Talk by Mark Gross (UCSD)
Date and Time: Tuesday, November 4, 2008, 12:00 PM in AP&M B412.
Title: Enumerative geometry
Abstract:
Enumerative geometry is a branch of geometry devoted
to counting geometric objects. For example, one could ask:
How many lines are there passing through two points? (Easy, that's one
line.) Or one could ask: given five lines in the plane, one could
ask: how many conic sections are there tangent to all five lines?(Harder, but the answer is still one.) Given a surface defined by
a cubic equation (say x3+y3+z3=1), how many straight lines
are contained in the surface? (This was determined in the mid-19th
century, and the answer is 27.) Even harder, given a three-dimensional
object defined by an equation like x5+y5+z5+w5=1, how
many plane conic sections are contained in this object? (Much harder,
the answer is 609,250.) I will give some examples and techniques,
and explain the history of how the field of enumerative geometry
had a rebirth when string theory started making predictions about
answers to such questions.