UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

October 19, 2010

939 Evans Hall

4:00PM: Geometric complexity theory

Ketan Mulmuley (University of Chicago)

Geometric complexity theory (GCT) is an appraoch towards the P vs. NP and related problems via algebraic geometry and representation theory. This talk will give a brief introduction to GCT. No background in complexity theory will be assumed. The references for GCT are available on the speakers home page. See also http://front.math.ucdavis.edu/0907.2850.

8:15PM: Galois groups in positive and mixed characteristic

Kiran Kedlaya (University of California, San Diego

It has been known for at least 30 years that there is an explicit isomorphism between the absolute Galois group of the field F_p((T)) and that of the field obtained from the p-adic field Q_p by adjoining all p-power roots of unity. A more precise way to say this is that there is an explicit recipe for passing back and forth between finite field extensions, or more generally finite etale algebras, over the two fields. I'll describe this recipe using Witt vectors (about which no previous knowledge will be assumed); this will also give a more general result of the same form, with the field F_p((T)) replaced by any field of characteristic p which is complete for a multiplicative norm (or even a somewhat more general ring). Such results are motivated by applications to p-adic Hodge theory, particularly the comparison between different cohomology theories attached to varieties over p-adic fields, but I'll say at most a few words about that.

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