Celebratio Mathematica

D. McDuff: “A countable infinity of \( \mathrm{II}_1 \) factors,” Ann. Math. (2) 90 : 2 (September 1969), pp. 361–371. This formed part of the author’s 1970 PhD thesis. MR 0256183 Zbl 0184.16901 article

Powers, in his paper on representations of uniformly hyperfinite algebras [1967], established the existence of an uncountable number of non-isomorphic type \( \mathrm{III} \) factors. However, although great progress has redently been made in the investigation of the isomorphism classes of \( \mathrm{II}_1 \) factors (see, in this connection, [Ching 1969; Dixmier and Lance 1969; Sakai 1968–1969; Zeller-Meier 1969]), the question of whether or not there exists an infinity of non-isomorphic \( \mathrm{II}_1 \) factors has remained unsolved. In this paper we construct a countable infinity of non-isomorphic \( \mathrm{II}_1 \) factors. The methods used here can in fact be generalised to give a set of non-isomorphic \( \mathrm{II}_1 \) factors which has cardinality \( c \). This will be the subject of a later paper by the same author.

D. McDuff: “Uncountably many \( \mathrm{II}_1 \) factors,” Ann. Math. (2) 90 : 2 (September 1969), pp. 372–377. This formed part of the author’s 1970 PhD thesis. MR 0259625 Zbl 0184.16902 article

È. H’juit and D. Makduf: “Certain pathological maximal ideals in the measure algebra of a compact group,” Dokl. Akad. Nauk SSSR 191 (1970), pp. 1241–1243. In Russian; English version in Sov. Math., Dokl. 11 (1970), 546–548.. MR 0257760 article

È. H’juitt and D. Makduff: “Some pathological maximal ideals in algebras of operators and algebras of measures on groups,” Mat. Sb. (N.S.) 83 (125) (1970), pp. 527–546. In Russian; English version in Math. USSR, Sb. 12:4 (1970), 525–541. MR 0273420 Zbl 0206.42603 article

D. McDuff: “Central sequences and the hyperfinite factor,” Proc. London Math. Soc. (3) 21 : 3 (1970), pp. 443–461. MR 0281018 Zbl 0204.14902 article

D. Makduff: “The structure of \( \mathrm{II}_1 \)-factors,” Uspehi Mat. Nauk 25 : 6(156) (1970), pp. 29–51. In Russian; English version in Russ. Math. Surv. 25:6 (1970), 29–50. This is a version of the author’s 1970 PhD thesis. MR 0306935 Zbl 0205.13802 article

M. D. Waddington: On the structure of \( \mathrm{II}_1 \) factors. Ph.D. thesis, Cambridge University, 1970. Advised by G. A. Reid. A version was reprinted in Uspehi Mat. Nauk 25:6(156) (1970). Earlier it was published in two parts in Ann. Math. 90:2 (1969) and Ann. Math. 90:2 (1969). phdthesis

D. McDuff: “On residual sequences in a \( \mathrm{II}_1 \) factor,” J. London Math. Soc. (2) 3 (1971), pp. 273–280. MR 0279597 Zbl 0208.38302 article

D. McDuff: “Configuration spaces of positive and negative particles,” Topology 14 : 1 (March 1975), pp. 91–107. MR 0358766 Zbl 0296.57001 article

The aim of this paper is to investigate the topology of two “configuration spaces” associated to a smooth manifold \( M \). The first is the space \( C(M) \) of all finite subsets of \( M \). Its points can be thought of as sets of indistinguishable particles moving about on \( M \), and it is topologised so that particles cannot collide. (cf. [Fadell and Neuwirth 1962], where configurations of distinguishable particles on a manifold are studied.) The second space, denoted \( C^{\pm}(M) \), has as its points pairs of finite subsets of \( M \), to be thought of as “positive” and “negative” particles. It is topologised so that particles of the same sign cannot collide, but a pair of particles of opposite sign can collide and annihilate each other. (More precise definitions are given in §2.)

D. McDuff and G. Segal: “Homology fibrations and the ‘group-completion’ theorem,” Invent. Math. 31 : 3 (1976), pp. 279–284. MR 0402733 Zbl 0306.55020 article

D. McDuff: “Configuration spaces,” pp. 88–95 in K-theory and operator algebras (Athens, GA, 21–25 April 1975). Edited by B. B. Morrel and I. M. Singer. Lecture Notes in Mathematics 575. Springer (Berlin), 1977. MR 0467734 Zbl 0352.55016 incollection

Isadore M. Singer	Related	Volume
Bernard B. Morrel	Related

D. McDuff: “The lattice of normal subgroups of the group of diffeomorphisms or homeomorphisms of an open manifold,” J. London Math. Soc. (2) 18 : 2 (1978), pp. 353–364. MR 509952 article

D. McDuff: “Foliations and monoids of embeddings,” pp. 429–444 in Geometric topology (Athens, GA, 1–12 August 1977). Edited by J. C. Cantrell. Academic Press (New York), 1979. MR 537744 Zbl 0473.57016 incollection

D. McDuff: “On the classifying spaces of discrete monoids,” Topology 18 : 4 (1979), pp. 313–320. MR 551013 Zbl 0429.55009 article

D. McDuff: “The homology of some groups of diffeomorphisms,” Comment. Math. Helv. 55 : 1 (1980), pp. 97–129. MR 569248 Zbl 0448.57015 article

D. McDuff: “On the group of volume-preserving diffeomorphisms of \( \mathbf{R}^n \),” Trans. Am. Math. Soc. 261 : 1 (September 1980), pp. 103–113. MR 576866 Zbl 0447.58012 article

D. McDuff: “\( C^1 \)-minimal subsets of the circle,” Ann. Inst. Fourier (Grenoble) 31 : 1 (1981), pp. 177–193. MR 613034 Zbl 0439.58020 article

D. McDuff: “On groups of volume-preserving diffeomorphisms and foliations with transverse volume form,” Proc. London Math. Soc. (3) 43 : 2 (1981), pp. 295–320. MR 628279 Zbl 0411.57028 article

D. McDuff: “On tangle complexes and volume-preserving diffeomorphisms of open 3-manifolds,” Proc. London Math. Soc. (3) 43 : 2 (1981), pp. 321–333. MR 628280 Zbl 0411.57029 article

D. McDuff: “Local homology of groups of volume preserving diffeomorphisms, I,” Ann. Sci. École Norm. Sup. (4) 15 : 4 (1982), pp. 609–648. Part II was published in Comment. Math. Helv. 58:1 (1983). MR 707329 Zbl 0577.58005 article

In this paper we prove the volume preserving analogue of the Mather–Thurston theorem, which relates the group of compactly supported \( C^{\infty} \)-diffeomorphisms of \( \mathbf{R}^n \) to the \( n \)-fold loop space of Haefliger’s classifying space for codimension \( n \) foliations ([Mather 1973; Thurston 1974]). The proof is based on a study of the behaviour of monoids of self-embeddings of manifolds much as in [McDuff 1979] and [McDuff 1980], where a proof is given of the original Mather–Thurston theorem. However, because every volume preserving self-embedding of a compact manifold is a diffeomorphism, we must extend our techniques to non-compact \( W \).

D. McDuff: “Some canonical cohomology classes on groups of volume preserving diffeomorphisms,” Trans. Am. Math. Soc. 275 : 1 (1983), pp. 345–356. MR 678355 Zbl 0522.57029 article

We discuss some canonical cohomology classes on the space \( \bar{B}\mathcal{D}\textit{iff}{\,}_{\omega 0}^cM \), where \( \mathcal{D}\textit{iff}{\,}_{\omega 0}^cM \) is the identity component of the group of compactly supported diffeomorphisms of the manifold \( M \) which preserve the volume form \( \omega \). We first look at some classes \( c_k(M) \), \( 1 \leq k \leq n = \textrm{dim} M \), which are defined for all \( M \), and show that the top class \[ c_n(M) \in H^n(\bar{B}\mathcal{D}\textit{iff}{\,}_{\omega 0}^cM;\mathbf{R}) \] is nonzero for \( M = S^n \), \( n \) odd, and is zero for \( M = S^n \), \( n \) even. When \( H_c^i(M;\mathbf{R}) = 0 \) for \( 0 \leq i < n \), the classes \( c_k(M) \) all vanish and a secondary class \[ s(M)\in H^{n-1}(\bar{B}\mathcal{D}\textit{iff}{\,}_{\omega 0}^cM;\mathbf{R}) \] may be defined. This is trivially zero when \( n \) is odd, and is twice the Calabi invariant for symplectic manifolds when \( n = 2 \). We prove that \( s(\mathbf{R}^n)\neq 0 \) when \( n \) is even by showing that it is one of a set of nonzero classes which were defined by Hurder in [1982].

