Measuring 
wild ramification using rigid 
geometry -- Kiran Kedlaya, February 29, 2008

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For a finite map between curves over an algebraically closed field of
characteristic p, it is well understood how to measure wild ramification
(Swan conductor) and use it to explain the shortfall in the Riemann-Hurwitz
formula (Grothendieck-Ogg-Shafarevich formula).  However, already for maps
between surfaces this becomes somewhat mysterious. I'll describe a
construction of a "differential Swan conductor" using ideas from the theory
of crystalline/rigid cohomology.  Although we cannot yet provide a global
cohomological interpretation, we can at least prove integrality of the
conductor (Hasse-Arf property), reconcile with an alternate definition due
to Abbes-Saito, and obtain some variational properties (e.g., comparing the
conductors along two transverse divisors with the conductor along the
exceptional divisor of a blowup).  Much of this is joint work with my
student Liang Xiao.