Abstract. The present authors introduced the notion of weakly unobstructed Lagrangian submanifolds and constructed their potential function PO purely in terms of A-model data in [FOOO3]. In this paper, we carry out explicit calculations involving PO on toric manifolds and study the relationship between this class of Lagrangian submanifolds with the earlier work of Givental [Gi1] which advocates that quantum cohomology ring is isomorphic to the Jacobian ring of a certain function, called the Landau-Ginzburg superpotential. Combining this study with the results from [FOOO3], we also apply the study to various examples to illustrate its implications to symplectic topology of Lagrangian fibers of toric manifolds. In particular we relate it to Hamiltonian displacement property of Lagrangian fibers and to Entov-Polterovich’s

In this work we construct Calabi quasi-morphisms on the universal cover ˜Ham(M) of the group of Hamiltonian diffeomorphisms for some non-monotone symplectic manifolds. This complements a result by Entov and Polterovich which applies in the monotone case. Moreover, in contrast with their work, we show that these quasi-morphisms descend to non-trivial homomorphisms on the fundamental group of Ham(M). 1

Abstract. We investigate the extrinsic topology of Lagrangian submanifolds and of their submanifolds in closed symplectic manifolds using Floer homological methods. The first result asserts that the homology class of a displaceable monotone Lagrangian submanifold vanishes in the homology of the ambient symplectic manifold. Combining this with spectral invariants we provide a new mechanism for proving Lagrangian intersection results e.g. entailing that any two simply connected Lagrangian submanifold in CP n ×CP n must intersect. Contents

Abstract. The chain complexes underlying Floer homology theories typically carry a real-valued filtration, allowing one to associate to each Floer homology class a spectral number defined as the infimum of the filtration levels of chains representing that class. These spectral numbers have been studied extensively in the case of Hamiltonian Floer homology by Oh, Schwarz, and others. We prove that the spectral number associated to any nonzero Floer homology class is always finite, and that the infimum in the definition of the spectral number is always attained. In the Hamiltonian case, this implies that what is known as the “nondegenerate spectrality ” axiom holds on all closed symplectic manifolds. Our proofs are entirely algebraic and rather elementary, and apply to any Floer-type theory (including Novikov homology) satisfying certain standard formal properties provided that one works with coefficients in a Novikov ring whose degree-zero part Λ0 is a field. The key ingredient is a theorem about linear transformations of finite-dimensional vector spaces over Novikov fields such as Λ0, which among other things asserts that the range of such a transformation is closed with respect to the ultrametric on Λn 0 induced by the valuation on Λ0. 1.

Abstract. Schwarz showed that when a closed symplectic manifold (M, ω) is symplectically aspherical (i.e. the symplectic form and the first Chern class vanish on π2(M)) then the spectral invariants, which are initially defined on the universal cover of the Hamiltonian group, descend to the Hamiltonian group Ham(M, ω). 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 Ham(M, ω). For example, they descend if the quantum multiplication of M is undeformed and H2(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 Ham(M, ω) has infinite diameter in the Hofer norm. Hence our results establish the infinite diameter of 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.

Abstract. This is a survey of a series of papers [FOOO3, FOOO4, FOOO5]. We discuss the calculation of the Floer cohomology of Lagrangian submanifold which is a T n orbit in a compact toric manifold. Applications to symplectic topology and to mirror symmetry are also discussed. Contents

For a class of symplectic manifolds, we introduce a functional which assigns a real number to any pair of continuous functions on the manifold. This functional has a number of interesting properties. On the one hand, it is Lipschitz with respect to the uniform norm. On the other hand, it serves as a measure of non-commutativity of functions in the sense of the Poisson bracket, the operation which involves first derivatives of the functions. Furthermore, the same functional gives rise to a non-trivial lower bound for the error of the simultaneous measurement of a pair of non-commuting Hamiltonians. These results manifest a link between the algebraic structure of the group of Hamiltonian diffeomorphisms and the function theory on a symplectic manifold. The above-mentioned functional comes from a special homogeneous quasi-morphism on the universal cover of the group, which is rooted in the Floer theory. a Partially supported by E. and J. Bishop Research Fund.

norms on groups of geometric origin

We review and streamline our previous results and the results of Y. Ostrover on the existence of Calabi quasi-morphisms and symplectic quasi-states on symplectic manifolds with semi-simple quantum homology. As an illustration, we discuss the case of symplectic toric Fano 4-manifolds. We present also new results due to D. McDuff: she observed that for the existence of quasi-morphisms/quasi-states it suffices to assume that the quantum homology contains a field as a direct summand, and she showed that this weaker condition holds true for one point blow-ups of non-uniruled symplectic manifolds. 1 Symplectic quasi-states and Calabi quasimorphisms Symplectic quasi-states: Let (M 2n, ω) be a closed connected 2n-dimensional symplectic manifold. A symplectic quasi-state on M is a (possibly non-linear!) real-valued functional ζ on the space C(M) of all continuous functions on M which satisfies the following conditions: Quasi-linearity: ζ(F + λG) = ζ(F) + λζ(G) for all λ ∈ R and for all functions F, G ∈ C ∞ (M) which commute with respect to the Poisson brackets:

Lie quasi-states on a real Lie algebra are functionals which are linear on any abelian subalgebra. We show that on the symplectic Lie algebra of rank at least 3 there is only one continuous non-linear Lie quasistate (up to a scalar factor, modulo linear functionals). It is related to the asymptotic Maslov index of paths of symplectic matrices. Contents 1 Introduction and main results 3 1.1 Lie quasi-states.......................... 3 1.2 Origins of Lie quasi-states.................... 4 1.3 Maslov quasi-state on sp (2n, R)................. 5