P. de la Harpe and D. McDuff: “Acyclic groups of automorphisms,” Comment. Math. Helv. 58 : 1 (1983), pp. 48–71. MR 699006 Zbl 0522.20034 article

D. McDuff: “Local homology of groups of volume-preserving diffeomorphisms, II,” Comment. Math. Helv. 58 : 1 (1983), pp. 135–165. Parts I and III were published in Ann. Sci. École Norm. 15:4 (1982) and Ann. Sci. École Norm. 16:4 (1983), respectively. MR 699012 Zbl 0598.57020 article

D. McDuff: “Local homology of groups of volume-preserving diffeomorphisms, III,” Ann. Sci. École Norm. Sup. (4) 16 : 4 (1983), pp. 529–540. Part II was published in Comment. Math. Helv. 58:1 (1983). MR 740589 Zbl 0619.58008 article

D. McDuff: “Symplectic diffeomorphisms and the flux homomorphism,” Invent. Math. 77 : 2 (1984), pp. 353–366. MR 752824 Zbl 0538.53041 article

D. McDuff: “Examples of simply-connected symplectic non-Kählerian manifolds,” J. Diff. Geom. 20 : 1 (1984), pp. 267–277. MR 772133 Zbl 0567.53031 article

D. McDuff: “Remarks on the homotopy type of groups of symplectic diffeomorphisms,” Proc. Am. Math. Soc. 94 : 2 (June 1985), pp. 348–352. MR 784191 Zbl 0569.57020 article

D. McDuff: “Examples of symplectic structures,” Invent. Math. 89 : 1 (1987), pp. 13–36. MR 892186 Zbl 0625.53040 article

D. McDuff: “Applications of convex integration to symplectic and contact geometry,” Ann. Inst. Fourier (Grenoble) 37 : 1 (1987), pp. 107–133. MR 894563 Zbl 0572.58010 article

D. McDuff: “Symplectic structures on \( \mathbf{R}^{2n} \),” pp. 87–94 in Aspects dynamiques et topologiques des groupes infinis transformation de la mécanique [Dynamical and topological apects of infinite group transformations in mechanics] (Lyon, 26–30 May 1986). Edited by P. Dazord, N. Desolneux-Moulis, and J.-M. Morvan. Travaux en Cours 25. Hermann (Paris), 1987. MR 906899 Zbl 0626.53025 incollection

J.-M. Morvan	Related
P. Dazord	Related
Nicole Desolneux-Moulis	Related

D. McDuff: “The moment map for circle actions on symplectic manifolds,” J. Geom. Phys. 5 : 2 (1988), pp. 149–160. Dedicated to I. M. Gelfand on the occasion of his 75th birthday. MR 1029424 Zbl 0696.53023 article

D. McDuff: “Book review: Mikhael Gromov, ‘Partial differential relations’,” Bull. Am. Math. Soc. (N.S.) 18 : 2 (1988), pp. 214–220. MR 1567681 article

D. McDuff: “The symplectic structure of Kähler manifolds of nonpositive curvature,” J. Diff. Geom. 28 : 3 (1988), pp. 467–475. MR 965224 Zbl 0632.53058 article

In this note we show that the Kähler form on a simply connected complete Kähler manifold \( W \) of nonpositive curvature is diffeomorphic to the standard symplectic form on \( \mathbf{R}^n \). This means in particular that the symplectic structure on a Hermitian symmetric space of noncompact type is standard. We also show that if \( L \) is a totally geodesic proper, connected Lagrangian submanifold of a complete Kähler manifold \( W \) of nonpositive curvature, then \( W \) is symplectomorphic to the cotangent bundle \( T^*L \) with its usual symplectic structure provided that the fundamental group \( \pi_1(W,L) \) vanishes. The proofs use a comparison theorem due to Greene & Wu and Sia & Yau.

“Dusa McDuff,” pp. 57 in American Men and Women of Science, 26th edition, vol. 5. R. R. Bowker (Farmington Hills, MI), 1989. incollection

D. McDuff: “Elliptic methods in symplectic geometry,” Bull. Am. Math. Soc. (N.S.) 23 : 2 (October 1990), pp. 311–358. MR 1039425 Zbl 0723.53018 article

The past few years have seen several exciting developments in the field of symplectic geometry, and a beginning has been made towards solving many important and hitherto inaccessible problems. The new techniques which have made this possible have come both from the calculus of variations and from the theory of elliptic partial differential operators. This paper describes some of the results that Gromov obtained using elliptic methods, and then shows how Floer applied these elliptic techniques to develop a new approach to Morse theory, which has important applications in the theory of 3- and 4-manifolds as well as in symplectic geometry. To give some idea of the context of their results, we begin with a section on symplectic geometry, which concentrates on questions about symplectic diffeomorphisms.

D. McDuff: “The structure of rational and ruled symplectic 4-manifolds,” J. Am. Math. Soc. 3 : 3 (July 1990), pp. 679–712. An erratum was published in J. Am. Math. Soc. 5:4 (1992). MR 1049697 Zbl 0723.53019 article

D. McDuff: Applications of PDE methods by Gromov, Floer, and others to symplectic geometry, 1990. 60 minute videocassette. Lecture recorded in Boulder, CO, 8 August 1989. MR 1109713 Zbl 0925.58025 misc

D. McDuff: “Rational and ruled symplectic 4-manifolds,” pp. 7–14 in Geometry of low-dimensional manifolds (Durham, UK, 11–21 July 1989), vol. 2: Symplectic manifolds and Jones–Witten theory. Edited by S. K. Donaldson and C. B. Thomas. London Mathematical Society Lecture Note Series 151. Cambridge University Press, 1990. MR 1171905 Zbl 0732.57012 incollection

Charles B. Thomas	Related
S. K. Donaldson	Related

D. McDuff: “Symplectic manifolds with contact type boundaries,” Invent. Math. 103 : 1 (1991), pp. 651–671. MR 1091622 Zbl 0719.53015 article

D. McDuff: “Blow ups and symplectic embeddings in dimension 4,” Topology 30 : 3 (November 1991), pp. 409–421. MR 1113685 Zbl 0731.53035 article

D. McDuff: “The local behaviour of holomorphic curves in almost complex 4-manifolds,” J. Diff. Geom. 34 : 1 (1991), pp. 143–164. MR 1114456 Zbl 0736.53038 article

D. McDuff: Some autobiographical notes, January 1991. Online notes. Notes taken from the acceptance speech that McDuff gave when she was awarded the AMS Satter Prize. misc

“1991 Ruth Lyttle Satter Prize in mathematics awarded in San Francisco,” Notices Am. Math. Soc. 38 : 3 (March 1991), pp. 185–187. article

D. McDuff: “Symplectic 4-manifolds,” pp. 541–548 in Proceedings of the International Congress of Mathematicians (Kyoto, 21–29 August 1990), vol. 1. Edited by I. Satake. Springer Japan (Tokyo), 1992. MR 1159241 Zbl 0744.53016 incollection

D. McDuff: “Immersed spheres in symplectic 4-manifolds,” Ann. Inst. Fourier (Grenoble) 42 : 1–2 (1992), pp. 369–392. MR 1162567 Zbl 0756.53021 article

D. McDuff: “Singularities of \( J \)-holomorphic curves in almost complex 4-manifolds,” J. Geom. Anal. 2 : 3 (1992), pp. 249–266. MR 1164604 Zbl 0758.53019 article

This note concerns the structure of singularities of maps \( f \) from a neighborhood of \( \{0\} \) in the complex plane \( \mathbb{C} \) to an almost complex manifold \( (V,J) \), which are \( J \)-holomorphic in the sense that \[ df \circ i = J \circ df \] and are singular (i.e., \( df = 0 \)) at \( \{0\} \). The main result is that when \( V \) has dimension 4, the topology of these singularities is the same as in the case when \( J \) is integrable. Thus, if the image \( \operatorname{Im}f = C \) is not multiply-covered, there is a neighborhood \( U \) of the point \( x = f(0) \), such that the pair \( (U, U \cap C) \) is homeomorphic to the cone over \( (S^3,K_x) \) where \( K_x \) is an algebraic knot in \( S^3 \) that depends only on the germ \( C \) at \( x \).

