Champs d'intérêt et thèmes de recherche
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Mes travaux les plus récents se rapportent à la topologie symplectique
et aux systèmes
hamiltoniens ainsi qu'à une théorie de Floer universelle des lagrangiennes, sujet qui a fait
l'objet d'un intense développement depuis une vingtaine
d'années.
La topologie (ou géométrie) symplectique est
l'étude mathématique des espaces courbes, de dimension paire
arbitraire, munis d'une forme symplectique, analogue
anti-symétrique d'une métrique riemannienne, qui donne à ces
espaces la structure qu'il faut pour donner
un sens aux lois de la physique aussi bien qu'aux procédés de
quantification (passage du classique au quantique). Ce sujet est
le versant mathématique de ce que les physiciens
appellent la théorie des super-cordes. Son développement
a attiré l'attention des physiciens (Witten, Vafa)
aussi bien que celle des mathématiciens (Gromov, Donaldson), et les méthodes employées
ont suivi une évolution rapide au cours des dernières années.
La plupart de mes travaux porte sur les aspects dits ``hard'' de la
topologie symplectique et des systèmes hamiltoniens, en se servant de
techniques topologiques, géométriques et analytiques, en particulier des
méthodes d'équations aux dérivées partielles elliptiques et de la
cohomologie
quantique.
Ces méthodes sont fondées sur l'étude du
comportement de différents espaces de modules de courbes pseudoholomorphes, qui sont
solutions d'équations de Cauchy-Riemann généralisées associées
à une structure presque complexe.
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Publications
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Preprints récents et articles depuis
1994
Note: The Cluster research project with O. Cornea is expected to spread over a few
years, there are a few papers related to this project: the preprint with O. Cornea below is one of them and will be re-written, the others are currently in preparation and not mentionned below. Most of the recent papers can be found on ArXiv.
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S. Hu and F. Lalonde, Homological Lagrangian Monodromy, preprint 2009, submitted.
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S. Hu and F. Lalonde, A relative Seidel morphism and the Albers map,
Trans. Amer. Math. Soc. 362 (2009), 1135-1168.
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S. Anjos, F. Lalonde and M. Pinsonnault, The homotopy type of the space of symplectic balls in rational ruled 4-manifolds,
Geometry and Topology 13 (2009), 1177-1227.
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S. Anjos and F. Lalonde, The full homotopy type of symplectic balls in S^2 x S^2 above the critical
value, preprint arXiv:math/0406129, 23 pages, 2008.
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S. Hu and F. Lalonde, Anti-symplectic involutions and Maslov indices, preprint,
18 pages, 2008
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S. Anjos and F. Lalonde, The topology of the space of symplectic balls in S^2 x S^2, C. R. Acad. Sci. Paris, Ser. I 345 (2007) 639-642.
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F. Lalonde, Lagrangian submanifolds: from the local model to the cluster complex, Proceedings of
the International Congress of Mathematicians, Madrid 2006, published by the European Mathematical
Society, 2006, pp 456-477.
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O. Cornea and F. Lalonde, Cluster Homology, ArXiv Math.SG/0508345, 56 pages, 2006.
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O. Cornea and F. Lalonde, Cluster homology: an overview of the construction and results, ERA-AMS
12 (2006), 1-12.
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F. Lalonde, A field theory for symplectic fibrations over surfaces with applications, Geometry and
Topology 8 (2004), 1189-1226.
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F. Lalonde and M. Pinsonnault, The topology of the space of symplectic balls in rational 4-manifolds, Duke Mathematical Journal 122 (2004), 347-397.
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E. Kerman and F. Lalonde, Length minimising paths for symplectically aspherical manifolds, Ann. Inst. Fourier 53 (2003), 1503-1526.
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F. Lalonde and D. McDuff, Symplectic structures on fiber bundles,
Topology 42 (2003), 309-347.
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F. Lalonde and M. Pinsonnault, Groupes d'automorphismes et plongements
symplectiques de boules dans les variétés rationelles, C.R. Acad. Sci. Paris, Ser. I 335 (2002), 931-934.
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F. Lalonde and D. McDuff, Cohomological properties of ruled symplectic structures on manifolds, Mirror Symmetry IV, AMS/IP Studies in Advanced Mathematics 33 (2002), 79- 99.
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D. Gatien and F. Lalonde, Holomorphic cylinders with
Lagrangian boundaries and Hamiltonian dynamics, Duke Mathematical Journal 102 (2000), 485 - 511.
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F. Lalonde, D. McDuff and L. Polterovich, Topological rigidity
of Hamiltonian loops and quantum homology, Inventiones Mathematicae 135 (1999), 369-385.
