Leonid Ryvkin

Leonid Ryvkin
Georg-August-Universität Göttingen | GAUG · Institute of Mathematics

Dr. rer. nat.

About

24
Publications
928
Reads
How we measure 'reads'
A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more
72
Citations
Additional affiliations
October 2018 - present
Paris Diderot University
Position
  • PostDoc Position
June 2018 - September 2018
Ruhr-Universität Bochum
Position
  • PostDoc Position
October 2014 - June 2018
Ruhr-Universität Bochum
Position
  • PhD Student

Publications

Publications (24)
Preprint
Full-text available
We introduce a novel symplectic reduction scheme for $C^\infty(M,\omega)$ that extends in a straightforward manner to the multisymplectic setting. Specifically, we exhibit a reduction of the $L_\infty$-algebra of observables on a premultisymplectic manifold $(M,\omega)$ in the presence of a compatible Lie algebra action $\mathfrak{g}\curvearrowrigh...
Article
Full-text available
We introduce the holonomy of a singular leaf L of a singular foliation as a sequence of group morphisms from πn(L)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _n...
Preprint
Full-text available
We discuss the notion of basic cohomology for Dirac structures and, more generally, Lie algebroids. We then use this notion to characterize the obstruction to a variational formulation of Dirac dynamics.
Preprint
We construct an $L_\infty$-algebra on the truncated canonical homology complex of a symplectic manifold, which naturally projects to the universal central extension of the Lie algebra of Hamiltonian vector fields.
Preprint
An important result for regular foliations is their formal semi-local triviality near simply connected leaves. We extend this result to singular foliations for all 2-connected leaves and a wide class of 1- connected leaves by proving a semi-local Levi-Malcev theorem for the semi-simple part of their holonomy Lie algebroid.
Preprint
Full-text available
We compare existence and equivariance phenomena for weak moment maps and homotopy moment maps in multisymplectic geometry.
Preprint
We introduce the holonomy of a singular leaf $L$ of a singular foliation as a sequence of group morphisms from $\pi_n(L)$ to the $\pi_{n-1}$ of the universal Lie $\infty$-algebroid of the transverse foliation of $L$. We include these morphisms in a long exact sequence, thus relating them to the holonomy groupoid of Androulidakis and Skandalis and t...
Preprint
Full-text available
We give a geometric description of the obstruction to the existence of homotopy comoment maps in multisymplectic geometry. We apply this description to determine the existence of comoments for multisymplectic compact group actions on spheres and provide explicit constructions for some of these comoments.
Preprint
Full-text available
We give a geometric description of the obstruction to the existence of comoments in multisymplectic geometry. We apply this description to determine the existence of comoments for multisymplectic compact group actions on spheres and provide explicit constructions for some of these comoments. MSC-classification (2010): 53D05, 53D20, 54H15, 57S15, 5...
Article
In this article we study multisymplectic geometry, i.e., the geometry of manifolds with a non-degenerate, closed differential form. First we describe the transition from Lagrangian to Hamiltonian classical field theories, and then we reformulate the latter in multisymplectic terms. Furthermore, we investigate basic questions on normal forms of mult...
Article
Given a vector field on a manifold M, we define a globally conserved quantity to be a differential form whose Lie derivative is exact. Integrals of conserved quantities over suitable submanifolds are constant under time evolution, the Kelvin circulation theorem being a well-known special case. More generally, conserved quantities are well-behaved u...
Article
In this note we complete the calculation of the number of $GL(\mathbb R^n)$-orbits on $\Lambda^k(\mathbb R^n)^*$, by treating the cases $(n,k)= (7,4)$ and $(8,5)$ not covered in the literature. We also calculate the number of of non-degenerate and stable orbits, as they are of special interest to multisymplectic and special geometry.
Article
The Darboux theorem in symplectic geometry implies that any two points in a connected symplectic manifold have neighbourhoods symplectomorphic to each other. The impossibility of such a theorem in the more general multisymplectic framework appears to be, at least, folkloristic, but no explicit counterexample seems to exist in the literature. In thi...
Chapter
Having constructed a notion of observables for n-plectic manifolds we will now try to generalize the notions of Hamiltonian group action and co-moment maps.
Chapter
Let K denote here a fixed ground field of characteristic 0. All vector spaces, linear maps and tensor products will be defined with respect to/taken over this field, unless we explicitly state otherwise. A good overview of the subject of L∞-algebras is provided in the n-lab ([12]).
Chapter
Throughout this and the next chapter we fix ℝ as our ground field.
Book
Leonid Ryvkin gives a motivated and self-sustained introduction to n-plectic geometry with a special focus on symmetries. The relevant algebraic structures from scratch are developed. The author generalizes known symplectic notions, notably observables and symmetries, to the n-plectic case, culminating in solving the existence question for co-momen...
Article
Given a multisymplectic manifold $(M,\omega)$ and a Lie algebra $\frak{g}$ acting on it by infinitesimal symmetries, Fregier-Rogers-Zambon define a homotopy (co-)moment as an $L_{\infty}$-algebra-homomorphism from $\frak{g}$ to the observable algebra $L(M,\omega)$ associated to $(M,\omega)$, in analogy with and generalizing the notion of a co-momen...

Network

Cited By