D. McDuff: “Erratum to: ‘The structure of rational and ruled symplectic 4-manifolds’,” J. Am. Math. Soc. 5 : 4 (October 1992), pp. 987–988. Erratum for an article published in J. Am. Math. Soc. 3:3 (1990). MR 1168961 Zbl 0799.53039 article

D. McDuff: “Remarks on the uniqueness of symplectic blowing up,” pp. 157–167 in Symplectic geometry (Warwick, UK, August 1990). Edited by D. Salamon. London Mathematical Society Lecture Note Series 192. Cambridge University Press, 1993. MR 1297134 Zbl 0822.53021 incollection

D. McDuff and L. Traynor: “The 4-dimensional symplectic camel and related results,” pp. 169–182 in Symplectic geometry (Warwick, UK, August 1990). Edited by D. Salamon. London Mathematical Society Lecture Note Series 192. Cambridge University Press, 1993. MR 1297135 Zbl 0821.53030 incollection

Dietmar Arno Salamon	Related
Lisa Traynor	Related

D. McDuff: “Notes on ruled symplectic 4-manifolds,” Trans. Am. Math. Soc. 345 : 2 (October 1994), pp. 623–639. MR 1188638 Zbl 0810.53020 article

A symplectic 4-manifold \( (V,\omega) \) is said to be ruled if it is the total space of a fibration whose fibers are 2-spheres on which the symplectic form does not vanish. This paper develops geometric methods for analysing the symplectic structure of these manifolds, and shows how this structure is related to that of a generic complex structure on \( V \). It is shown that each \( V \) admits a unique ruled symplectic form up to pseudo-isotopy (or deformation). Moreover, if the base is a sphere or if \( V \) is the trivial bundle over the torus, all ruled cohomologous forms are isotopic. For base manifolds of higher genus this remains true provided that a cohomological condition of the form is satisfied: one needs the fiber to be “small” relative to the base. These results correct the statement of Theorem 1.3 in The structure of rational and ruled symplectic manifolds, J. Amer. Math. Soc. 3 (1990), 679–712, and give more details of some of the proofs.

D. McDuff and L. Polterovich: “Symplectic packings and algebraic geometry,” Invent. Math. 115 : 1 (1994), pp. 405–429. With an appendix by Yael Karshon. MR 1262938 Zbl 0833.53028 article

Leonid V. Polterovich	Related
Yael Karshon	Related

D. McDuff: “Singularities and positivity of intersections of \( J \)-holomorphic curves,” pp. 191–215 in Holomorphic curves in symplectic geometrErratumy. Edited by M. Audin and J. Lafontaine. Progress in Mathematics 117. Birkhäuser (Basel), 1994. With an appendix by Gang Liu. MR 1274930 incollection

Jacques Lafontaine	Related
Gang Liu	Related
Michèle Audin	Related

D. McDuff and D. Salamon: \( J \)-holomorphic curves and quantum cohomology. University Lecture Series 6. American Mathematical Society (Providence, RI), 1994. MR 1286255 Zbl 0809.53002 book

A. Jackson: “Academy of Arts and Science elections,” Notices Am. Math. Soc. (August 1995), pp. 887. article

F. Lalonde and D. McDuff: “The geometry of symplectic energy,” Ann. Math. (2) 141 : 2 (March 1995), pp. 349–371. MR 1324138 Zbl 0829.53025 article

F. Lalonde and D. McDuff: “Local non-squeezing theorems and stability,” Geom. Funct. Anal. 5 : 2 (March 1995), pp. 364–386. Dedicated to Misha Gromov on the occasion of his 50th birthday. MR 1334871 Zbl 0837.58014 article

Mikhael Leonidovich Gromov	Related
François Lalonde	Related

D. McDuff: “An irrational ruled symplectic 4-manifold,” pp. 545–554 in The Floer memorial volume. Edited by H. Hofer, C. H. Taubes, A. Weinstein, and E. Zehnder. Progress in Mathematics 133. Birkhäuser (Basel), 1995. MR 1362840 Zbl 0836.57017 incollection

Andreas Floer	Related
Helmut Hofer	Related
Eduard J. Zehnder	Related
Clifford Henry Taubes	Related
Alan Weinstein	Related

D. McDuff and D. Salamon: Introduction to symplectic topology. Oxford Mathematical Monographs. Clarendon Press (Oxford), 1995. A 2nd edition was published in 1998. MR 1373431 Zbl 0844.58029 book

F. Lalonde and D. McDuff: “Erratum for ‘Hofer’s \( L^{\infty} \)-geometry: Energy and stability of Hamiltonian flows, II’,” Invent. Math. 123 : 3 (1996), pp. 613. Erratum for an article published in Invent. Math. 122:1 (1996). article

F. Lalonde and D. McDuff: “Erratum for ‘Hofer’s \( L^{\infty} \)-geometry: Energy and stability of Hamiltonian flows, I’,” Invent. Math. 123 : 3 (1996), pp. 613. Erratum for an article published in Invent. Math. 122:1 (1996). MR 1383964 article

D. McDuff and D. Salamon: “A survey of symplectic 4-manifolds with \( b^+ = 1 \),” pp. 47–60 in Proceedings of 4th Gökova geometry-topology conference (Gökova, Turkey, 29 May–2 June 1995), published as Turk. J. Math. 20 : 1. Issue edited by S. Akbulut, T. Önder, and R. J. Stern. Tübitak (Ankara), 1996. MR 1392662 Zbl 0870.57023 incollection

Turgut Önder	Related
Selman Akbulut	Related
Dietmar Arno Salamon	Related
Ronald John Stern	Related

D. McDuff and M. Symington: “Associativity properties of the symplectic sum,” Math. Res. Lett. 3 : 5 (1996), pp. 591–608. MR 1418574 Zbl 0877.53025 article

F. Lalonde and D. McDuff: “The classification of ruled symplectic 4-manifolds,” Math. Res. Lett. 3 : 6 (1996), pp. 769–778. MR 1426534 Zbl 0874.57019 article

F. Lalonde and D. McDuff: “\( J \)-curves and the classification of rational and ruled symplectic 4-manifolds,” pp. 3–42 in Contact and symplectic geometry (Cambridge, UK, July–December 1994). Edited by C. B. Thomas. Publications of the Newton Institute 8. Cambridge University Press, 1996. MR 1432456 Zbl 0867.53028 incollection

Charles B. Thomas	Related
François Lalonde	Related

F. Lalonde and D. McDuff: “Hofer’s \( L^{\infty} \)-geometry: Energy and stability of Hamiltonian flows, I,” Invent. Math. 122 : 1 (1996), pp. 1–33. Errata were published in Invent. Math. 123:3 (1996). MR 1354953 Zbl 0844.58020 article

Consider the group \( \mathrm{Ham}^c(M) \) of compactly supported Hamiltonian symplectomorphisms of the symplectic manifold \( (M,\omega) \) with the Hofer \( L^{\infty} \)-norm. A path in \( \mathrm{Ham}^c(M) \) will be called a geodesic if all sufficiently short pieces of it are local minima for the Hofer length functional \( \mathscr{L} \). In this paper, we give a necessary condition for a path \( \gamma \) to be a geodesic. We also develop a necessary condition for a geodesic to be stable, that is, a local minimum for \( \mathscr{L} \). This condition is related to the existence of periodic orbits for the linearization of the path, and so extends Ustilovsky’s work on the second variation formula. Using it, we construct a symplectomorphism of \( S^2 \) which cannot be reached from the identity by a shortest path. In later papers in this series, we will use holomorphic methods to prove the sufficiency of the condition given here for the characterisation of geodesics as well as the sufficiency of the condition for the stability of geodesics. We will also investigate conditions under which geodesics are absolutely length-minimizing.

F. Lalonde and D. McDuff: “Hofer’s \( L^{\infty} \)-geometry: Energy and stability of Hamiltonian flows, II,” Invent. Math. 122 : 1 (1996), pp. 35–69. An erratum was published in Invent. Math. 123:3 (1996). Zbl 0844.58021 article

In this paper we first show that the necessary condition introduced in our previous paper is also a sufficient condition for a path to be a geodesic in the group \( \mathrm{Ham}^c(M) \) of compactly supported Hamiltonian symplectomorphisms. This applies with no restriction on \( M \). We then discuss conditions which guarantee that such a path minimizes the Hofer length. Our argument relies on a general geometric construction (the gluing of monodromies) and on an extension of Gromov’s non-squeezing theorem both to more general manifolds and to more general capacities. The manifolds we consider are quasi-cylinders, that is spaces homeomorphic to \( M\times D^2 \) which are symplectically ruled over \( D^2 \). When we work with the usual capacity (derived from embedded balls), we can prove the existence of paths which minimize the length among all homotopic paths, provided that \( M \) is semi-monotone. (This restriction occurs because of the well-known difficulty with the theory of \( J \)-holomorphic curves in arbitrary \( M \).) However, we can only prove the existence of length-minimizing paths (i.e. paths which minimize length amongst all paths, not only the homotopic ones) under even more restrictive conditions on \( M \), for example when \( M \) is exact and convex or of dimension 2. The new difficulty is caused by the possibility that there are non-trivial and very short loops in \( \mathrm{Ham}^c(M) \). When such length minimizing paths do exist, we can extend the Bialy–Polterovich calculation of the Hofer norm on a neighbourhood of the identity (\( C^1 \)-flatness).

Although it applies to a more restricted class of manifolds, the Hofer–Zehnder capacity seems to be better adapted to the problem at hand, giving sharper estimates in many situations. Also the capacity-area inequality for split cylinders extends more easily to quasi-cylinders in this case. As applications, we generalise Hofer’s estimate of the time for which an autonomous flow is length-minimizing to some manifolds other than \( \mathbf{R}^{2n} \), and derive new results such as the unboundedness of Hofer’s metric on some closed manifolds, and a linear rigidity result.

