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SUMMARY:Bounding Betti numbers of real hypersurfaces near the tropical lim
 it - Kristin Shaw\, University of Oslo
DTSTART:20191106T141500Z
DTEND:20191106T151500Z
UID:TALK125794@talks.cam.ac.uk
CONTACT:Dhruv Ranganathan
DESCRIPTION:Almost 150 years ago Harnack proved a tight upper bound on the
  number of connected components of a real planar algebraic curve of degree
  d. However\, in higher dimensions we know very little about the topology 
 of real algebraic hypersurfaces. For example\, we do not know the maximal 
 number of connected components of real quintic surfaces in projective spac
 e. \n\nIn this talk I will explain the proof of a conjecture of Itenberg w
 hich\, for a particular class of real algebraic projective hypersurfaces\,
  bounds all Betti numbers\, not only the number of connected components\, 
 in terms of the Hodge numbers of the complexification. The real hypersurfa
 ces we consider arise from Viro’s patchworking construction\, which is a
 n effective and combinatorial method for constructing topological types of
  real algebraic varieties. Today these real hypersurfaces can be thought o
 f as near the tropical limit. To prove the bounds conjectured by Itenberg 
 we develop a real analogue of tropical homology and use a spectral sequenc
 e to relate these groups to tropical homology and their dimensions to Hodg
 e numbers. Lurking in the spectral sequence of the proof are the keys to h
 aving combinatorial control of the topology of the real hypersurfaces near
  the tropical limit in any toric variety.\n\nThis is joint work with Arthu
 r Renaudineau.
LOCATION:CMS MR13
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