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UID:8039@i2m.univ-amu.fr
DTSTART;TZID=Europe/Paris:20150622T000000
DTEND;TZID=Europe/Paris:20150703T000000
DTSTAMP:20241120T205633Z
URL:https://www.i2m.univ-amu.fr/evenements/moduli-space-of-symplectic-ball
 s-in-4-manifolds-and-packing-morlet-chair-francois-lalonde/
SUMMARY:Pairs (CIRM\, Luminy\, Marseille): Moduli Space of Symplectic Balls
  in 4-Manifolds and Packing (Morlet Chair - François Lalonde)
DESCRIPTION:Pairs: \n\n\n\n CIRM - Jean-Morlet Chair \n François Lalonde &
 amp\; Andrei Teleman\n\nModuli Spaces in Symplectic Topology and Gauge T
 heory\n\nEspaces de modules en topologie symplectique et théorie de Jauge
  \n\n\n 2015 - Semester 2 \n\n\n\n\n\n\n\n\n\nRESEARCH IN PAIRS\nModuli Sp
 ace of Symplectic Balls in 4-manifolds and Packing (1578)\nDates : 22 Jun
 e - 3 July at CIRM (Marseille Luminy\, France)\n\n\n\n\n\n\n\n\n\nDESCRIP
 TION\n\nWe investigated one of the most elusive questions in symplectic to
 pology: is there a notion of "statistical symplectic topology"? What does 
 happen when we take various limits\, say on the dimension of the configura
 tion space or on the radius of embedded symplectic balls in a given fixed 
 symplectic manifold? In terms of physics\, this boils down to understandin
 g if there is any reasonable notion of probabilistic String theory. Note t
 hat it might perfectly be that this notion does not make sense from the St
 ring approach\, but does make sense for the point of view of Symplectic To
 pology.\nThere are evidences that support both points of view. Indeed\, th
 e inexistence of a statistical Symplectic Topology is supported by the fac
 t that the moduli space of k symplectically embedded balls of equal radius
  seems to loose all of its hard characteristics as k goes to infinity and 
 the radius goes to zero. That is to say\, it behaves like volume-perservin
 g balls. This was probably best exemplified in a series of papers by Anjos
 -Lalonde-Pinsonnault that showed\, for the first time\, the existence of a
  critical value in moduli spaces in Symplectic Topology: below that critic
 al value (of the radius of a symplectic ball in a rational 4-manifold M)\,
  the full moduli space retracts to the space of frames\, that is to say to
  a compact finite dimensional CW complex. But above that value\, the space
  is genuinely infinite dimensional and contains homology and homotopy in d
 imensions as high as one wishes. That proof uses heavy machinery\, relying
  on hard elliptic methods\, on a (cofinite) stratification of the infinite
  dimensional space of almost complex structures on M tamed by the symplect
 ic form on which various groups act\, and on a comparison between the stra
 tification for the J-structures on the blow-up of M at the given ball and 
 the stratification for the J-structures on M itself. In any case\, one nee
 ds hard methods to prove that dichotomy between hard and soft moduli space
 s.\nMore recently\, an isoperimetric conjecture of Viterbo that states tha
 t the round ball achieves the minimum for some symplectic capacities\, has
  been proved by Ostrover and Artstein-Avidan. Even if the conjecture is st
 ated in finite dimensions\, their proof relies on the asymptotic behaviour
  of these capacities. This supports the idea that some limits\, when caref
 ully taken\, can carry more information than what one would find in the so
 ft world of volume-preserving maps.\nNote that the space of embeddings of 
 balls and ellipsoids lead in general\, if no limit is taken\, to an extrao
 rdinarily rich corpus\, where Number theory intervenes in an essential way
 . This was best illustrated by the recent work of McDuff and Schlenk. Olgu
 ta Buse has also recently done substantial work in proving that limits lea
 d to soft objects.\n\n\n \n\nPARTICIPANTS\n\n\n 	Silvia Anjos (IST Lisbo
 n)\n 	François Lalonde (Université de Montréal &amp\; Aix-Marseille Un
 iversité)\n 	Jordan Payette (Université de Montréal)\n 	Martin Pinsonn
 ault (University of Western Ontario)\n\n\n\n\n\n  \n\n\n\n\n\n\n\n\n\n\n\
 n
CATEGORIES:Manifestation scientifique,Morlet Chair Semester,Morlet
 Research in Pairs
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