F. Lalonde and D. McDuff: “Positive paths in the linear symplectic group,” pp. 361–387 in The Arnold–Gelfand mathematical seminars. Edited by V. I. Arnold, I. M. Gelfand, M. Smirnov, and V. S. Retakh. Birkhäuser (Boston), 1997. MR 1429901 Zbl 0868.58031 incollection

A positive path in the linear symplectic group \( \mathrm{Sp}(2n) \) is a smooth path which is everywhere tangent to the positive cone. These paths are generated by negative definite (time-dependent) quadratic Hamiltonian functions on Euclidean space. A special case are autonomous positive paths, which are generated by time-independent Hamiltonians, and which all lie in the set \( \mathcal{U} \) of diagonalizable matrices with eigenvalues on the unit circle. However, as was shown by Krein, the eigenvalues of a general positive path can move off the unit circle. In this paper, we extend Krein’s theory: we investigate the general behavior of positive paths which do not encounter the eigenvalue 1, showing, for example, that any such path can be extended to have endpoint with all eigenvalues on the circle. We also show that in the case \( 2n = 4 \) there is a close relation between the index of a positive path and the regions of the symplectic group that such a path can cross. Our motivation for studying these paths came from a geometric squeezing problem [Lalonde and McDuff, 1995] in symplectic topology. However, they are also of interest in relation to the stability of periodic Hamiltonian systems [Gelfand and Lidskii 1958] and in the theory of geodesics in Riemannian geometry [Bott 1956].

Yuriĭ Mikhaĭlovich Smirnov	Related
Vladimir Solomonovich Retakh	Related
Vladimir Igorevich Arnold	Related
François Lalonde	Related
Israïl Moiseevich Gelfand	Related

D. McDuff: “Lectures on Gromov invariants for symplectic 4-manifolds,” pp. 175–210 in Gauge theory and symplectic geometry (Montreal, 3–14 July 1995). Edited by J. Hurtubise and F. Lalonde. NATO ASI Series C: Mathematical and Physical Sciences 488. Kluwer (Dordrecht), 1997. Based on notes taken by Wladyslav Lorek. MR 1461573 Zbl 0881.57037 incollection

Taubes’s recent spectacular work setting up a correspondence between \( J \)-holomorphic curves in symplectic 4-manifolds and solutions of the Seiberg–Witten equations counts \( J \)-holomorphic curves in a somewhat new way. The “standard” theory concerns itself with moduli spaces of connected curves, and gives rise to Gromov–Witten invariants: see, for example, [McDuff and Salamon 1994; Ruan and Tian 1995, 1996]. However, Taubes’s curves arise as zero sets of sections and so need not be connected. These notes are in the main expository. We first discuss the invariants as Taubes defined them, and then discuss some alternatives, showing, for example, a way of dealing with multiply-covered exceptional spheres. We also calculate some examples, in particular finding the Gromov invariant of the fiber class of an elliptic surface by counting \( J \)-holomorphic curves, rather than going via Seiberg–Witten theory.

Wladyslav Lorek	Related
François Lalonde	Related
Jacques Hurtubise	Related

P. C. Kenschaft: “Dusa Waddington McDuff,” pp. 137–142 in Notable women in mathematics: A biographical dictionary. Edited by C. Morrow and T. Perl. Greenwood Press (Westport, CN), 1998. incollection

Charlene Morrow	Related
Patricia Clark Kenschaft	Related
Teri Hoch Perl	Related

F. Lalonde, D. McDuff, and L. Polterovich: “On the flux conjectures,” pp. 69–85 in Geometry, topology, and dynamics (Montreal, 26–30 June 1995). Edited by F. Lalonde. CRM Proceedings Lecture Notes 15. American Mathematical Society (Providence, RI), 1998. MR 1619124 Zbl 0974.53062 ArXiv dg-ga/9706015 incollection

François Lalonde	Related
Leonid V. Polterovich	Related

D. McDuff: “From symplectic deformation to isotopy,” pp. 85–99 in Topics in symplectic 4-manifolds (Irvine, CA, 28–30 March 1996). Edited by R. J. Stern. First International Press Lecture Series 1. International Press (Cambridge, MA), 1998. MR 1635697 Zbl 0928.57018 ArXiv dg-ga/9606004 incollection

D. McDuff: “Symplectic structures — A new approach to geometry,” Notices Am. Math. Soc. 45 : 8 (1998), pp. 952–960. Based on the author’s AWM Emmy Noether Lecture given in Baltimore, January 1998. MR 1644353 Zbl 0906.53023 article

D. McDuff: “Recent developments in symplectic topology,” pp. 28–42 in European Congress of Mathematics (Budapest, 22–26 July 1996), vol. II. Edited by A. Balog, G. O. H. Katona, A. Recski, and D. Sza’sz. Progress in Mathematics 169. Birkhäuser (Basel), 1998. MR 1645817 Zbl 0913.58024 incollection

A. Balog	Related
András Recski	Related
D. Szász	Related
Gyula O. H. Katona	Related

D. McDuff: “Fibrations in symplectic topology,” pp. 339–357 in Proceedings of the International Congress of Mathematicians (Berlin, 18–27 August 1998), published as Doc. Math. Extra Volume ICM 1998 : I. Issue edited by G. Fischer and U. Rehmann. 1998. MR 1648038 Zbl 0907.53022 incollection

Every symplectic form on a \( 2n \)-dimensional manifold is locally the Cartesian product of \( n \) area forms. This local product structure has global implications in symplectic topology. After briefly reviewing the most important achievements in symplectic topology of the past 4 years, the talk will discuss several different situations in which one can see this influence: for example, the use of fibered mappings in the construction of efficient symplectic embeddings of fat ellipsoids into small balls, and the theory of Hamiltonian fibrations (work of Lalonde, Polterovich, Salamon and the speaker). The most spectacular example is Donaldson’s recent work, showing that every compact symplectic manifold admits a symplectic Lefschetz pencil.

Ulf Rehmann	Related
Gerd Fischer	Related

D. McDuff and D. Salamon: Introduction to symplectic topology, 2nd edition. Oxford Mathematical Monographs. Clarendon Press (Oxford), 1998. Republication of 1995 original. MR 1698616 Zbl 1066.53137 book

L. Love: Dusa McDuff, 1999. Online article. Part of the series “Biographies of Women Mathematicians”. misc

D. McDuff: “Symplectic topology and capacities,” pp. 69–81 in Prospects in mathematics (Princeton, NJ, 17–21 March 1996). Edited by H. Rossi. American Mathematical Society (Providence, RI), 1999. MR 1660473 Zbl 0947.53043 incollection

F. Lalonde, D. McDuff, and L. Polterovich: “Topological rigidity of Hamiltonian loops and quantum homology,” Invent. Math. 135 : 2 (1999), pp. 369–385. MR 1666763 Zbl 0907.58004 article

This paper studies the question of when a loop \( \phi = \{\phi_t\}_{0\leq t\leq 1} \) in the group \( \mathrm{Symp}(M,\omega) \) of symplectomorphisms of a symplectic manifold \( (M,\omega) \) is isotopic to a loop that is generated by a time-dependent Hamiltonian function. (Loops with this property are said to be Hamiltonian.) Our main result is that Hamiltonian loops are rigid in the following sense: if \( \phi \) is Hamiltonian with respect to \( \omega \), and if \( \phi^{\prime} \) is a small perturbation of \( \phi \) that preserves another symplectic form \( \omega^{\prime} \), then \( \phi^{\prime} \) is Hamiltonian with respect to \( \omega^{\prime} \). This allows us to get some new information on the structure of the flux group, i.e. the image of \( \pi_1(\mathrm{Symp}(M,\omega)) \) under the flux homomorphism. We give a complete proof of our result for some manifolds, and sketch the proof in general. The argument uses methods developed by Seidel for studying properties of Hamiltonian loops via the quantum homology of \( M \).

François Lalonde	Related
Leonid V. Polterovich	Related

D. McDuff: “Introduction to symplectic topology,” pp. 7–33 in Symplectic geometry and topology (Providence, RI). Edited by Y. Eliashberg and L. Traynor. IAS/Park City Mathematics Series 7. American Mathematical Society, 1999. MR 1702941 Zbl 0978.53120 incollection

Lisa Traynor	Related
Yakov M. Eliashberg	Related

D. McDuff: “The virtual moduli cycle,” pp. 73–102 in Northern California symplectic geometry seminar (Berkeley and Stanford, CA, 1989–1998). Edited by Y. Eliashberg, D. Fuchs, D. Ratiu, and A. Weinstein. AMS Translation Series 2 196. American Mathematical Society (Providence, RI), 1999. MR 1736215 Zbl 0958.53060 incollection

This article is an attempt to describe one possible construction of the virtual moduli cycle that is used as a tool in the construction of Gromov–Witten invariants for a general symplectic manifold. There are many different versions of this construction. Here I will in the main follow [Liu and Tian 1998; 1999], since their approach (when modified by an idea of Siebert’s) seems to involve the least amount of analysis. However, it does involved quite a bit of topology, some of which they only outline. The aim here is to flesh out the picture and to explain the different ingredients that are needed to make the construction work. We do not try to give full proofs, nor do we work out all the ideas in full generality.

