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Jean-Pierre Magnot

Publications and source records attributed to Jean-Pierre Magnot.

At least 55 records · Page 3Linked to original sources
arXiv2020

On the differential geometry of numerical schemes and weak solutions of functional equations

We exhibit differential geometric structures that arise in numerical methods, based on the construction of Cauchy sequences, that are currently used to prove explicitly the existence of weak solutions to functional equations. We describe the geometric framework, highlight several examples and describe how two well-known proofs fit with our setting. The first one is a re-interpretation of the classical proof of an implicit functions theorem in an ILB setting, for which our setting enables us to state an implicit functions theorem without additional norm estimates, and the second one is the finite element method of the Dirichlet problem where the set of triangulations appear as a smooth set of parameters. In both case, smooth dependence on the set of parameters is established. Before that, we develop the necessary theoretical tools, namely the notion of Cauchy diffeology on spaces of Cauchy sequences and a new generalization of the notion of tangent space to a diffeological space.

math.FA↗
arXiv2020

On the geometry of $Diff(S^1)-$pseudodifferential operators based on renormalized traces

In this article, we examine the geometry of a group of Fourier-integral operators, which is the central extension of $Diff(S^1)$ with a group of classical pseudo-differential operators of any order. Several subgroups are considered, and the corresponding groups with formal pseudodifferential operators are defined. We investigate the relationship of this group with the restricted general linear group $GL_{res},$ we define a right-invariant pseudo-Riemannian metric on it that extends the Hilbert-Schmidt Riemannian metric by the use of renormalized traces of pseudo-differential operators, and we describe classes of remarkable connections.

math.DG↗
arXiv2020

On the Cauchy problem for a Kadomtsev-Petviashvili hierarchy on non-formal operators and its relation with a group of diffeomorphisms

We establish a rigorous link between infinite-dimensional regular Frölicher Lie groups built out of non-formal pseudodifferential operators and the Kadomtsev-Petviashvili hierarchy. We introduce a version of the Kadomtsev-Petviashvili hierarchy on a regular Frölicher Lie group of series of non-formal odd-class pseudodifferential operators. We solve its corresponding Cauchy problem, and we establish a link between the dressing operator for our hierarchy and the action of diffeomorphisms and non-formal Sato-like operators on jet spaces. In appendix, we describe the group of Fourier integral operators in which this correspondence seems to take place. Also, motivated by Mulase's work on the KP hierarchy, we prove a group factorization theorem for our group of Fourier integral operators.

math-ph↗
arXiv2020

Well-posedness of the Kadomtsev-Petviashvili hierarchy, Mulase factorization, and Frölicher Lie groups

We recall the notions of Frölicher and diffeological spaces and we build regular Frölicher Lie groups and Lie algebras of formal pseudo-differential operators in one independent variable. Combining these constructions with a smooth version of the Mulase factorization of infinite dimensional groups based on formal pseudo-differential operators, we present two proofs of the well-posedness of the Cauchy problem for the Kadomtsev-Petviashvili (KP) hierarchy in a smooth category. We also generalize these results to a KP hierarchy modelled on formal pseudo-differential operators with coefficients which are series in formal parameters, describe a rigorous derivation of the Hamiltonian interpretation of the KP hierarchy, and discuss how solutions depending on formal parameters can lead to sequences of functions converging to a class of solutions of the standard KP-I equation.

math-ph↗
arXiv2018

On algebras and groups of formal series over a groupoid and application to some spaces of cobordism

We develop here a concept of deformed algebras and their related groups through two examples. Deformed algebras are obtained from a fixed algebra by deformation along a family of indexes, through formal series. We show how the example of deformed algebra used in \cite{Ma2013} is only an example among others, and how they often give rise to regular Frölicher Lie groups, and sometimes to Fréchet Lie groups, that are regular. The first example, indexed by $\mathbb{N},$ is obtained by deformations of the group of bounded classical pseudo-differential operators $Cl^{0,*}$ by algebras of (maybe unbounded) classical pseudo-differential operators. In the second one, the set of indexes is a $\N-$graded groupoïd, which is made of manifolds with boundary that are understood as morphisms of cobordisms. Here again, we get regular Lie groups, and we show how this setting applies to a class of examples that are derived of so-called stochastic cosurfaces.

math.GR↗
arXiv2018

A Geometric Framework for the Inconsistency in Pairwise Comparisons

In this study, a pairwise comparison matrix is generalized to the case when coefficients create Lie group $G$, non necessarily abelian. A necessary and sufficient criterion for pairwise comparisons matrices to be consistent is provided. Basic criteria for finding a nearest consistent pairwise comparisons matrix (extended to the class of group $G$) are proposed. A geometric interpretation of pairwise comparisons matrices in terms of connections to a simplex is given. Approximate reasoning is more effective when inconsistency in data is reduced.

math.LO↗
arXiv2017

The group of diffeomorphisms of a non compact manifold is not regular

We show that a group of diffeomorphisms $\D$ on the open unit interval $I,$ equipped with the topology of uniform convergence on any compact set of the derivatives at any order, is non regular: the exponential map is not defined for some path of the Lie algebra. this result extends to the group of diffeomorphisms of finite dimensional, non compact manifold $M.$

math.DG↗
arXiv2017

On the domain of implicit functions in a projective limit setting without additional norm estimates