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F. Lalonde et C. Pestieau, Stabilisation of symplectic inequalities
and applications, in Amer. Math. Soc. Translations, Series 2, Volume
196 (1999) pp. 63-72.
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F. Lalonde, D. McDuff and L. Polterovich, On the Flux
conjectures, in
CRM Proceedings and Lecture Notes, American Mathematical Society,
Volume 15 (1998), 69-86.
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F. Lalonde and L. Polterovich, Symplectic diffeomorphisms as isometries of
Hofer's norm, Topology 36 (1997), 711-728
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F. Lalonde and D. McDuff, Positive paths in the linear symplectic group,
The Arnold-Gelfand seminar, Birkhauser, 1997, pp 1-20.
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F. Lalonde, Energy and capacities in symplectic topology,
in: W.H. Kazez (ed.), Geometric Topology, Studies in Advanced Mathematics, vol. 2,
American Mamthematical Society and International Press, 1997, 328-374.
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F. Lalonde, New trends in symplectic geometry,
invited survey in the new series of
C.R. Math. Rep. Acad. Sci. Canada, vol. 19 (2), 1997, 33-50.
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F. Lalonde, J-curves and symplectic invariants,
in: J. Hurtubise and F. Lalonde (eds), Proceedings of the NATO Summer Advanced Institute (SMS) on
Gauge Theory and Symplectic Geometry
Université de Montréal 1995, Kluwer Academic Publishers, Dordrecht, 1997.
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F. Lalonde and D. McDuff, The classification of ruled symplectic 4-manifolds,
Mathematical Research Letters 3 (1996), 769-778.
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F. Lalonde and D. McDuff, J-holomorphic curves and the classification of rational
and ruled symplectic 4-manifolds, in: C.B. Thomas (ed.), Symplectic and Contact Geometry,
Proceedings of the Newton Institute Special Year on Symplectic Geometry,
Cambridge University Press, 1996, pp 1-40.
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F. Lalonde and D. McDuff, Local Non-Squeezing Theorems and
Stability, Geometric and Functional Analalysis 5 (Special volume in the honour of M. Gromov) (1995),
364 - 386.
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F. Lalonde and D. McDuff, Hofer's $L^{\infty}$-geometry:
energy and stability of Hamiltonian flows part I, Inventiones Mathematicae
122 (1995), 1 - 34.
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F. Lalonde and D. McDuff, Hofer's $L^{\infty}$-geometry:
energy and stability of Hamiltonian flows part II, Inventiones Mathematicae 122
(1995), 35 - 69.
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F. Lalonde and D. McDuff, The geometry of symplectic energy,
Annals of Mathematics 141 (1995), 349 - 371.
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F. Lalonde, Isotopy of symplectic balls, Gromov's radius,
and the structure of irrational ruled symplectic 4-manifolds,
Mathematische Annalen 300 (1994), 273-296.
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M. Audin, F. Lalonde and L. Polterovich, Symplectic rigidity:
Lagrangian submanifolds,
in: M. Audin and J. Lafontaine (eds.), Holomorphic Curves in
Symplectic Geometry, Progress in Mathematics, vol. 117, Birkhauser, 1994, pp. 271-322.
Monographies et édition de proceedings depuis 1994
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M. Abreu, F. Lalonde, L.Polterovich eds, New Perspectives and Challenges in Symplectic Field Theory, The CRM Proceedings and Lecture Notes'', vol. 49, American Mathematical Society, 2009, 342 pages.
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P. Biran, O. Cornea and F. Lalonde (eds), Morse theoretical methods in symplectic topology and
non-linear analysis , Proceedings of the NATO Advanced Study Institute (Montréal, 2004), Kluwer
Academic Publishers, Dordrecht, 2005.
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Y. Eliashberg, B. Khesin and F. Lalonde eds, Symplectic and Contact
Topology: Interactions and Perspectives (Proceedings of the workshop on ''Symplectic topology and
higher dimensional Gauge invariants'' held at the Fields Institute in
March-April 2001), Fields Institute Communications 35, AMS, 2003.
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F. Lalonde (ed.), Proceedings of the CRM Workshop on
Geometry, Topology and Dynamics Montréal 1995, CRM Proceedings and Lecture Notes 15, AMS 1998.
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J. Hurtubise and F. Lalonde (eds), Gauge Theory and Symplectic Geometry,
Proceedings of the NATO Summer Advanced Institute on
Gauge Theory and Symplectic Geometry
(Montréal 1995), Kluwer Academic Publishers, Dordrecht, 1997.
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