Related papers: From Euclidean field theory to hyperk\"ahler Floer…
In $n$-dimensional classical field theory one studies maps from $n$-dimensional manifolds in such a way that classical mechanics is recovered for $n=1$. In previous papers we have shown that the standard polysymplectic framework in which…
It is the goal of this paper to present the first steps for defining the analogue of Hamiltonian Floer theory for covariant field theory, treating time and space relativistically. While there already exist a number of competing geometric…
Under natural restrictions it is known that a nonlinear Schr\"odinger equation is a Hamiltonian PDE which defines a symplectic flow on a symplectic Hilbert space preserving the Hilbert norm. When the potential is one-periodic in time and…
We construct Hamiltonian Floer complexes associated to continuous, and even lower semi-continuous, time dependent exhaustion functions on geometrically bounded symplectic manifolds. We further construct functorial continuation maps…
In symplectic geometry, Floer theory is the most important tool to prove the existence of time-periodic solutions in Hamiltonian mechanics. The core observation is that the $L^2$-gradient lines of the symplectic action functional are…
We study a Floer-theoretic approach to harmonic maps from the two-torus into non-flat K\"ahler manifolds. Building on the complex-regularized polysymplectic (CRPS) formalism of [BF24], which provides a Hamiltonian description of harmonic…
By coupling a Hamiltonian mechanical system with a linear Hamiltonian field theory one obtains an infinite-dimensional Hamiltonian system with regularizing nonlinearity, where the underlying phase space is given by the product of a…
This paper is concerned with the rational symplectic field theory in the Floer case. For this observe that in the general geometric setup for symplectic field theory the contact manifolds can be replaced by mapping tori of symplectic…
The purpose of this mostly expository paper is to discuss a connection between Nielsen fixed point theory and symplectic Floer homology theory for symplectomorphisms of surface and a calculation of Seidel's symplectic Floer homology for…
We prove the existence of infinitely many time-periodic solutions of nonlinear Schr\"odinger equations using pseudo-holomorphic curve methods from Hamiltonian Floer theory. For the generalization of the Gromov-Floer compactness theorem to…
We extend the pseudoholomorphic curve methods from Floer theory to infinite-dimensional phase spaces and use our results to prove the existence of a forced time-periodic solution to a general Hamiltonian PDE with regularizing nonlinearity.…
Floer invented his theory in the mid eighties in order to prove the Arnol'd conjectures on the number of fixed point of Hamiltonian diffeomorphisms and Lagrangian intersections. Over the last thirty years, many versions of Floer homology…
According to the Arnold conjectures and Floer's proofs, there are non-trivial lower bounds for the number of periodic solutions of Hamiltonian differential equations on a closed symplectic manifold whose symplectic form vanishes on spheres.…
In symplectic topology one uses elliptic methods to prove rigidity results about symplectic manifolds and solutions of Hamiltonian equations on them, where the most basic example is given by geodesics on Riemannian manifolds. Harmonic maps…
This review paper is devoted to presenting the standard multisymplectic formulation for describing geometrically classical field theories, both the regular and singular cases. First, the main features of the Lagrangian formalism are…
We develop a family of deformations of the differential and of the pair-of-pants product on the Hamiltonian Floer complex of a symplectic manifold (M,\omega) which upon passing to homology yields ring isomorphisms with the big quantum…
In a first part we propose an introduction to multisymplectic formalisms, which are generalisations of Hamilton's formulation of Mechanics to the calculus of variations with several variables: we give some physical motivations, related to…
This is one in a series of papers devoted to the foundations of Symplectic Field Theory sketched in [Y Eliashberg, A Givental and H Hofer, Introduction to Symplectic Field Theory, Geom. Funct. Anal. Special Volume, Part II (2000) 560--673].…
We prove that when Hodge theory survives on non-compact symplectic manifolds, a compact symplectic Lie group action having fixed points is necessarily Hamiltonian, provided the associated almost complex structure preserves the space of…
On one side, from the properties of Floer cohomology, invariant associated to a symplectic manifold, we define and study a notion of symplectic hyperbolicity and a symplectic capacity measuring it. On the other side, the usual notions of…