arXiv subjects
Publications and source records attributed to Oliver Fabert.
We use Hamiltonian Floer theory to prove a cuplength result about the existence of periodic solutions of particle-field systems with a Gaussian random field. As a concrete model we study stochastic electrodynamics, which approximates quantum electrodynamics up to first order in $\hbar$, and is even exact in the case of low-order Hamiltonians or when the particles are treated classically.
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 pseudo-holomorphic curves, enabling the use of elliptic PDE methods. Multisymplectic geometry is the geometric framework underlying Hamiltonian field theory, where the time line is replaced by higher-dimensional manifolds. In the case of two dimensions and using complex structures, we introduce a novel multisymplectic framework that is fit for the generalization of the elliptic methods from symplectic geometry. Besides proving a Darboux theorem, we show that the $L^2$-gradient lines of our multisymplectic action functional are now pseudo-Fueter curves defined using a compatible almost hyperk\"ahler structure.
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 from surfaces are the natural 2-dimensional generalizations of geodesics. In this paper, we give the corresponding generalization of symplectic manifolds and Hamiltonian equations, leading to a class of partial differential equations that share properties similar to Hamiltonian (ordinary) differential equations. Two rigidity results are discussed: a non-squeezing theorem and a version of the cuplength result for quadratic Hamiltonians on cotangent bundles. The proof of the latter uses a generalization of Floer curves, for which the necessary Fredholm and compactness results will be proven.
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 field theory is described, is not suitable for variational techniques. In this paper, we introduce for $n=2$ a Lagrange-Hamilton formalism that allows us to define a generalization of Hamiltonian Floer theory. As an application, we prove a cuplength estimate for our Hamiltonian equations that yields a lower bound on the number of solutions to Laplace equations with nonlinearity. We also discuss the relation with holomorphic Floer theory.
Hamiltonian Floer theory plays an important role for finding periodic solutions of Hamilton's equation, which can be seen as a generalization of Newton's equation. Generalizing Newton's equation to Laplace's equation with non-linearity, we show, building on the work of Ginzburg and Hein, that this role is taken over by the hyperk\"ahler Floer theory of Hohloch, Noetzel, and Salamon. Apart from establishing $C^0$-bounds in order to be able to deal with noncompact hyperk\"ahler manifolds, the core ingredient is a regularization scheme for the polysymplectic formalism due to Bridges, which allows us to link Euclidean field theory with hyperk\"ahler Floer theory. As a concrete result, we prove a cuplength estimate.
We show how methods from Hamiltonian Floer theory can be used to establish lower bounds for the number of different time-periodic measures of time-periodic Hamiltonian systems with diffusion. After proving the existence of closed random periodic solutions and of the corresponding Floer curves for Hamiltonian systems with random walks with step width $1/n$ for every $n\in\mathbb{N}$, we show that, after passing to a subsequence, they converge in probability distribution as $n\to\infty$. Besides using standard results from Hamiltonian Floer theory and about convergence of tame probability measures, we crucially use that sample paths of Brownian motion are almost surely Hölder continuous with Hölder exponent $0<α<\frac{1}{2}$.
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 frameworks for covariant field theory generalizing symplectic geometry, none of them are readily suitable for variational techniques such as Hamiltonian Floer theory, since the corresponding action functionals are too degenerate. Instead, we show how a regularization procedure introduced by Bridges leads to a new geometric framework for which we can show that the finite energy $L^2$-gradient lines of the corresponding action functional, called Floer curves, converge asymptotically to space-time periodic solutions. As a concrete example we prove the existence of Floer curves, and hence also of space-time periodic solutions, for a class of coupled particle-field systems defined in this new framework.
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 finite-dimensional symplectic manifold with an infinite-dimensional linear symplectic Hilbert space. After our compactness results we continue our program for defining a Floer homology theory for this class of infinite-dimensional Hamiltonian systems. Based on the presence of small divisors we introduce a new notion of nondegeneracy for time-periodic solutions which allows us to prove that the linearization of the nonlinear Floer operator is Fredholm when viewed as a map between suitable Sobolev space completions.
