Annual Report 2005 - Fields Institute - University of Toronto

outstanding question in this field – that m(P) is bounded
below by c > 0 for noncyclotomic polynomials P with integral
coefficients.
A little later, he showed that the logarithmic Mahler
measure of a polynomial in n variables (its average over
the n-torus) is the limit of the Mahler measures of suitable
polynomials in one variable, at least for certain P – for
example m(1 + x + y ) is the limit of m(1 + x + x n ) as n → ∞.
This was proved later for general P, and made it apparent
that it is natural to look at the larger set of (logarithmic)
Mahler measures m(P) for polynomials of one or more
variables. This led C. Smyth to show that the logarithmic
Mahler measures of certain polynomials of 2 or 3 variables
were given by the values of L-functions – for example that
m(1 + x + y ) = L'(χ,-1) where χ is the quadratic character
of conductor 3. This in turn led C. Deninger (in 1997) to
show that, if P(z 1 , …, z n ) does not vanish on the n-torus,
then m(P) is a Deligne period of the motive associated to
the variety defined by P = 0. At the same time, he conjectured
further connections between Mahler measures and
the values of L-functions, this time of elliptic curves. Boyd
followed this by finding families of (conjectural) formulas
of this general type relating m(P) to L(E,2) for certain P(z 1 ,
z 2 ), where E is a factor of the Jacobian variety of the curve
P = 0.
More recently David has established, in a sequence of
papers (two of which are joint with F. Rodriguez-Villegas),
a fascinating and deep connection between the Mahler
measure of certain polynomials of degree 2 and the values
of the Bloch-Wigner dilogarithm D(z) and of the zeta functions
of number fields, in particular for “A-polynomials”
which arise from one-cusped hyperbolic 3-manifolds.
In this case, as in so many others, David’s pursuit of
the problem through extensive numerical calculations
has led to startling new insights. A fuller description of
Boyd’s work on which this note was based in part may
be found at www.pims.math.ca/Scientific_Programme/
CRM-Fields--PIMS_Prize /David_Boyd:_Recipient_of_
the_2005_CRM-Fields_Prize/
PUBLIC LECTURES
RCI/Fields Institute Public Lectures
Rob Almgren and Agnes Tourin
November 7, 2004
Held at the University of Toronto
The Mathematics of Glider Racing
L e c t u r e s a n d S p e c i a l E v e n t s
Racing glider
For over 150 years, the Royal Canadian Institute for the
Advancement of Science has been sponsoring weekend
public lectures in Science. In recent years, the Fields Institute
has, with the help of Chandler Davis, co-sponsored
one or two talks each year on mathematical themes. On
November 7, 2004, Rob Almgren (Toronto) and Agnes
Tourin (McMaster) together delivered a public lecture on
the Mathematics of Glider Racing.
Almgren, who is also a glider pilot, introduced the elements
of the sport of glider racing over distances of hundreds of
kilometers. A graph of the glider’s vertical elevation would
actually be an irregular, saw-tooth shaped function. The
reason is that the glider descends gradually till it reaches an
updraft (a column of rising air, typically with a cloud sitting
at the top), whereupon it can again gain altitude. Since
the strength and exact location of such updrafts are not
entirely predictable, optimizing the speed of travel turns
out to be a problem in stochastic control theory – a singular
problem in fact. What one controls is the angle of descent
(steeper is faster). For example, it may be preferable to
bypass a weak updraft for a stronger but more distant one,
Fields Institute 2005 ANNUAL REPORT 39