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CATEGORIES:Differential Geometry and Topology Seminar
SUMMARY:Knots\, minimal surfaces and J-holomorphic curves
- Joel Fine\, Université Libre de Bruxelles
DTSTART;TZID=Europe/London:20220126T160000
DTEND;TZID=Europe/London:20220126T170000
UID:TALK167468AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/167468
DESCRIPTION:I will describe work in progress\, parts of which
are joint with Marcelo Alves. Let K be a knot or l
ink in the 3-sphere. I will explain how one can co
unt minimal surfaces in hyperbolic 4-space which h
ave ideal boundary equal to K\, and in this way ob
tain a link invariant. In other words the number o
f minimal surfaces doesn’t depend on the isotopy c
lass of the link. These counts of minimal surfaces
can be organised into a two-variable polynomial w
hich is perhaps a known polynomial invariant of th
e link\, such as HOMFLYPT .\n\n“Counting minimal s
urfaces” needs to be interpreted carefully here\,
similar to how Gromov-Witten invariants “count” J-
holomorphic curves. Indeed I will explain how thes
e minimal surface invariants can be seen as Gromov
-Witten invariants for the twistor space of hyperb
olic 4-space. This leads naturally to a new class
of infinite-volume 6-dimensional symplectic manifo
lds with well behaved counts of J-holomorphic curv
es. This gives more potential knot invariants\, fo
r knots in 3-manifolds other than the 3-sphere. It
also enables the counting of minimal surfaces in
more general Riemannian 4-manifolds\, besides hype
rbolic space.
LOCATION:MR13
CONTACT:Henry Wilton
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