We consider a natural class of time-periodic infinite-dimensional nonlinear Hamiltonian systems modelling the interaction of a classical mechanical system of particles with a scalar wave field. When the field is defined on a space torus $\mathbb{T}^d=\mathbb{R}^d/(2π\mathbb{Z})^d$ and the coordinates of the particles are constrained to a submanifold $Q\subset\mathbb{T}^d$, we prove that the number of $T$-periodic solutions of the coupled Hamiltonian particle-field system is bounded from below by the $\mathbb{Z}_2$-cuplength of the space of contractible loops in $Q$, provided that the square of the ratio $T/2π$ of time period $T$ and space period $X=2π$ is a Diophantine irrational number. The latter condition is necessary since for the infinite-dimensional version of Gromov-Floer compactness as well as for the $C^0$-bounds we need to deal with small divisors.
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. In particular, when the nonlinearity is sufficiently regularizing, bounded and time-periodic, we prove an infinite-dimensional version of Gromov-Floer compactness by using ideas from the theory of Diophantine approximations to overcome the small divisor problem. Furthermore, in the case when the infinite-dimensional phase space is a product of a finite-dimensional closed symplectic manifold with linear symplectic Hilbert space, we prove a cup-length estimate for the number of periodic solutions.
We prove the existence of infinitely many time-periodic solutions of nonlinear Schrödinger equations using pseudo-holomorphic curve methods from Hamiltonian Floer theory. For the generalization of the Gromov-Floer compactness theorem to infinite dimensions, we show how to solve the arising small divisor problem by combining elliptic methods with results from the theory of diophantine approximations.
We construct the chain level $L_\infty$-structure that extends the Lie bracket on symplectic cohomology.
This is the first of two papers devoted to showing how the rich algebraic formalism of Eliashberg-Givental-Hofer's symplectic field theory (SFT) can be used to define higher algebraic structures on the symplectic cohomology of open symplectic manifolds. Using the SFT of Hamiltonian mapping tori we show how to define a homotopy extension of the well-known Lie bracket on symplectic cohomology. Apart from discussing applications to the existence of closed Reeb orbits, we outline how the $L_{\infty}$-structure is conjecturally related via mirror symmetry to the extended deformation theory of complex structures.
In this paper we show how the rich algebraic formalism of Eliashberg-Givental-Hofer's symplectic field theory (SFT) can be used to define higher algebraic structures in Hamiltonian Floer theory. Using the SFT of Hamiltonian mapping tori we show how to define a homotopy extension of the well-known Lie bracket and discuss how it can be used to prove the existence of multiple closed Reeb orbits. Furthermore we show how to define the analogue of rational Gromov-Witten theory in the Hamiltonian Floer theory of open symplectic manifolds. More precisely, we introduce a so-called cohomology F-manifold structure in Hamiltonian Floer theory and prove that it generalizes the well-known Frobenius manifold structure in rational Gromov-Witten theory.
Under natural restrictions it is known that a nonlinear Schrödinger 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 after passing to the projectivization, it makes sense to ask whether the natural analogue of the Arnold conjecture holds. By employing methods from non-standard model theory we show how Hamiltonian Floer theory can be generalized from finite to infinite dimensions. While our proof entirely builds on finite-dimensional results, we do not ask for any prior knowledge of non-standard model theory.
We prove a non-squeezing result for infinite-dimensional Hamiltonian flows using non-standard model theory. For this we prove the existence of a corresponding family of pseudoholomorphic spheres and characterize the maximal time in terms of a limiting behaviour for these curves. While our proof is based on the finite-dimensional results from Gromov's original proof, we do not ask for any prior knowledge of non-standard model theory.
This paper defines symplectic scale manifolds based on Hofer-Wysocki-Zehnder's scale calculus. We introduce Hamiltonian vector fields and flows on these by narrowing down sc-smoothness to what we denote by strong sc-smoothness, a concept which effectively formalizes the desired smoothness properties for Hamiltonian functions. We show the concept to be invariant under sc-smooth symplectomorphisms, whence it is compatible with Hofer's scale manifolds. We develop and verify the theory at the hand of the free Schrödinger equation.
Polyfold theory was developed by Hofer-Wysocki-Zehnder by finding commonalities in the analytic framework for a variety of geometric elliptic PDEs, in particular moduli spaces of pseudoholomorphic curves. It aims to systematically address the common difficulties of compactification and transversality with a new notion of smoothness on Banach spaces, new local models for differential geometry, and a nonlinear Fredholm theory in the new context. We shine meta-mathematical light on the bigger picture and core ideas of this theory. In addition, we compiled and condensed the core definitions and theorems of polyfold theory into a streamlined exposition, and outline their application at the example of Morse theory.