Oliver Fabert

Vrije Universiteit Amsterdam | VU · Department of Mathematical & Computer Science

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 Td=Rd/(2πZ)d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepac...

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 va...

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 va...

After our compactness result, we continue our program for defining a Floer homology theory for Hamiltonian partial differential equations with regularizing nonlinearities. Despite the presence of small divisors, in this paper we prove that for generic choices the linearization of the nonlinear Floer operator is Fredholm when viewed as a map between...

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 $\mathbb{T}^d=\mathbb{R}^d/(2\pi\mathbb{Z})^d$ and the coordinates of the particles are constrained to a submanif...

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 define a homotopy extension of the well-known Lie bracket and discuss how it can be used to prove the existenc...

We prove the existence of a forced time-periodic solution to general nonlinear Hamiltonian PDEs by using pseudoholomorphic curves as in symplectic homology theory. With a number of assumptions on the nonlinearity, we prove that for a generic time period a solution exists, thereby complementing a result by Rabinowitz from 1978. In order to extend th...

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 infinite dimensions, we show how to solve the arising small divisor problem by combining elliptic methods with...

We construct the chain level $L_\infty$-structure that extends the Lie bracket on symplectic cohomology.

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...

Under natural restrictions it is known that a nonlinear Schrodinger 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 projectization, it makes sense to ask whether the natural analogue of the Arnold conjecture h...

Believing in the axiom of choice, we show how to deduce symplectic non-squeezing in infinite dimensions from the corresponding result for all finite dimensions. Employing methods from model theory of mathematical logic and motivated by analogous constructions in non-standard stochastic analysis, we use that every infinite-dimensional symplectic Hil...

We show how the nontriviality of the Lie-infinity structure on symplectic cohomology leads to the existence of (multiple simple) closed Reeb orbits. Based on our results we postulate a new mirror symmetry principle for open Calabi-Yau manifolds: Manifolds with a highly singular (extended) moduli space of complex structures are mirror to manifolds w...

The classical mirror symmetry conjecture for closed Calabi-Yau manifolds can be formulated as an isomorphism of Frobenius manifolds. Passing from closed to open Calabi-Yau manifolds, we show that one expects to get an isomorphism of cohomology F-manifolds.

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...

Following the work of Piunikhin-Salamon-Schwarz, the Floer cohomology for Hamiltonian symplectomorphisms with its pair-of-pants product is ring isomorphic to the small quantum cohomology ring of the underlying symplectic manifold. In this paper we complete the correspondence between rational Gromov-Witten theory and the Floer theory of symplectomor...

Based on the localization result for descendants in rational SFT moduli spaces from our last joint paper, we prove topological recursion relations for the Hamiltonian in SFT of symplectic mapping tori and in local SFT. Combined with the dilaton equation in SFT, we use them to prove a reconstruction theorem for descendants from primaries. While it t...

In this paper we define a local version of symplectic field theory which generalizes local Gromov-Witten theory in the same way as standard symplectic field theory generalizes standard Gromov-Witten theory. While local symplectic field theory assigns invariants to closed Reeb orbits in contact manifolds, we show that nicely-embedded curves in four-...

It was pointed out by Eliashberg in his ICM 2006 plenary talk that the integrable systems of rational Gromov-Witten theory very naturally appear in the rich algebraic formalism of symplectic field theory (SFT). Carefully generalizing the definition of gravitational descendants from Gromov-Witten theory to SFT, one can assign to every contact manifo...

This survey wants to give a short introduction to the transversality problem in symplectic field theory and motivate to approach it using the new Fredholm theory by Hofer, Wysocki and Zehnder. With this it should serve as a lead-in for the user's guide to polyfolds, which will appear soon and is the result of a working group organized by J. Fish, R...

Infinite dimensional Hamiltonian systems appear naturally in the rich algebraic structure of Symplectic Field Theory. Carefully defining a generalization of gravitational descendants and adding them to the picture, one can produce an infinite number of symmetries of such systems . As in Gromov-Witten theory, the study of the topological meaning of...

It was pointed out by Y. Eliashberg in his ICM 2006 plenary talk that the rich algebraic formalism of symplectic field theory leads to a natural appearance of quantum and classical integrable systems, at least in the case when the contact manifold is the prequantization space of a symplectic manifold. In this paper we generalize the definition of g...

Branched covers of orbit cylinders are the basic examples of holomorphic curves studied in symplectic field theory. Since all curves with Fredholm index one can never be regular for any choice of cylindrical almost complex structure, we generalize the obstruction bundle technique of Taubes for determining multiple cover contributions from Gromov-Wi...

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 manifolds with symplectomorphisms. While the cylindrical contact homology is given by the Floer homologies of powers...

In this paper we present results from modeling the Earth's gravitational field over the northern part of South-America using spherical wavelets. We have applied our analysis to potential data that we derived from CHAMP using the energy balance method, and to terrestrial gravity anomalies. Our approach provides a regional correction to the EGM96 ref...

The gravitational potential of the Earth is usually modeled by means of a series expansion in terms of spherical harmonics. However, the computation of the series coefficients requires preferably homogeneous distributed global data sets. Since one of the most important features of wavelet functions is the ability to localize both in the spatial and...

Usually the gravity field of the Earth is modeled by means of a series expansion in terms of spherical harmonics. However, the computation of the series coefficients requires preferably homogeneous distributed global data sets. Since wavelet functions localize both in the spatial and in the frequency domain, regional and local structures may be mod...

In this paper we present results from modeling the Earth’s gravitational field over the northern part of South-America using spherical wavelets. We have applied our analysis to potential data that we derived from CHAMP using the energy balance method, and to terrestrial gravity anomalies. Our approach provides a regional correction to the EGM96 ref...

The determination of equipotential surfaces above the Earth's surface from satellite data is closely related to the ill-posed downward continuation of the gravity potential. Spherical wavelets can be used for the required regularization as well as many standard regularizations, e.g. Tikhonov-Philipps, can be reformulated as a spherical multiresolut...

Polar motion consists of two main signal components: the Chandler wobble and the annual oscillation. In particular the Chandler wobble is characterized by a time-varying energy behavior, i.e. the amplitude and the frequency are time-dependent functions. Since wavelet analysis is an appropriate tool for the detection of signal components with time-v...

Polar motion consists of two main oscillations, the Chandler wobble and the annual wobble. In order to extract one of these oscillations the time-dependent energy be- havior has to be taken into account. Wavelet analysis is an appropriate tool for the detection of signal components with time-varying frequencies and/or amplitudes. Due to the fact th...

For many geodetic and geophysical applications a measured time series has to be filtered, i.e., for the extraction of certain desired signal components. In general, the filtering procedure consists of three steps: (1) computation of the time-frequency- components, (2) extraction of the desired components, (3) computation of the filtered signal from...

In this paper, we explore the feasibility of the use of radially symmetric spherical wavelet functions (1) for the multi-resolution representation of the geopotential and (2) for the estimation of the wavelet-represented gravity field model using data from advanced gravity missions such as CHAMP, GRACE, and GOCE. The representation approach can be...

In this paper we present results for modeling the Earth's gravitational field using spherical wavelets and applying method-ologies for the estimation of the corresponding coefficients. The observation types in our techniques could either be gravity gradient tensor measurements from the Goce gradiometer, or other gravity mapping mission data such as...

Although the definition of symplectic field theory suggests that one has to count holomorphic curves in cylindrical manifolds equipped with a cylindrical almost complex structure, it is already well-known from Gromov-Witten theory that, due to the presence of multiply-covered curves, we in general cannot achieve transversality for all moduli spaces...