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SUMMARY:Global existence and convergence of smooth solutions to Yang-Mills
  gradient flow over compact four-manifolds - Paul Feehan
DTSTART:20140616T140000Z
DTEND:20140616T150000Z
UID:TALK52197@talks.cam.ac.uk
CONTACT:Prof. Clément Mouhot
DESCRIPTION:Given a compact Lie group and a principal bundle over a closed
  Riemannian manifold\, the quotient space of connections\, modulo the acti
 on of the group of gauge transformations\, has fundamental significance fo
 r algebraic geometry\, low-dimensional topology\, the classification of sm
 ooth four-dimensional manifolds\, and high-energy physics.\n\nThe quotient
  space of connections is equipped with the Yang-Mills energy functional an
 d Atiyah and Bott (1983) had proposed that its gradient flow with respect 
 to the natural Riemannian metric on the quotient space should prove to be 
 an important tool for understanding the topology of the quotient space via
  an infinite-dimensional Morse theory. The critical points of the energy f
 unctional are gauge-equivalence classes of Yang-Mills connections. However
 \, thus far\, smooth solutions to the Yang-Mills gradient flow have only b
 een known to exist for all time and converge to critical points\, as time 
 tends to infinity\, in relatively few cases\, including (1) when the base 
 manifold has dimension two or three (Rade\, 1991 and 1992\, in dimension t
 wo and three\; G. Daskalopoulos\, 1989 and 1992\, in dimension two)\, (2) 
 when the base manifold is a complex algebraic surface (Donaldson\, 1985)\,
  and (3) in the presence of rotational symmetry in dimension four (Schlatt
 er\, Struwe\, and Tahvildar-Zadeh\, 1998). Global existence of solutions w
 ith up to finitely many point singularities (caused by the ``bubbling'' ph
 enomenon) was proved independently by Struwe (1994) and Kozono\, Maeda\, a
 nd Naito (1995). However\, the question of global existence of smooth solu
 tions over general compact\, Riemannian\, four-dimensional base manifolds 
 has thus far remained unresolved.\n\nIn this talk we shall describe our pr
 oof of the following result: Given a compact Lie group and a smooth initia
 l connection on a principal bundle over a compact\, Riemannian\, four-dime
 nsional manifold\, there is a unique\, smooth solution to the Yang-Mills g
 radient flow which exists for all time and converges to a smooth Yang-Mill
 s connection on the given principal bundle as time tends to infinity.
LOCATION:CMS\, MR13
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