The classical theory of analytic functions of one complex
variable is a beautiful tapestry woven by the hands of such
giants of nineteenth-century mathematics as Cauchy, Riemann, and
Weierstrass. The subject has applications in many branches of
mathematics and engineering. Complex analysis in higher
dimensions is a younger field of study, one developed during the
twentieth century. It interacts with algebraic and differential
geometry, harmonic analysis, and partial differential equations,
and it has found applications in mathematical physics and control
theory. The multi-dimensional theory of analytic functions
differs from the one-dimensional theory in many fascinating ways:
for example, holomorphic functions of several variables never
have isolated singularities, there is no Riemann mapping theorem
in higher dimensions, and the Cauchy-Riemann equations in several
variables form an over-determined system of partial differential
equations. In my research, I am particularly interested in how
the shape of a domain in multi-dimensional complex space
influences the function theory on the domain.