D. Rațiu	Related
Alan Weinstein	Related
Yakov M. Eliashberg	Related
Dmitry Fuchs	Related	Volume

D. McDuff: “Almost complex structures on \( S^2\times S^2 \),” Duke Math. J. 101 : 1 (2000), pp. 135–177. MR 1733733 Zbl 0974.53020 article

D. McDuff: “A glimpse into symplectic geometry,” pp. 175–187 in Mathematics: Frontiers and perspectives. Edited by V. I. Arnol’d, M. Atiyah, P. Lax, and B. Mazur. American Mathematical Society (Providence, RI), 2000. MR 1754776 Zbl 0964.53003 incollection

M. F. Atiyah	Related	Volume
Vladimir Igorevich Arnold	Related
Barry C. Mazur	Related
Peter D. Lax	Related

M. Abreu and D. McDuff: “Topology of symplectomorphism groups of rational ruled surfaces,” J. Am. Math. Soc. 13 : 4 (2000), pp. 971–1009. MR 1775741 Zbl 0965.57031 article

Let \( M \) be either \( S^2\times S^2 \) or the one point blow-up \( \mathbb{C}P^2\# \overline{\mathbb{C}P}^2 \) of \( \mathbb{C}P^2 \). In both cases \( M \) carries a family of symplectic forms \( \omega_{\lambda} \), where \( \lambda > -1 \) determines the cohomology class \( [\omega_\lambda] \). This paper calculates the rational (co)homology of the group \( G_\lambda \) of symplectomorphisms of \( (M,\omega_\lambda) \) as well as the rational homotopy type of its classifying space \( BG_\lambda \). It turns out that each group \( G_\lambda \) contains a finite collection \( K_k \), \( k = 0 \),\( \dots,\ell = \ell(\lambda) \), of finite dimensional Lie subgroups that generate its homotopy. We show that these subgroups “asymptotically commute”, i.e. all the higher Whitehead products that they generate vanish as \( \lambda\to \infty \). However, for each fixed \( \lambda \) there is essentially one nonvanishing product that gives rise to a “jumping generator” \( w_\lambda \) in \( H^*(G_\lambda) \) and to a single relation in the rational cohomology ring \( H^*(BG_\lambda) \). An analog of this generator \( w_\lambda \) was also seen by Kronheimer in his study of families of symplectic forms on 4-manifolds using Seiberg–Witten theory. Our methods involve a close study of the space of \( \omega_\lambda \)-compatible almost complex structures on \( M \).

D. McDuff: “Quantum homology of fibrations over \( S^2 \),” Int. J. Math. 11 : 5 (2000), pp. 665–721. MR 1780735 Zbl 1110.53307 article

This paper studies the (small) quantum homology and cohomology of fibrations \( p: P\to S^2 \) whose structural group is the group of Hamiltonian symplectomorphisms of the fiber \( (M,\omega) \). It gives a proof that the rational cohomology splits additively as the vector space tensor product \[ H^*(M)\otimes H^*(S^2) ,\] and investigates conditions under which the ring structure also splits, thus generalizing work of Lalonde–McDuff–Polterovich and Seidel. The main tool is a study of certain operations in the quantum homology of the total space \( P \) and of the fiber \( M \), whose properties reflect the relations between the Gromov–Witten invariants of \( P \) and \( M \). In order to establish these properties we further develop the language introduced in [22] to describe the virtual moduli cycle (defined by Liu–Tian, Fukaya–Ono, Li–Tian, Ruan and Siebert).

D. McDuff and J. Slimowitz: “Hofer–Zehnder capacity and length minimizing Hamiltonian paths,” Geom. Topol. 5 : 2 (2001), pp. 799–830. MR 1871405 Zbl 1002.57056 article

We use the criteria of Lalonde and McDuff to show that a path that is generated by a generic autonomous Hamiltonian is length minimizing with respect to the Hofer norm among all homotopic paths provided that it induces no non-constant closed trajectories in \( M \). This generalizes a result of Hofer for symplectomorphisms of Euclidean space. The proof for general \( M \) uses Liu–Tian’s construction of \( S^1 \)-invariant virtual moduli cycles. As a corollary, we find that any semifree action of \( S^1 \) on \( M \) gives rise to a nontrivial element in the fundamental group of the symplectomorphism group of \( M \). We also establish a version of the area-capacity inequality for quasicylinders.

D. McDuff: “Symplectomorphism groups and almost complex structures,” pp. 527–556 in Essays on geometry and related topics: Mémoires dédies à André Haefliger [Essays on geometry and related topics: Memoirs dedicated to André Haefliger], vol. 2. Edited by É. Ghys. Monographie de lÉnseignement Mathématique 38. Kundig (Geneva), 2001. MR 1929338 Zbl 1010.53064 ArXiv math/0010274 incollection

André Haefliger	Related
Étienne Ghys	Related

D. McDuff: “Geometric variants of the Hofer norm,” J. Sympl. Geom. 1 : 2 (2002), pp. 197–252. MR 1959582 Zbl 1037.37033 article

This note discusses some geometrically defined seminorms on the group \( \mathrm{Ham}(M,\omega) \) of Hamiltonian diffeomorphisms of a closed symplectic manifold \( (M,\omega) \), giving conditions under which they are nondegenerate and explaining their relation to the Hofer norm. As a consequence we show that if an element in \( \mathrm{Ham}(M,\omega) \) is sufficiently close to the identity in the \( C^2 \)-topology then it may be joined to the identity by a path whose Hofer length is minimal among all paths, not just among paths in the same homotopy class relative to endpoints. Thus, true geodesics always exist for the Hofer norm. The main step in the proof is to show that a “weighted” version of the nonsqueezing theorem holds for all fibrations over \( S^2 \) generated by sufficiently short loops. Further, an example is given showing that the Hofer norm may differ from the sum of the one sided seminorms.

F. Lalonde and D. McDuff: “Cohomological properties of ruled symplectic structures,” pp. 79–99 in Mirror symmetry IV: Proceedings of the conference on strings, duality, and geometry (Montreal, March 2000). Edited by E. D’Hoker, S.-T. Yau, and D. H. Phong. AMS/IP Studies in Advanced Mathematics 33. American Mathematical Society (Providence, RI), 2002. MR 1968218 Zbl 1090.53071 ArXiv math/0010277 incollection

François Lalonde	Related
Duong H. Phong	Related
Eric D'Hoker	Related
Shing-Tung Yau	Related

F. Lalonde and D. McDuff: “Symplectic structures on fiber bundles,” Topology 42 : 2 (March 2003), pp. 309–347. Errata were published in Topology 44:6 (2005). MR 1941438 Zbl 1032.53077 article

Let \( \pi:P\to B \) be a locally trivial fiber bundle over a connected \( \mathrm{CW} \) complex \( B \) with fiber equal to the closed symplectic manifold \( (M,\omega) \). Then \( \pi \) is said to be a symplectic fiber bundle if its structural group is the group of symplectomorphisms \( \mathrm{Symp}(M,\omega) \), and is called Hamiltonian if this group may be reduced to the group \( \mathrm{Ham}(M,\omega) \) of Hamiltonian symplectomorphisms. In this paper, building on prior work by Seidel and Lalonde, McDuff and Polterovich, we show that these bundles have interesting cohomological properties. In particular, for many bases \( B \) (for example when \( B \) is a sphere, a coadjoint orbit or a product of complex projective spaces) the rational cohomology of \( P \) is the tensor product of the cohomology of \( B \) with that of \( M \). As a consequence the natural action of the rational homology \( H_k(\mathrm{Ham}(M)) \) on \( H_*(M) \) is trivial for all \( M \) and all \( k > 0 \).

