arXiv research

Study of Hamiltonian boundary value problems with symplectic symmetries.

New boundary conditions improve Hamiltonian analysis in GR.

We analyse the definition of quasi-local energy in GR based on a Hamiltonian analysis of the Einstein-Hilbert action initiated by Brown-York. The role of the constraint equations, in particular the Hamiltonian constraint on the timelike boundary, neglected in previous studies, is emphasized here. We argue that a consis…

Analyzes perturbed contact instantons with Legendrian boundary conditions using geometric analysis.

We develop a theory of "quasi"-Hamiltonian G-spaces for which the moment map takes values in the group G itself rather than in the dual of the Lie algebra. The theory includes counterparts of Hamiltonian reductions, the Guillemin-Sternberg symplectic cross-section theorem and of convexity properties of the moment map. …

We develop a method for preserving pseudoholomorphic curves in contact 3-manifolds under surgery along transverse links. This makes use of a geometrically natural boundary value problem for holomorphic curves in a 3-manifold with stable Hamiltonian structure, where the boundary conditions are defined by 1-parameter fam…

Geometric Hydrodynamics tackles open problems in fluid dynamics.

The study finds conditions for free boundary Hamiltonian stationary discs in complex 2-space.

Introduces Lax-Kirchhoff moduli spaces for quivers and Lie groups.

Hamiltonian dynamical systems tend to have infinitely many periodic orbits. For example, for a broad class of symplectic manifolds almost all levels of a proper smooth Hamiltonian carry periodic orbits. The Hamiltonian Seifert conjecture is the existence problem for regular compact energy levels without periodic orbits…

This paper is a discussion of relations between some free-boundary problems and infinite dimensional Lie groups; particularly a version of Nahm's equations for the group of Hamiltonian diffeomorphisms in two dimensions.

Paper presents a new approach to continuum mechanics using port-Hamiltonian framework.

Quantum algorithm solves financial option pricing using Hamiltonian simulation.

Study scattering rigidity for Hamiltonian systems, proving lens rigidity for non-trapping Finsler manifolds.

The space of positive Lagrangians in an almost Calabi-Yau manifold is an open set in the space of all Lagrangian submanifolds. A Hamiltonian isotopy class of positive Lagrangians admits a natural Riemannian metric $Υ$, which gives rise to a notion of geodesics. We study geodesics of positive $O_n(\mathbb{R})$ invariant…

A normal form for edge metrics is derived under the necessary conditions that the metric be normalized and exact. The normal forms for such an edge metric are shown to be in 1-1 correspondence with representative metrics for a reduced conformal infinity on the boundary. The normal form is constructed via solution of a …

Researchers found a way to measure energy in black hole perturbations.

Clarifies global structure of Stokes-Dirac structures on manifolds.

Paper develops a new fluid flow model with energy exchange through boundaries.

For simple and simply-connected complex algebraic group G, we conjecture the existence of a functor eta_G from the category of 2-bordisms to the category of holomorphic symplectic varieties with Hamiltonian action, such that gluing of boundaries corresponds to the holomorphic symplectic quotient with respect to the dia…

Hamiltonian RNN controls hidden states gradient for long-term dependencies.

We define a class of boundary value problems on manifolds with fibered boundary. This class is in a certain sense a deformation between the classical boundary value problems and the Atiyah-Patodi-Singer problems in subspaces. The boundary conditions in this theory are taken as elements of the C^*-algebra generated by p…

We recover the higher order terms for the acoustic wave equation from measurements of the modulus of the solution. The recovery of these coefficients is reduced to a question of stability for inverting a Hamiltonian flow transform, not the geodesic X-ray transform encountered in other inverse boundary problems like the…

It is shown that the new Poisson brackets proposed in Part I of this work (J. Math. Phys. 34, 5747(hep-th/9305133)) arise naturally in an extension of the formal variational calculus incorporating divergences. The linear spaces of local functionals, evolutionary vector fields, functional forms, multi-vectors and differ…

The paper explores deformations of quasi-Hamiltonian spaces to Hamiltonian spaces.

We consider the problem $\mathbf{P_{curve}}$ of minimizing $\int \limits_0^L \sqrt{ξ^2 + κ^2(s)} \, {\rm d}s$ for a curve $\mathbf{x}$ on $\mathbb R$ with fixed boundary points and directions. Here the total length $L\geq 0$ is free, $s$ denotes the arclength parameter, $κ$ denotes the absolute curvature of $\mathbf{x}…

Method learns molecular Hamiltonian for accurate electron dynamics predictions.

Study boundary value problems for elliptic operators on manifolds.

Boundary value problems for operators of Dirac type arise naturally in connection with the conformal geometry of surfaces immersed in Euclidean 3--space. Recently such boundary value problems have been successfully applied to a variety of problems from computer graphics. Here we investigate under which conditions these…

Elliptic boundary value problem for G2 structures on manifolds.

Solves steering problem with continuous time, Hilbert-Schmidt cost, and matrix ODEs.

Maps with many singularities found in complex space.

Neural networks solve high-dimensional HJB PDEs with asymptotic guarantees.

We prove a spectral flow formula for one-parameter families of Hamiltonian systems under homoclinic boundary conditions, which relates the spectral flow to the relative Maslov index of a pair of curves of Lagrangians induced by the stable and unstable subspaces, respectively. Finally, we deduce sufficient conditions fo…

Paper extends port-Hamiltonian model to include internal energy for compressible and incompressible flow.

Study of boundary value problems in 7, 4, and 3 dimensions with G2 and holonomy structures.

The Teichmüller space $\mathcal{T}_S(\mathbf{b})$ of hyperbolic metrics on a surface $S$ with fixed lengths at the boundary components is symplectic. We prove that any sum of infinitesimal earthquakes on $S$ that is tangent to $\mathcal{T}_S(\mathbf{b})$ is Hamiltonian, by providing a Hamiltonian $\mathbb{L}$. Such fun…

Machine learning mirrors statistical data assimilation, offering new optimization methods.

Transforms input design for probabilistic models into optimal control of a Hamiltonian system.

Paper connects minimal and maximal surfaces' boundary value problems.

In this paper we give a proof of the existence of an orthogonal geodesic chord on a Riemannian manifold homeomorphic to a closed disk and with concave boundary. This kind of study is motivated by the link of the multiplicity problem with the famous Seifert conjecture (formulated in 1948) about multiple brake orbits for…

In this paper, we study the Lagrangian F-stability and Hamiltonian F-stability of Lagrangian self-shrinkers. We prove a characterization theorem for the Hamiltonian F-stability of $n$-dimensional complete Lagrangian self-shrinkers without boundary, with polynomial volume growth and with the second fundamental form sati…

Develops control and observer methods for complex systems.

A guide for solving first-order elliptic boundary value problems.

Introduces HGN for learning Hamiltonian dynamics from images.

Method proves ellipticity of vacuum spacetime boundary problems.

In the case of a compact real analytic symplectic manifold M we describe an approach to the complexification of Hamiltonian flows [Se, Do1, Th1] and corresponding geodesics on the space of Kahler metrics. In this approach, motivated by recent work on quantization, the complexified Hamiltonian flows act, through the Gro…

We study boundary value problems for the Dirac operator on Riemannian Spin$^c$ manifolds of bounded geometry and with noncompact boundary. This generalizes a part of the theory of boundary value problems by C. Bär and W. Ballmann for complete manifolds with closed boundary. As an application, we derive the lower bound …