The future of computation in algebraic geometry

   The ability to represent and manipulate objects in algebraic
geometry on a computer has had a great effect on the subject.
It has facilitated research by enabling the study of more intricate
examples.  It has altered the course of research by opening up 
new areas of study.  One such new area is in applications of 
algebraic geometry in which computed examples play a major role.
This computation has typically been symbolic.  However, current 
and likely future trends in computer technology, specifically
increased parallelism, will lead to a diminished role for symbolic
computation, which is not parallelizable, and an increased role
for parallelizable numerical computation.

   My talk will discuss computation in algebraic geometry,
both symbolic and numeric.  After an overview which includes the
role of parallelism in symbolic and numeric computation for 
algebraic geometry, I will discuss some of the basics of 
numerical computation in algebraic geometry.  The talk will 
conclude with some recent work that is making the numerical 
future for computation in algebraic geometry possible.