D. McDuff and D. Salamon: \( J \)-holomorphic curves and symplectic topology. AMS Colloquium Publications 52. American Mathematical Society (Providence, RI), 2004. Republished in 2012. MR 2045629 Zbl 1064.53051 book

D. McDuff: “Lectures on groups of symplectomorphisms,” pp. 43–78 in The proceedings of the 23rd winter school “Geometry and physics” (Srní, Czech Republic, 18–25 January 2003). Edited by J. Slovák. Supplemento ai Rendiconti del Circolo Matemàtico di Palermo. Serie II 72. 2004. MR 2069395 Zbl 1078.53088 ArXiv math/0201032 incollection

D. McDuff: “A survey of the topological properties of symplectomorphism groups,” pp. 173–193 in Topology, geometry and quantum field theory: Proceedings of the 2002 Oxford symposium in honour of the 60th birthday of Graeme Segal (Oxford, 24–29 June 2002). Edited by U. Tillmann. London Mathematical Society Lecture Note Series 308. Cambridge University Press, 2004. MR 2079375 Zbl 1102.57013 ArXiv SG/0404340 incollection

The special structures that arise in symplectic topology (particular Gromov–Witten invariants and quantum homology) place as yet rather poorly understood restrictions on the topological properties of symplectomorphism groups. This article surveys some recent work by Abreu, Lalonde, McDuff, Polterovich and Seidel, concentrating particularly on the homotopy properties of the action of the group of Hamiltonian symplectomorphisms on the underlying manifold \( M \). It sketches the proof that the evaluation map \[ \pi_1(\mathrm{Ham}(M))\to\pi_1(M) \] given by \[ \{\phi_t\}\mapsto\{\phi_t(x_0)\} \] is trivial, as well as explaining similar vanishing results for the action of the homology of \( \mathrm{Ham}(M) \) on the homology of \( M \). Applications to Hamiltonian stability are discussed.

Ulrike Tillmann	Related
Graeme Bryce Segal	Related

J. Kedra and D. McDuff: “Homotopy properties of Hamiltonian group actions,” Geom. Topol. 9 (2005), pp. 121–162. MR 2115670 Zbl 1077.53072 article

F. Lalonde and D. McDuff: “Errata to ‘Symplectic structures on fiber bundles’,” Topology 44 : 6 (2005), pp. 1301–1303. Errata for article published in Topology 42:2 (2003). MR 2168577 article

D. McDuff: “Enlarging the Hamiltonian group,” J. Symp. Geom. 3 : 4 (2005), pp. 481–530. MR 2235852 Zbl 1109.53078 article

This paper investigates ways to enlarge the Hamiltonian subgroup \( \mathrm{Ham} \) of the symplectomorphism group \( \mathrm{Symp} \) of a symplectic manifold \( (M,\omega) \) to a group that both intersects every connected component of \( \mathrm{Symp} \) and characterizes symplectic bundles with fiber \( M \) and closed connection form. As a consequence, it is shown that bundles with closed connection form are stable under appropriate small perturbations of the symplectic form. Further, the manifold \( (M,\omega) \) has the property that every symplectic \( M \)-bundle has a closed connection form if and only if the flux group vanishes and the flux homomorphism extends to a crossed homomorphism defined on the whole group \( \mathrm{Symp} \). The latter condition is equivalent to saying that a connected component of the commutator subgroup \( [\mathrm{Symp},\mathrm{Symp}] \) intersects the identity component of \( \mathrm{Symp} \) only if it also intersects \( \mathrm{Ham} \). It is not yet clear when this condition is satisfied. We show that if the symplectic form vanishes on 2-tori, the flux homomorphism extends to the subgroup of \( \mathrm{Symp} \) acting trivially on \( \pi_1(M) \). We also give an explicit formula for the Kotschick–Morita extension of the flux homomorphism in the monotone case. The results in this paper belong to the realm of soft symplectic topology, but raise some questions that may need hard methods to answer.

D. McDuff and S. Tolman: On nearly semifree circle actions. Preprint, March 2005. ArXiv math/0503467 techreport

Recall that an effective circle action is semifree if the stabilizer subgroup of each point is connected. We show that if \( (M,\omega) \) is a coadjoint orbit of a compact Lie group \( G \) then every element of \( \pi_1(G) \) may be represented by a semifree \( S^1 \)-action. A theorem of McDuff–Slimowitz then implies that \( \pi_1(G) \) injects into \( \pi_1(\operatorname{Ham}(M, \omega)) \), which answers a question raised by Weinstein. We also show that a circle action on a manifold \( M \) which is semifree near a fixed point \( x \) cannot contract in a compact Lie subgroup \( G \) of the diffeomorphism group unless the action is reversed by an element of \( G \) that fixes the point \( x \). Similarly, if a circle acts in a Hamiltonian fashion on a manifold \( (M,\omega) \) and the stabilizer of every point has at most two components, then the circle cannot contract in a compact Lie subgroup of the group of Hamiltonian symplectomorphism unless the circle is reversed by an element of \( G \).

D. McDuff: “Symplectomorphism groups and quantum cohomology,” pp. 457–471 in The unity of mathematics: In honor of the ninetieth birthday of I. M. Gelfand (Cambridge, MA, 31 August–4 September 2003). Edited by P. Etingof, V. Retakh, and I. M. Singer. Progress in Mathematics 244. Birkhäuser (Boston), 2006. MR 2181814 Zbl 1099.53057 incollection

Vladimir Solomonovich Retakh	Related
Pavel Ilyich Ètingof	Related
Israïl Moiseevich Gelfand	Related
Isadore M. Singer	Related	Volume

D. McDuff: “Floer theory and low dimensional topology,” Bull. Am. Math. Soc. (N.S.) 43 : 1 (2006), pp. 25–42. MR 2188174 Zbl 1084.57028 article

The new 3- and 4-manifold invariants recently constructed by Ozsváth and Szabó are based on a Floer theory associated with Heegaard diagrams. The following notes try to give an accessible introduction to their work. In the first part we begin by outlining traditional Morse theory, using the Heegaard diagram of a 3-manifold as an example. We then describe Witten’s approach to Morse theory and how this led to Floer theory. Finally, we discuss Lagrangian Floer homology. In the second part, we define the Heegaard Floer complexes, explaining how they arise as a special case of Lagrangian Floer theory. We then briefly describe some applications, in particular the new 4-manifold invariant, which is conjecturally just the Seiberg–Witten invariant.

D. McDuff and S. Tolman: “Topological properties of Hamiltonian circle actions,” Int. Math. Res. Pap. 2006 (2006), pp. 1–77. MR 2210662 Zbl 1123.53044 article

This paper studies Hamiltonian circle actions, that is, circle subgroups of the group \( \mathrm{Ham}(M,\omega) \) of Hamiltonian symplectomorphisms of a closed symplectic manifold \( (M,\omega) \). Our main tool is the Seidel representation of \( \pi_1(\mathrm{Ham}(M,\omega)) \) in the units of the quantum homology ring. We show that if the weights of the action at the points at which the moment map is a maximum are sufficiently small, then the circle represents a nonzero element of \( \pi_1(\mathrm{Ham}(M,\omega)) \). Further, if the isotropy has order at most two and the circle contracts in \( \mathrm{Ham}(M,\omega) \), then various symmetry properties hold. For example, the image of the normalized moment map is a symmetric interval \( [-a,a] \). If the action is semifree (i.e., the isotropy weights are 0 or \( \pm 1 \)), then we calculate the leading order term in the Seidel representation, an important technical tool in understanding the quantum cohomology of manifolds that admit semifree Hamiltonian circle actions. If the manifold is toric, we use our results about this representation to describe the basic multiplicative structure of the quantum cohomology ring of an arbitrary toric manifold. There are two important technical ingredients; one relates the equivariant cohomology of \( M \) to the Morse flow of the moment map, and the other is a version of the localization principle for calculating Gromov–Witten invariants on symplectic manifolds with \( S^1 \)-actions.

D. Auroux, S. Donaldson, V. Guillemin, T. Mrowka, and G. Tian: “Dedication to Dusa McDuff,” J. Symplectic Geom. 5 : 1 (2007), pp. i. MR 2371180 article

Gang Tian	Related
Tomasz Stanislaw Mrowka	Related
Denis S. Auroux	Related
Victor William Guillemin	Related
S. K. Donaldson	Related

D. McDuff: “The symplectomorphism group of a blow up,” Geom. Dedicata 132 (2008), pp. 1–29. MR 2396906 Zbl 1155.53055 ArXiv SG/0509664 article

We study the relation between the symplectomorphism group \( \operatorname{Symp} M \) of a closed connected symplectic manifold \( M \) and the symplectomorphism and diffeomorphism groups \( \operatorname{Symp} \tilde{M} \) and \( \operatorname{Diff} \tilde{M} \) of its one point blow up \( \tilde{M} \). There are three main arguments. The first shows that for any oriented \( M \) the natural map from \( \pi_1(M) \) to \( \pi_0(\operatorname{Diff} \tilde{M}) \) is often injective. The second argument applies when \( M \) is simply connected and detects nontrivial elements in the homotopy group \( \pi_1(\operatorname{Diff} \tilde{M}) \) that persist into the space of self-homotopy equivalences of \( \tilde{M} \). Since it uses purely homological arguments, it applies to \( c \)-symplectic manifolds \( (M,a) \), that is, to manifolds of dimension \( 2n \) that support a class \( a \in H^2(M;\mathbb{R}) \) such that \( a^n \neq 0 \). The third argument uses the symplectic structure on \( M \) and detects nontrivial elements in the (higher) homology of \( B\operatorname{Symp} M \) using characteristic classes defined by parametric Gromov–Witten invariants. Some results about many point blow ups are also obtained. For example we show that if \( M \) is the four-torus with \( k \)-fold blow up \( \tilde{M}_k \) (where \( k > 0 \)) then \( \pi_1(\operatorname{Diff} \tilde{M}_k) \) is not generated by the groups \( \pi_1(\operatorname{Symp}(\tilde{M}_k,\tilde{\omega})) \) as \( \tilde{\omega} \) ranges over the set of all symplectic forms on \( \tilde{M}_k \).

