  The difficulties of detecting association, measuring correlation, and
establishing cause and effect have fascinated mankind since time immemorial.
Democritus, the Greek philosopher, underscored well the importance and the
difficulty of proving causality when he wrote, "I would rather discover one
cause than gain the kingdom of Persia." To address the difficulties of relating
cause and effect, statisticians have developed many inferential techniques.
Perhaps the most well-known method stems from Karl Pearson's coefficient of
correlation, which Pearson introduced in the late 19th century based on ideas
of Francis Galton.
  I will describe in this lecture the recently-devised distance correlation
coefficient and describe its advantages over the Pearson and other classical
measures of correlation. We will examine an application of the distance
correlation coefficient to data drawn from large astrophysical databases, where
it is desired to classify galaxies according to various types. Further, the
lecture will analyze data arising in the ongoing national discussion of the
relationship between state-by-state homicide rates and the stringency of state
laws governing firearm ownership.
  The lecture will also describe a remarkable singular integral which lies at
the core of the theory of the distance correlation coefficient. We will see
that this singular integral admits generalizations to the truncated Maclaurin
expansions of the cosine function and to the theory of spherical functions on
symmetric cones.
