Departmental Colloquium
Ravi Vakil
Stanford University
Generalizing the cross ratio: The moduli space of n points on the projective line up to projective equivalence
Abstract: Four ordered points on the projective line, up to projective
equivalence, are classified by the cross ratio, a notion introduced by
Cayley. This theory can be extended to more points, leading to one of
the first important examples of an invariant theory problem, studied
by Kempe, Hilbert, and others. Instead of the cross ratio (a point on
the projective line), we get a point in a larger projective space, and
the equations necessarily satisfied by such points exhibit classical
combinatorial and geometric structure. For example, the case of six
points is intimately connected to the outer automorphism of $S_6$. We
extend this picture to an arbitrary number of points, completely
describing the equations of the moduli space. This is joint work with
Ben Howard, John Millson, and Andrew Snowden. This talk is intended
for a general mathematical audience, and much of the talk will be
spent discussing the problem, and an elementary graphical means of
understanding it.
Friday April 2, 2010 at 3:00 PM in SEO 636