D. McDuff: “Hamiltonian \( S^1 \)-manifolds are uniruled,” Duke Math. J. 146 : 3 (2009), pp. 449–507. MR 2484280 Zbl 1183.53080 article

D. McDuff: “Symplectic embeddings of 4-dimensional ellipsoids,” J. Topol. 2 : 1 (2009), pp. 1–22. MR 2499436 Zbl 1166.53051 article

We show how to reduce the problem of symplectically embedding one 4-dimensional rational ellipsoid into another to a problem of embedding disjoint unions of balls into \( \mathbb{C}P^2 \). For example, the problem of embedding the ellipsoid \( E(1,k) \) into a ball \( B \) is equivalent to that of embedding \( k \) disjoint equal balls into \( \mathbb{C}P^2 \), and so can be solved by the work of Gromov, McDuff–Polterovich, and Biran. (Here \( k \) is the ratio of the area of the major axis to that of the minor axis.) As a consequence we show that the ball may be fully filled by the ellipsoid \( E(1,k) \) for \( k = 1,\,4 \) and all \( k\geq 9 \), thus answering a question raised by Hofer.

D. McDuff: “Some 6-dimensional Hamiltonian \( S^1 \)-manifolds,” J. Topol. 2 : 3 (2009), pp. 589–623. MR 2546587 Zbl 1189.53073 article

In an earlier paper we explained how to convert the problem of symplectically embedding one 4-dimensional ellipsoid into another into the problem of embedding a certain set of disjoint balls into \( \mathbb{C}P^2 \) by using a new way to desingularize orbifold blow-ups \( Z \) of the weighted projective space \( \mathbb{C}P_{1,m,n}^2 \). We now use a related method to construct symplectomorphisms of these spaces \( Z \). This allows us to construct some well-known Fano 3-folds (including the Mukai–Umemura 3-fold) in purely symplectic terms using a classification by Tolman of a particular class of Hamiltonian \( S^1 \)-manifolds. We also show that (modulo scaling) these manifolds are uniquely determined by their fixed-point data up to equivariant symplectomorphism. As part of this argument, we show that the symplectomorphism group of a certain weighted blow-up of a weighted projective plane is connected.

D. McDuff: “Symplectic embeddings and continued fractions: A survey,” Jpn. J. Math. 4 : 2 (2009), pp. 121–139. MR 2576029 Zbl 1222.53081 article

D. McDuff: “Monodromy in Hamiltonian Floer theory,” Comment. Math. Helv. 85 : 1 (2010), pp. 95–133. MR 2563682 Zbl 1222.53092 article

Schwarz showed that when a closed symplectic manifold \( (M,\omega) \) is symplectically aspherical (i.e. the symplectic form and the first Chern class vanish on \( \pi_2(M) \)) then the spectral invariants, which are initially defined on the universal cover of the Hamiltonian group, descend to the Hamiltonian group \( \mathrm{Ham}(M,\omega) \). In this note we describe less stringent conditions on the Chern class and quantum homology of \( M \) under which the (asymptotic) spectral invariants descend to \( \mathrm{Ham}(M,\omega) \). For example, they descend if the quantum multiplication of \( M \) is undeformed and \( H_2(M) \) has rank \( > 1 \), or if the minimal Chern number is at least \( n + 1 \) (where \( \dim M = 2n \)) and the even cohomology of \( M \) is generated by divisors. The proofs are based on certain calculations of genus zero Gromov–Witten invariants. As an application, we show that the Hamiltonian group of the one point blow up of \( T^4 \) admits a Calabi quasimorphism. Moreover, whenever the (asymptotic) spectral invariants descend it is easy to see that \( \mathrm{Ham}(M,\omega) \) has infinite diameter in the Hofer norm. Hence our results establish the infinite diameter of \( \mathrm{Ham} \) in many new cases. We also show that the area pseudonorm–a geometric version of the Hofer norm–is nontrivial on the (compactly supported) Hamiltonian group for all noncompact manifolds as well as for a large class of closed manifolds.

D. McDuff: “Loops in the Hamiltonian group: A survey,” pp. 127–148 in Symplectic topology and measure preserving dynamical systems (Snowbird, UT, 1–5 July 2007). Edited by A. Fathi, Y.-G. Oh, and C. Viterbo. Contemporary Mathematics 512. American Mathematical Society (Providence, RI), 2010. MR 2605315 Zbl 1202.53083 ArXiv 0711.4086 incollection

This note describes some recent results about the homotopy properties of Hamiltonian loops in various manifolds, including toric manifolds and one point blow ups. We describe conditions under which a circle action does not contract in the Hamiltonian group, and construct an example of a loop \( \gamma \) of diffeomorphisms of a symplectic manifold \( M \) with the property that none of the loops smoothly isotopic to \( \gamma \) preserve any symplectic form on \( M \). We also discuss some new conditions under which the Hamiltonian group has infinite Hofer diameter. Some of the methods used are classical (Weinstein’s action homomorphism and volume calculations), while others use quantum methods (the Seidel representation and spectral invariants).

Yong-Geun Oh	Related
Claude Viterbo	Related
Albert Fathi	Related

D. McDuff: “What is symplectic geometry?,” pp. 33–53 in European women in mathematics: Proceedings of the 13th general meeting (Cambridge, UK, 3–6 September 2007). Edited by C. Hobbs and S. Paycha. World Scientific Publishers (Hackensack, NJ), 2010. MR 2605361 Zbl 1192.53001 incollection

Catherine Ann Hobbs	Related
Sylvie Paycha	Related

D. McDuff and S. Tolman: “Polytopes with mass linear functions, I,” Int. Math. Res. Not. 2010 : 8 (2010), pp. 1506–1574. MR 2628835 Zbl 1202.52010 ArXiv 0807.0900 article

Let \( \Delta \) be an \( n \)-dimensional polytope that is simple, that is, exactly \( n \) facets meet at each vertex. An affine function is “mass linear” on \( \Delta \) if its value on the center of mass of \( \Delta \) depends linearly on the positions of the supporting hyperplanes. On the one hand, we show that certain types of symmetries of \( \Delta \) give rise to nonconstant mass linear functions on \( \Delta \). On the other hand, we show that most polytopes do not admit any nonconstant mass linear functions. Further, if every affine function is mass linear on \( \Delta \), then \( \Delta \) is a product of simplices. Our main result is a classification of all smooth polytopes of dimension \( \leq 3 \) which admit nonconstant mass linear functions. In particular, there is only one family of smooth three-dimensional polytopes — and no polygons — that admit “essential mass linear functions,” that is, mass linear functions that do not arise from the symmetries described above. In part II, we will complete this classification in the four-dimensional case. These results have geometric implications. Fix a symplectic toric manifold \( (M,\omega,T,\Phi) \) with moment polytope \( \Delta = \Phi(M) \). Let \[ \mathrm{Symp}_0(M,\omega) \] denote the identity component of the group of symplectomorphisms of \( (M,\omega) \). Any linear function \( H \) on \( \Delta \) generates a Hamiltonian \( \mathbb{R} \) action on \( M \) whose closure is a subtorus \( T_H \) of \( T \). We show that if the map \[ \pi_1(T_H)\to\pi_1(\mathrm{Symp}_0(M,\omega)) \] has finite image, then \( H \) is mass linear. Combining this fact and the claims described above, we prove that in most cases, the induced map \[ \pi_1(T)\to\pi_1(\mathrm{Symp}_0(M,\omega)) \] is an injection. Moreover, the map does not have finite image unless \( M \) is a product of projective spaces. Note also that there is a natural maximal compact connected subgroup \[ \mathrm{Isom}_0(M)\subset\mathrm{Symp}_0(M,\omega) ;\] there is a natural compatible complex structure \( J \) on \( M \), and \( \mathrm{Isom}_0(M) \) is the identity component of the group of symplectomorphisms that also preserve this structure. We prove that if the polytope \( \Delta \) supports no essential mass linear functions, then the induced map \[ \pi(\mathrm{Isom}_0(M))\to\pi_1(\mathrm{Symp}_0(M,\omega)) \] is injective. Therefore, this map is injective for all four-dimensional symplectic toric manifolds and is injective in the six-dimensional case unless \( M \) is a \( \mathbb{C}P^2 \) bundle over \( \mathbb{C}P^1 \).

