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Published in Comptes Rendus Mathématique de l' Académie des Sciences de Paris, 2002
Nous étudions l’espace $Pl(c,\lambda)$ des plongements symplectiques de la boule fermée $B^4(c) \subset\mathbf{R}^4$ de capacité $c$ dans $(S^2 \times S^2,(1+\lambda)\omega_{st}\oplus\omega_{st})$.
Recommended citation: Lalonde, F., Pinsonnault, M., Groupes d' automorphismes et plongements symplectiques de boules dans les variétés rationnelles. C. R. Math. Acad. Sci. Paris 335 (2002), no.11, 931–934. https://doi.org/10.1016/S1631-073X(02)02583-9
Published in Duke Mathematical Journal, 2004
We study in this paper the rational homotopy type of the space of symplectic embeddings of the standard ball $B^4(c) \subset \mathbf{R}^4$ into 4-dimensional rational symplectic manifolds. We compute the rational homotopy groups of that space when the 4-manifold has the form $M_{\mu}=(S^2\times S^2,\mu\omega_0\oplus\omega_0)$ where $\omega_0$ is the area form on the sphere with total area 1 and $\mu$ belongs to the interval $[1,2]$.
Recommended citation: Lalonde, F., Pinsonnault, M., The topology of the space of symplectic balls in rational 4-manifolds. Duke Math. J. 122 (2004), no.2, 347–397. https://doi.org/10.1215/S0012-7094-04-12223-7
Published in Compositio Mathematica, 2006
In this paper, we study the homotopy type of the symplectomorphism group $\mathrm{Symp}(\tilde X_\iota)$, simplifying and extending the results of math.SG/0207096. This allows us to compute the rational homotopy groups of the space $\Im\mathrm{Emb}(B_{c},X)$ of unparametrized symplectic embeddings of $B_{c}$ into $X$.
Recommended citation: Pinsonnault, M., Symplectomorphism groups and embeddings of balls into rational ruled 4-manifolds. Compos. Math. 144 (2008), no.3, 787–810. https://doi.org/10.1112/S0010437X0700334X
Published in The Journal of Symplectic Geometry, 2006
Let $(M,\omega)$ be a four dimensional compact connected symplectic manifold. We prove that $(M,\omega)$ admits only finitely many inequivalent Hamiltonian effective 2-torus actions.
Recommended citation: Karshon, Y., Kessler, L., Pinsonnault, M., A compact symplectic four-manifold admits only finitely many inequivalent toric actions. J. Symplectic Geom. 5 (2007), no.2, 139–166. http://projecteuclid.org/euclid.jsg/1202004454
Published in Journal of Modern Dynamics, 2006
We prove that the group of Hamiltonian automorphisms of a symplectic 4-manifold contains only finitely many conjugacy classes of maximal compact tori with respect to the action of the full symplectomorphism group.
Recommended citation: Pinsonnault, M., Maximal compact tori in the Hamiltonian group of 4-dimensional symplectic manifolds. J. Mod. Dyn. 2 (2008), no.3, 431–455. https://doi.org/10.3934/jmd.2008.2.431
Published in Geometry & Topology, 2008
In this paper, we compute the rational homotopy type, the minimal model, and the cohomology with rational coefficients of the embedding space $\mathrm{Emb}_{\omega}(B^{4}(c),M)$ for a “large” ball of capacity $c \in [c_{\mathrm{crit}},w_{M})$. In particular, we show that it does not have the homotopy type of a finite CW-complex.
Recommended citation: Anjos, S., Lalonde, F., Pinsonnault, M., The homotopy type of the space of symplectic balls in rational ruled 4-manifolds. Geom. Topol. 13 (2009), no.2, 1177–1227. https://doi.org/10.2140/gt.2009.13.1177
Published in The Journal of Symplectic Geometry, 2011
We completely solve the symplectic packing problem with equally sized balls for any rational, ruled, symplectic 4-manifolds. We give explicit formulae for the packing numbers, the generalized Gromov widths, the stability numbers, and the corresponding obstructing exceptional classes.
Recommended citation: Buse, O., Pinsonnault, M., Packing numbers of rational ruled four-manifolds. J. Symplectic Geom. 11 (2013), no.2, 269–316. https://doi.org/10.4310/JSG.2013.v11.n2.a5
Published in Mathematische Zeitschrift, 2012
In this paper, we compute the rational homotopy Lie algebra of symplectomorphism groups of the 3-point blow-up of the projective plane (with an arbitrary symplectic form) and show that in some cases, depending on the sizes of the blow-ups, it is infinite dimensional.
Recommended citation: Anjos, S., Pinsonnault, M., The homotopy Lie algebra of symplectomorphism groups of 3-fold blow-ups of the projective plane. Math. Z. 275 (2013), no.1-2, 245–292. https://doi.org/10.1007/s00209-012-1134-5
Published in J. Math. Phys., 2014
We obtain the semi-classical expansion of the kernels and traces of Toeplitz operators with $\mathcal{C}^k$-symbol on a symplectic manifold. We also give a semi-classical estimate of the distance of a Toeplitz operator to the space of self-adjoint and multiplication operators.
