/pdf/introduction-to-the-modern-theory-of-dynamical-systems-1eced70mum.pdf

Introduction to the Modern Theory of Dynamical Systems

Introduction I. FOUNDATIONS 1. From classical to modern 2. Linear symplectic geometry 3. Symplectic manifolds 4. Almost complex structures II. SYMPLECTIC MANIFOLDS 5. Symplectic group actions 6. Symplectic fibrations 7. Constructing symplectic manifolds III. SYMPLECTOMORPHISMS 8. Area-preserving diffeomorphisms 9. Generating functions 10. The group of symplectomorphisms IV. SYMPLECTIC INVARIANTS 11. The Arnold conjecture 12. Symplectic capacities 13. New directions

https://tspace.library.utoronto.ca/bitstream/1807/9754/1/symplectic_topology_1995.pdf

Introduction to symplectic topology

A group theoretical approach to hydrodynamics considers hydrodynamics to be the differential geometry of diffeomorphism groups. The principle of least action implies that the motion of a fluid is described by the geodesics on the group in the right-invariant Riemannian metric given by the kinetic energy. Investigation of the geometry and structure of such groups turns out to be useful for describing the global behavior of fluids for large time intervals.

Topological methods in hydrodynamics

Proposition 1.4. (1) Any symplectic vector space has even dimension (2) Any isotropic subspace is contained in a Lagrangian subspace and Lagrangians have dimension equal to half the dimension of the total space. (3) If (V1,ω1), (V2,ω2) are symplectic vector spaces with L1, L2 Lagrangian subspaces, and if dim(V1) = dim(V2), then there is a linear isomorphism φ : V1 → V2 such that φω2 = ω1 and φ(L1) = L2.

/pdf/an-introduction-to-symplectic-topology-3zecmb5805.pdf

An introduction to symplectic topology

The results in this paper concern computations of Floer cohomology using generating functions. The first part proves the isomorphism between Floer cohomology and Generating function cohomology introduced by Lisa Traynor. The second part proves that the Floer cohomology of the cotangent bundle (in the sense of Part I), is isomorphic to the cohomology of the loop space of the base. This has many consequences, some of which were given in Part I, others will be given in forthcoming papers. The results in this paper had been announced (with indications of proof) in a talk at the ICM 94 in Z¨ urich.

/pdf/functors-and-computations-in-floer-homology-with-3jf43z2m66.pdf

Functors and Computations in Floer homology with Applications Part II

Symplectic topology as the geometry of generating functions

Introduction 411 1. Some basic results in symplectic topology 413 2. Capacity and symplectic reduction 414 3. Volume estimates for Lagrange submanifolds 416 3.1. The case of R. 417 3.2. Deformations of the zero-section in cotangent bundles 419 3.3. Generalization to the case of CP 422 4. An application to billiards 423 5. Geometry of convex sets and periodic orbits 425 6. Compensated compactness and closure of the symplectic group 426 Appendix A. A summary of symplectic geometry through generating functions 427 Appendix B. John’s ellipsoid 428 References 430

https://www.jstor.org/stable/info/2646224

Metric and isoperimetric problems in symplectic geometry

The aim of this paper is twofold: first of all, we show that the C 0 limit of a pair of commuting Hamiltonians commute. This means on one hand that if the limit of the Hamiltonians is smooth, the Poisson bracket of their limit still vanishes, and on the other hand that we may define "commutation" for C 0 functions. The second part of the paper deals with solving "multi-time" Hamilton- Jacobi equations using variational solutions. This extends the work of Barles and Tourin in the viscosity case to include the case of C 0 Hamiltonians, and removes their convexity assumption, provided we are in the framework of "variational solutions".

/pdf/commuting-hamiltonians-and-hamilton-jacobi-multi-time-2jomgpzza1.pdf

Commuting Hamiltonians and Hamilton-Jacobi multi-time equations

Accurate asymptotic expressions are given for the exponentially small eigenvalues of Witten Laplacians acting on p-forms. The key ingredient, which replaces explicit formulas for global quasimodes in the case p = 0, is Barannikov’s presentation of Morse theory in Barannikov (Adv Soviet Math 21:93–115, 1994).

/pdf/precise-arrhenius-law-for-p-forms-the-witten-laplacian-and-4d7chevw8v.pdf

Precise Arrhenius law for p-forms: The Witten Laplacian and Morse-Barannikov complex.

We prove that if a sequence of pairs of smooth commuting Hamiltonians converge in the $C^0$ topology to a pair of smooth Hamiltonians, these commute. This allows us define the notion of commuting continuous Hamiltonians. As an application we extend some results of Barles and Tourin on multi-time Hamilton-Jacobi equations to a more general setting.

Commuting Hamiltonians and multi-time Hamilton-Jacobi equations

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

Claude Viterbo

Author Tools

Chat about Author