L. Polterovich: “Focus on the scientist: Dusa McDuff,” MSRI Emissary (Spring 2010), pp. 3. article

D. J. Albers: “Dusa McDuff,” Chapter 12, pp. 215–239 in Fascinating mathematical people: Interviews and memoirs. Edited by D. J. Albers and G. L. Alexanderson. Princeton University Press, 2011. incollection

Donald J. Albers	Related
Gerald Lee Alexanderson	Related

D. McDuff: “Displacing Lagrangian toric fibers via probes,” pp. 131–160 in Low-dimensional and symplectic topology (Athens, GA, 18–29 May 2009). Edited by M. Usher. Proceedings of Symposia in Pure Mathematics 82. American Mathematical Society (Providence, RI), 2011. MR 2768658 Zbl 1255.53063 ArXiv 0904.1686 incollection

This note studies the geometric structure of monotone moment polytopes (the duals of smooth Fano polytopes) using probes. The latter are line segments that enter the polytope at an interior point of a facet and whose direction is integrally transverse to this facet. A point inside the polytope is displaceable by a probe if it lies less than half way along it. Using a construction due to Fukaya–Oh–Ohta–Ono, we show that every rational polytope has a central point that is not displaceable by probes. In the monotone case, this central point is its unique interior integral point, and we show that every other point is displaceable by probes if and only if the polytope satisfies the star Ewald condition. (This is a strong version of the Ewald conjecture concerning the integral symmetric points in the polytope.) Further, in dimensions up to and including three every monotone polytope is star Ewald. These results are closely related to the Fukaya–Oh–Ohta–Ono calculations of the Floer homology of the Lagrangian fibers of a toric symplectic manifold, and have applications to questions introduced by Entov–Polterovich about the displaceability of these fibers.

D. McDuff: “The topology of toric symplectic manifolds,” Geom. Topol. 15 : 1 (2011), pp. 145–190. MR 2776842 Zbl 1218.14045 article

This is a collection of results on the topology of toric symplectic manifolds. Using an idea of Borisov, we show that a closed symplectic manifold supports at most a finite number of toric structures. Further, the product of two projective spaces of complex dimension at least two (and with a standard product symplectic form) has a unique toric structure. We then discuss various constructions, using wedging to build a monotone toric symplectic manifold whose center is not the unique point displaceable by probes, and bundles and blow ups to form manifolds with more than one toric structure. The bundle construction uses the McDuff–Tolman concept of mass linear function. Using Timorin’s description of the cohomology algebra via the volume function we develop a cohomological criterion for a function to be mass linear, and explain its relation to Shelukhin’s higher codimension barycenters.

D. McDuff: “The Hofer conjecture on embedding symplectic ellipsoids,” J. Diff. Geom. 88 : 3 (2011), pp. 519–532. MR 2844441 Zbl 1239.53109 article

D. McDuff and F. Schlenk: “The embedding capacity of 4-dimensional symplectic ellipsoids,” Ann. Math. (2) 175 : 3 (2012), pp. 1191–1282. MR 2912705 Zbl 1254.53111 article

This paper calculates the function \( c(a) \) whose value at \( a \) is the infimum of the size of a ball that contains a symplectic image of the ellipsoid \( E(1,a) \). (Here \( a\geq 1 \) is the ratio of the area of the large axis to that of the smaller axis.) The structure of the graph of \( c(a) \) is surprisingly rich. The volume constraint implies that \( c(a) \) is always greater than or equal to the square root of \( a \), and it is not hard to see that this is equality for large \( a \). However, for \( a \) less than the fourth power \( \tau^4 \) of the golden ratio, \( c(a) \) is piecewise linear, with graph that alternately lies on a line through the origin and is horizontal. We prove this by showing that there are exceptional curves in blow ups of the complex projective plane whose homology classes are given by the continued fraction expansions of ratios of Fibonacci numbers. On the interval \( [\tau^4,7] \) we find \( c(a)=(a+1)/3 \). For \( a\geq 7 \), the function \( c(a) \) coincides with the square root except on a finite number of intervals where it is again piecewise linear.

The embedding constraints coming from embedded contact homology give rise to another capacity function \( c^{}_{\text{ECH}} \) which may be computed by counting lattice points in appropriate right angled triangles. According to Hutchings and Taubes, the functorial properties of embedded contact homology imply that \( c^{}_{\text{ECH}}(a)\leq c(a) \) for all \( a \). We show here that \( c^{}_{\text{ECH}}(a) \geq c(a) \) for all \( a \).

D. McDuff and D. Salamon: \( J \)-holomorphic curves and symplectic topology, 2nd edition. AMS Colloquium Publications 52. American Mathematical Society (Providence, RI), 2012. Republication of 2004 original. MR 2954391 Zbl 1272.53002 book

D. McDuff and S. Tolman: “Polytopes with mass linear functions, II: The four-dimensional case,” Int. Math. Res. Not. 2013 : 15 (2013), pp. 3509–3599. MR 3089735 ArXiv 1106.1623 article

This paper continues the analysis begun in part I of the structure of smooth moment polytopes \( \Delta\subset t^* \) that support a mass linear function \( H\in t \). As explained there, besides its purely combinatorial interest, this question is relevant to the study of the homomorphism \[ \pi_1(T^n)\to \pi_1(\mathrm{Symp}(M_{\Delta},\omega_{\Delta})) \] from the fundamental group of the torus \( T^n \) to that of the group of symplectomorphisms of the \( 2n \)-dimensional symplectic toric manifold \( (M_{\Delta},\omega_{\Delta})) \) associated to \( \Delta \). In Part I, we made a general investigation of this question and classified all mass linear pairs \( (\Delta,H) \) in dimensions up to 3. The main result of the current paper is a classification of all four-dimensional examples. Along the way, we investigate the properties of general constructions such as fibrations, blow ups, and expansions (or wedges), describing their effect on both moment polytopes and mass linear functions. We end by discussing the relation of mass linearity to Shelukhin’s notion of full mass linearity. The two concepts agree in dimensions up to and including 4. However, full mass linearity may be the more natural concept when considering the question of which blowups preserve mass linearity.

J. Latschev, D. McDuff, and F. Schlenk: “The Gromov width of 4-dimensional tori,” Geom. Topol. 17 : 5 (2013), pp. 2813–2853. MR 3190299 Zbl 1277.57024 article

Felix Schlenk	Related
Janko Latschev	Related

Prof. Dusa McDuff honored by the American Philosophical Society and American Mathematical S ociety, 18 November 2013. Barnard College online news article. misc

D. McDuff, S. Tabachnikov, and M. Saul: “Israel Moiseevich Gelfand, Part II.” Edited by V. Retakh. Notices Am. Math. Soc. 60 : 2 (2013), pp. 162–171. Zbl 1290.01026 article

Vladimir Solomonovich Retakh	Related
Israïl Moiseevich Gelfand	Related
Mark E. Saul	Related
Serge Lvovich Tabachnikov	Related

M. Abreu, M. S. Borman, and D. McDuff: “Displacing Lagrangian toric fibers by extended probes,” Algebr. Geom. Topol. 14 : 2 (2014), pp. 687–752. MR 3159967 Zbl 1288.53076 article

In this paper we introduce a new way of displacing Lagrangian fibers in toric symplectic manifolds, a generalization of McDuff’s original method of probes. Extended probes are formed by deflecting one probe by another auxiliary probe. Using them, we are able to displace all fibers in Hirzebruch surfaces except those already known to be nondisplaceable, and can also displace an open dense set of fibers in the weighted projective space \( \mathbb{P}(1,3,5) \) after resolving the singularities. We also investigate the displaceability question in sectors and their resolutions. There are still many cases in which there is an open set of fibers whose displaceability status is unknown.

Matthew Strom Borman	Related
Miguel Tribolet Abreu	Related

D. McDuff and E. Opshtein: “Nongeneric \( J \)-holomorphic curves and singular inflation,” Algebr. Geom. Topol. 15 : 1 (2015), pp. 231–286. MR 3325737 Zbl pre06425403 article

This paper investigates the geometry of a symplectic 4-manifold \( (M,\omega) \) relative to a \( J \)-holomorphic normal crossing divisor \( \mathscr{S} \). Extending work by Biran, we give conditions under which a homology class \( A \in H_2(M;\mathbb{Z}) \) with nontrivial Gromov invariant has an embedded \( J \)-holomorphic representative for some \( \mathscr{S} \)-compatible \( J \). This holds for example if the class \( A \) can be represented by an embedded sphere, or if the components of \( S \) are spheres with self-intersection \( -2 \). We also show that inflation relative to \( \mathscr{S} \) is always possible, a result that allows one to calculate the relative symplectic cone. It also has important applications to various embedding problems, for example of ellipsoids or Lagrangian submanifolds.

D. McDuff: “Corrigendum: ‘Symplectic embeddings of 4-dimensional ellipsoids’,” J. Topol. 8 : 4 (2015), pp. 1119–1122. Corrigendum to an article published in J. Topol. 2:1 (2009). MR 3431670 article

Interview: Dusa McDuff, 4 June 2015. Online video, 10:53. Interview at the Centre International de Rencontres Mathématiques. misc

Dusa McDuff

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