I plan to report on a joint paper with Jakob Stix concerning the p-adic section for curves. I will formulate Grothendieck's section conjecture and its p-adic variant, as well as its valuation theoretical version. Finally I will present a new result, which reduces the p-adic section conjecture for curves to a completely local valuation theoretic problem.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/node/246