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SUMMARY:H_{4g-6}(M_g) - Soren Galatius (University of Copenhagen)
DTSTART:20181203T100000Z
DTEND:20181203T110000Z
UID:TALK115240@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:The set of isomorphism classes of genus g Riemann surfaces car
 ries a natural topology in which it may be locally parametrized by 3g-3 co
 mplex parameters.  The resulting space is denoted M_g\, the moduli space o
 f Riemann surfaces\, and is more precisely a complex orbifold of that dime
 nsion.  The study of this space has a very long history involving many are
 as of mathematics\, including algebraic geometry\, group theory\, and stab
 le homotopy theory.  The space M_g is not compact\, essentially because a 
 family of Riemann surface may degenerate into a non-smooth object\, and ma
 y be compactified in several interesting ways.  I will discuss a compactif
 ication due to Harvey\, which looks like a compact real (6g-6)-dimensional
  manifold with corners\, except for orbifold singularities.  The combinato
 rics of the corner strata in this compactification may be encoded using gr
 aphs.  Using this compactification\, I will explain how to define a chain 
 map from Kontsevich&#39\;s graph complex to a chain complex calculating th
 e rational homology of M_g.  The construction is particularly interesting 
 in degree 4g-6\, where our methods give rise to many non-zero classes in H
 _{4g-6}(M_g)\, contradicting some predictions.  This is joint work with Ch
 an and Payne (arXiv:1805.10186).
LOCATION:Seminar Room 1\, Newton Institute
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