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CATEGORIES:Junior Geometry Seminar
SUMMARY:A proof of Donaldson's Theorem - Samuel Muñoz Echá
niz\, University of Cambridge
DTSTART;TZID=Europe/London:20211105T160000
DTEND;TZID=Europe/London:20211105T170000
UID:TALK165538AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/165538
DESCRIPTION:Donaldson’s Diagonalization Theorem states that if
the intersection form of a closed oriented smooth
4-manifold X is definite\, then it is diagonaliza
ble over the integers. We present an outline of th
e original proof by Donaldson\, which appeared in
his celebrated 1983 paper. \n\nThe proof relies on
some geometric features of the moduli space of an
ti-self-dual (ASD) connections on an SU(2)-princip
al bundle over X. More concretely\, it provides an
oriented cobordism between X and a disjoint union
of complex projective spaces. This gives an estim
ate on the signature of X\, which is the key step
to the proof. \n\nWe will introduce the ASD modul
i space and discuss some of its properties\, such
as dimension\, its singularities and its compactif
ication. If time permits\, we will also discuss so
me of the consequences of the theorem in Freedman’
s work on 4-dimensional topology\, such as the exi
stence of an exotic differentiable structure on th
e 4-dimensional Euclidean space.\n
LOCATION:MR13
CONTACT:Macarena Arenas
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