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CATEGORIES:Algebraic Geometry Seminar
SUMMARY:Bounding Betti numbers of real hypersurfaces near
the tropical limit - Kristin Shaw\, University of
Oslo
DTSTART;TZID=Europe/London:20191106T141500
DTEND;TZID=Europe/London:20191106T151500
UID:TALK125794AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/125794
DESCRIPTION:Almost 150 years ago Harnack proved a tight upper
bound on the number of connected components of a r
eal planar algebraic curve of degree d. However\,
in higher dimensions we know very little about the
topology of real algebraic hypersurfaces. For exa
mple\, we do not know the maximal number of connec
ted components of real quintic surfaces in project
ive space. \n\nIn this talk I will explain the pro
of of a conjecture of Itenberg which\, for a parti
cular class of real algebraic projective hypersurf
aces\, bounds all Betti numbers\, not only the num
ber of connected components\, in terms of the Hodg
e numbers of the complexification. The real hypers
urfaces we consider arise from Viro’s patchworking
construction\, which is an effective and combinat
orial method for constructing topological types of
real algebraic varieties. Today these real hypers
urfaces can be thought of as near the tropical lim
it. To prove the bounds conjectured by Itenberg we
develop a real analogue of tropical homology and
use a spectral sequence to relate these groups to
tropical homology and their dimensions to Hodge nu
mbers. Lurking in the spectral sequence of the pro
of are the keys to having combinatorial control of
the topology of the real hypersurfaces near the t
ropical limit in any toric variety.\n\nThis is joi
nt work with Arthur Renaudineau.
LOCATION:CMS MR13
CONTACT:Dhruv Ranganathan
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