Departmental Colloquium
János Kollár
Princeton University
Cremona transformations and homeomorphisms of topological surfaces.
Abstract: The simplest Cremona transformation of projective 3-space
is the involution
$\sigma:(x_0:x_1:x_2:x_3)\mapsto
\left(\frac1{x_0}:\frac1{x_1}:\frac1{x_2}:\frac1{x_3}\right),$
which is a homeomorphism outside the "coordinate tetrahedron"
$(x_0x_1x_2x_3=0)$.
By studying the action of $\sigma$ on real quadric surfaces,
we show that $\sigma$ and its conjugates generate a dense subgroup
of $Homeo(S^2)$, the group of homeomorphisms
of the 2-sphere.
Then we show that the same holds if the 2-sphere is replaced
by the torus or by any non-orientable surface
and explain why there can not be similar results for
orientable surfaces of genus $\geq 2$.
(Joint work with Frédéric Mangolte.)
Friday October 3, 2008 at 3:00 PM in SEO 636