### A 2-nilpotent real section conjecture

- Series
- Algebra Seminar
- Time
- Thursday, March 17, 2011 - 15:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Kirsten Wickelgren – Harvard University

Grothendieck's anabelian conjectures say that hyperbolic curves over
certain fields should be K(pi,1)'s in algebraic geometry. It follows
that points on such a curve are conjecturally the sections of etale pi_1
of the structure map. These conjectures are analogous to equivalences
between fixed points and homotopy fixed points of Galois actions on
related topological spaces. This talk will start with an introduction to
Grothendieck's anabelian conjectures, and then present a 2-nilpotent
real section conjecture: for a smooth curve X over R with negative Euler
characteristic, pi_0(X(R)) is determined by the maximal 2-nilpotent
quotient of the fundamental group with its Galois action, as the kernel
of an obstruction of Jordan Ellenberg. This implies that the set of real
points equipped with a real tangent direction of the smooth
compactification of X is determined by the maximal 2-nilpotent quotient
of Gal(C(X)) with its Gal(R) action, showing a 2-nilpotent birational
real section conjecture.