We examine how implicit functions on ILB-Fréchet spaces can be obtained without metric or norm estimates which are classically assumed. We obtain implicit functions defined on a domain $D$ which is not necessarily open, but which contains the unit open ball of a Banach space. The corresponding inverse functions theorem is obtained, and we finish with an open question on the adequate (generalized) notion of differentiation, needed for the corresponding version of the Fröbenius theorem.

math.FA↗
arXiv2017

The diffeology of Milnor's classifying space

We define a diffeology on the Milnor classifying space of a diffeological group $G$, constructed in a similar fashion to the topological version using an infinite join. Besides obtaining the expected classification theorem for smooth principal bundles, we prove the existence of a diffeological connection on any principal bundle (with mild conditions on the bundles and groups), and apply the theory to some examples, including some infinite-dimensional groups, as well as irrational tori.

math.GT↗
arXiv2017

Remarks on a new possible discretization scheme for gauge theories

We propose here a new discretization method for a class continuum gauge theories which action functionnals are polynomials of the curvature. Based on the notion of holonomy, this discretization procedure appears gauge-invariant for discretized analogs of Yang-Mills theories, and hence gauge-fixing is fully rigorous for these discretized action functionnals. Heuristic parts are forwarded to the quantization procedure via Feynman integrals and the meaning of the heuristic infinite dimensional Lebesgue integral is questionned.

math-ph↗
arXiv2016

Differentiation on spaces of triangulations and optimized triangulations

We describe a smooth structure, called Frölicher space, on CW complexes and spaces of triangulations. This structure enables differential methods for e.g. minimization of functionnals. As an application, we exhibit how an optimized triangulation can be obtained in order to solve a standard PDE.

math.DG↗
arXiv2015

Remarks on the geometry and the topology of the loop spaces $H^{s}(S^1, N),$ for $s\leq 1/2.$

We first show that, for a fixed locally compact manifold $N,$ the space $L^2(S^1,N)$ has not the homotopy type odf the classical loop space $C^\infty(S^1,N),$ by two theorems: - the inclusion $C^\infty(S^1,N) \subset L^2(S^1,N)$ is null homotopic if $N $ is connected, - the space $L^2(S^1,N)$ is contractible if $N$ is compact and connected. After this first remark, we show that the spaces $H^s(S^1,N)$ carry a natural structure of Frölicher space, equipped with a Riemannian metric, which motivates the definition of Riemannian Frölicher space.

math.DG↗
arXiv2015

On Diff(M)-pseudo-differential operators and the geometry of non linear grassmannians

We consider two principal bundles of embeddings with total space $Emb(M,N),$ with structure groups $Diff(M)$ and $Diff_+(M),$ where $Diff_+(M)$ is the groups of orientation preserving diffeomorphisms. The aim of this paper is to describe the structure group of the tangent bundle of the two base manifolds: $$ B(M,N) = Emb(M,N)/Diff(M) \hbox{ and } B_+(M,N)= Emb(M,N)/Diff_+(M).$$ From the various properties described, an adequate group seems to be a group of Fourier integral operators, which is carefully studied. This is the main goal of this paper to analyze this group, which is a central extension of a group of diffeomorphisms by a group of pseudo-differential operators which is slightly different from the one developped in \cite{OMYK4}. We show that these groups are regular, and develop the necessary properties for applications to the geometry of $ B(M,N) .$ A case of particular interest is $M=S^1,$ where connected components of $B_+(S^1,N)$ are deeply linked with homotopy classes of oriented knots. In this example, the structure group of the tangent space $TB_+(S^1,N)$ is a subgroup of some group $GL_{res},$ following the classical notations of \cite{PS}. These constructions suggest some approaches in the spirit of \cite{Ma2006} that could lead to knot invariants through a theory of Chern-Weil forms.

math.DG↗
arXiv2015

From infinitesimal symmetries to deformed symmetries of Lax-type equations

Using the procedure initiated in \cite{Ma2013}, we deform Lax-type equations though a scaling of the time parameter. This gives an equivalent (deformed) equation which is integrable in terms of power series of the scaling parameter. We then describe a regular Frölicher Lie group of symmetries of this deformed equation

math.AP↗
arXiv2014

Deformed algebras: examples and application to Lax equations

We develop here a concept of deformed algebras through three examples and an application. Deformed algebras are obtained from a fixed algebra by deformation along a family of indexes, through formal series. We show how the example of deformed algebra used in \cite{Ma2013} is only an example among others, and how they often give rise to regular Frölicher Lie groups. Then, we show how such deformed algebras arise in a formal integration of Lax equations through time scaling. The infinite dimensional groups under consideration enables to state the uniqueness of the formal solutions, their smooth dependance under perturbation, and to study some of the symmetries.

math.FA↗
arXiv2013

From configurations to branched configurations and beyond

We propose here a geometric and topological setting for the study of branching effects arising in various fields of research, e.g. in statistical mechanics and turbulence theory. We describe various aspects that appear key points to us, and finish with a limit of such a construction which stand in the dynamics on probability spaces where it seems that branching effects can be fully studied without any adaptation of the framework.

math.DG↗