Recommended citation: Barron, T., Ma, X., Marinescu, G., Pinsonnault, M. "Semi-classical properties of Berezin-Toeplitz operators with $\mathcal{C}^k$-symbol." J. Math. Phys.55 (2014), no.4, 042108, 25 pp. https://doi-org.proxy1.lib.uwo.ca/10.1063/1.4870869
Published in C. R. Math. Acad. Sci. Soc. R. Can., 2014
Given a symplectic manifold, we ask in how many different ways can a torus act on it.
Recommended citation: Karshon, Y., Kessler, L., Pinsonnault, M. "Counting toric actions on symplectic four-manifolds." C. R. Math. Acad. Sci. Soc. R. Can. 37 (2015), no.1, 33–40. https://arxiv.org/abs/1409.6061
Published in Journal of Symplectic Geometry, 2016
We establish connections between contact isometry groups of certain contact manifolds and compactly supported symplectomorphism groups of their symplectizations.
Recommended citation: Hind, R., Pinsonnault, M., and Wu, W. (2024). "Symplectormophism groups of non-compact manifolds, orbifold balls, and a space of Lagrangians." J. Symplectic Geom.14 (2016), no.1, 203–226. https://doi.org/10.4310/JSG.2016.v14.n1.a8
Published in arxiv.org, 2022
In this paper, we compute the homotopy type of the group of equivariant symplectomorphisms of $S^2\times S^2$ and $\mathbf{C}P^2\sharp\overline{\mathbf{C}P^2}$ under the presence of Hamiltonian group actions of the circle $S^1$. We prove that the group of equivariant symplectomorphisms are homotopy equivalent to either a torus, or to the homotopy pushout of two tori depending on whether the circle action extends to a single toric action or to exactly two non-equivalent toric actions.
Recommended citation: Pranav V. Chakravarthy, Martin Pinsonnault. (2023). "Centralizers of Hamiltonian circle actions on rational ruled surfaces." arxiv:2202.08255 . https://arxiv.org/abs/2202.08255
Published in Canadian Journal of Mathematics. Journal Canadien de Mathématiques, 2023
We study generators of the fundamental group of the group of symplectomorphisms $\mathrm{Symp}(\mathbf{C}P^2\sharp 5\overline{\mathbf{C}P^2},\omega)$ for some particular symplectic forms.
Recommended citation: Anjos, S., Barata, M., Pinsonnault, M., Reis, A. Loops in the fundamental group of $\mathbf{Symp}(\overline{\mathbf{C}P^2\sharp 5\mathbf{C}P^2},\omega)$ which are not represented by circle actions. Canad. J. Math.75 (2023), no.4, 1226–1271. https://doi.org/10.4153/s0008414x22000323
Published in Mathematische Annalen, 2023
We apply Zhang’s almost Kähler Nakai–Moishezon theorem and Li–Zhang’s comparison of J-symplectic cones to establish a stability result for the symplectomorphism group of a rational 4-manifold $M$ with Euler number up to 12.
Recommended citation: Anjos, S., Li, J., Li, TJ. et al. Stability of the symplectomorphism groups of rational surfaces. Math. Ann. 389, 85–119 (2024). https://doi-org.proxy1.lib.uwo.ca/10.1007/s00208-023-02643-5 https://doi.org/10.1007/s00208-023-02643-5
Published in Transactions of the AMS, 2023
Let $M=(M,\omega)$ be either the product $S^2\times S^2$ or the non-trivial $S^2$ bundle over $S^2$ endowed with any symplectic form $\omega$. Suppose a finite cyclic group $\mathbf{Z}_n$ is acting effectively on $(M,\omega)$ through Hamiltonian diffeomorphisms, that is, there is an injective homomorphism $\mathbf{Z}_n\hookrightarrow \mathrm{Ham}(M,\omega)$. In this paper, we investigate the homotopy type of the group $\mathrm{Symp}^{\mathbf{Z}_n}(M,\omega)$ of equivariant symplectomorphisms.
Recommended citation: Pranav V. Chakravarthy, Martin Pinsonnault. (2023). "Centralizers of Hamiltonian finite cyclic group actions on rational ruled surfaces." https://doi.org/10.1090/tran/9223. https://doi.org/10.1090/tran/9223
Published in arxiv.org, 2024
We investigate spaces of symplectic embeddings of $n\leq 4$ balls into the complex projective plane. We prove that they are homotopy equivalent to explicitly described algebraic subspaces of the configuration spaces of $n$ points.
Recommended citation: Sílvia Anjos, Jarek Kędra, Martin Pinsonnault. (2024). "Embeddings of symplectic balls into the complex projective plane." arxiv:2307.00556. https://arxiv.org/abs/2307.00556
Published in arXiv.org, 2024
We give a complete and self-contained exposition of the $J$-tame inflation lemma: Given any tame almost complex structure $J$ on a symplectic 4-manifold $(M,\omega)$, and given any compact, embedded, $J$-holomorphic submanifold $Z$, it is always possible to construct a deformation of symplectic forms $\omega_t$ in classes $[\omega_t]=[\omega]+tPD(Z)$, for $0\leq t$ less than an upper bound $0< T$ that only depends on the self-intersection $Z\cdot Z$.
Recommended citation: Pranav Chakravarthy, Jordan Payette, Martin Pinsonnault. (2024). "Remarks on $J$-tame inflation." arXiv:2403.19110. https://arxiv.org/abs/2403.19110
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Graduate course, The University of Western Ontario, Department of Mathematics, 2024
Instructor: Martin Pinsonnault