Annual Report 2005 - Fields Institute - University of Toronto

Dawson’s title “Stochastic dynamics of evolving populations”
reflects his interest in stochastic processes with both
a spatial and a branching or genealogical component. He
reviewed classical probabilistic models from population
genetics, and their generalization to superprocesses, with
particular attention to the Fleming-Viot infinitely many
alleles model (IMA). Using a representation due to Donnelly
and Kurtz, he showed how the duals of these processes
give rise to coalescents – now a major object of study in and
of themselves.
This led naturally to the principal focus of his talk, namely
the large-time evolution and analysis of spatial structure
in these models. This can be understood from the point
of view of a tagged particle, using coalescent ideas, or in
terms of block averages that describe the behaviour of the
process at multiple scales. The former approach allows
one to decompose the IMA process into immortal super-
Brownian clans, for which one can prove self-similarity
(e.g. using results of Dawson and Greven). On the other
hand, the analysis in terms of block averages in principle
allows one to represent a host of different models as fixed
points of a certain non-linear mapping, including both the
Fleming-Viot and mutually catalytic branching models. He
expressed the hope that similar ideas will succeed for models
of the evolution of interacting species, shedding light on
such topics as coevolution, symbiosis, and speciation.
2005 CRM–FIELDS PRIZE
David W. Boyd
The Centre de recherches mathématiques (CRM) of
l’Université de Montréal, and the Fields Institute jointly
established the CRM–Fields Prize in 1994 with the goal of
recognizing exceptional work in the mathematical sciences.
Recipients are chosen by the Advisory Committee of the
CRM together with the Scientific Advisory Panel of the
Fields Institute. The main selection criterion is outstanding
contribution to the advancement of research.
David Boyd of the University of British Columbia is this
year’s recipient of the CRM–Fields Prize, for his seminal
contributions to analysis, number theory, geometry and
mathematical computation. David has also played a
prominent role in Canadian mathematical life through
his involvement in the CMS, PIMS, and NSERC – serving
in 1977-78 and 2001-02 as chair of the Grant Selection
Committee and on the Steacie Prize Selection Committee
during 1990-92 – and on the boards of both the Centre de
recherches mathématiques and the Fields Institute. He will
L e c t u r e s a n d S p e c i a l E v e n t s
David Boyd
deliver a lecture on his work in Fall 2005
Established in 1994, the CRM–Fields Prize has become the
premier prize in Mathematics in Canada. It is awarded in
recognition of exceptional achievement in the Mathematical
Sciences, for research conducted primarily in Canada
or in affiliation with a Canadian university. Boyd is the last
recipient of the CRM-Fields prize; starting in 2006 it will be
renamed the CRM-Fields-PIMS prize.
One of the principal themes of Boyd’s research has been
Mahler measure. The logarithmic Mahler measure of a polynomial
P(z) with complex coefficients is defined to be the
average of log |P| over the circle,
m(P) = log |P(e(t))| dt
where e(t) = exp(2πit). If P(z) = a 0 (z – α 1 )…(z – α d ) (with
a 0 > 0), it follows from Jensen’s formula that the Mahler
measure
M(P) := exp m(P) = a 0 i max{| α i |, 1}.
The so-called Pisot and Salem numbers are of considerable
interest in the subject because the Mahler measure of their
monic irreducible polynomials is often very small. Boyd, in
a series of papers in the 1970s, single-handedly revitalized
this field. For example in a remarkable paper in the Duke
Mathematics Journal in 1977, he showed that the construction
used by Salem to prove that every Pisot number is
a limit point of Salem numbers, can be used to produce
every Salem number. He then exploited this construction
to determine the smallest Salem numbers known, lending
credence to Lehmer’s conjecture – the most prominent
Fields Institute 2005 ANNUAL REPORT 38