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SUMMARY:Hypersymplectic structures on 4-manifolds and the G2 Laplacian flo
 w - Joel Fine\, ULB
DTSTART:20171018T150000Z
DTEND:20171018T160000Z
UID:TALK72461@talks.cam.ac.uk
CONTACT:Ivan Smith
DESCRIPTION:A hypersymplectic structure on a 4-manifold is a triple w_1\, 
 w_2\, w_3 of symplectic forms such that any non-zero linear combination of
  these forms is again symplectic. The prototypical example is a the triple
  of Kähler forms of a hyperkähler metric. Donaldson has conjectured that
  up to isotopy\, this is the only example. More precisely\, Donaldson conj
 ectures that on a compact 4-manifold\, any hypersymplectic triple is isoto
 pic through cohomologous hypersymplectic triples to a hyperkähler triple.
  This is a special case of a famous folklore conjecture: a compact symplec
 tic 4-manifold with c_1=0 and b_+=3 admits a compatible integrable complex
  structure making it hyperkähler. I will describe an approach to Donaldso
 n’s conjecture which goes via G2 geometry. It gives a natural flow of hy
 persymplectic structures which tries to deform a given triple into a hyper
 kähler one. It can be thought of as an analogue of Ricci flow adapted to 
 this context. I will then explain joint work with Chengjian Yao\, which sh
 ows that the hypersymplectic flow exists as long as the scalar curvature o
 f the associated G2 metrics remains bounded. It is intriguing that this is
  a stronger existence result than what is currently known for Ricci flow. 
 I will not assume any prior knowledge of Ricci flow\, or G2 geometry. \n
LOCATION:MR13
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