecture).
Slides from some lectures

Computing Apolynomials using Puiseux expansions
Slides from a lecture at CNTA VIII, Toronto, June 24, 2004 (.5 Mbytes)

The Apolynomials of families of symmetric knots,
Slides from a lecture at Knots and Manifolds, Vancouver,
July 23, 2004
(1.0 Mbytes)

Mahler's measure and Lfunctions of elliptic curves evaluated at s=3,
Slides from a lecture at the SFU/UBC number theory seminar, December 7, 2006 (250 Kbytes)
Selected Publications on vanishing of Littlewood polynomials at x = 1

On a problem of Byrnes concerning polynomials with
restricted coefficients ,
Mathematics of Computation 66 (1997), 16971703.

On a problem of Byrnes concerning polynomials
with restricted coefficients, II ,
Mathematics of Computation ,
71 (2002), 12051217.

Littlewood polynomials with high order zeros ,
D. Berend & S. Golan,
Mathematics of Computation , 75 (2006), 15411552.
BIRS Workshops

The Many Aspects of Mahler's Measure , April 26  May 1, 2003.

Explicit Methods in Number Theory , November 13  18, 2004.
 Number Theory Inspired by Cryptography , November 5  10, 2005.
 Analytic Methods for Diophantine Equations , May 13  18, 2006.
 Lowdimensional topology and number theory , October 21  26, 2007.
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