arXiv research

We introduce a version of Aubry-Mather theory for the length functional of causal curves in compact Lorentzian manifolds. Results include the existence of maximal invariant measures, calibrations and calibrated curves. We prove two versions of the Mather's graph theorem. A class of examples, the Lorentzian Hedlund exam…

The paper characterizes gaps in minimal foliations on tori using energy criteria.

Develops a Thom-Mather theory for corank 1 frontals.

We consider Aubry-Mather theory for a subclass of class A spacetimes, i.e. compact vicious spacetimes with globally hyperbolic Abelian cover. In this subclass, called class A_1, we obtain improved results on timelike maximizers and Lipschitz continuity of the time separation of the Abelian cover on the i.g. optimal sub…

This paper shows stable mappings are never dense on non-compact manifolds.

The paper studies maximal stretch and Lipschitz maps on negatively curved manifolds.

Study calculates Mather β-function for ellipses and applies it to rigidity problems.

Adapts a short argument to derive a stability theorem for smooth maps.

The study proves the existence of many geodesics on complex manifolds.

The paper explores density of stable mappings and their properties.

The paper (in French) exemplifies graphically a solution of the heat equation which is a 1-dimensional unfolding of an elliptic umbilic catastrophe. The example is due to James Damon and adapts Thom-Mather's singularity theory to multiscale models of scale-space analysis in image processing.

This article is devoted to the study of smooth desingularization, which are customary employed in the definition of De Rham Intersection Cohomology with differential forms. In this paper we work with the category of Thom-Mather simple spaces. We construct a functor which sends each Thom-Mather simple space into a smoot…

We establish a form of the h-principle for the existence of foliations quasi-complementary to a given one; the same methods also provide a proof of the classical Mather-Thurston theorem.

The paper refines Mather-Thurston theorems for flat connections in manifolds.

The paper extends Thurston's method to new variants of Mather-Thurston theorem.

Non-compact manifolds prevent $C^0$-stable mappings from being dense.

We give the characterization of Arnol'd-Mather type for stable singular Legendre immersions. The most important building block of the theory is providing a module structure on the space of infinitesimal integral deformations by means of the notion of natural liftings of differential systems and of contact Hamiltonian v…

It is a celebrated result of Mather that the group of $C^k$--diffeomorphisms of an $n$--manifold is simple, provided that a mild isotopy condition is satisfied, with the possible exception of $k=n+1$. The purpose of this article is mostly expository, and in it we give a detailed account of Mather's proof in the case wh…

In a previous work we proved the uniqueness and functoriality of primary unfoldings on simple Thom-Mather spaces, which is a functor to the category of smooth manifolds. In this article we extend these results for any stratified Thom-Mather pseudomanifold with arbitary finite length, through a new kind of intermediate …

In this paper we give three applications of a method to prove h-principles on closed manifolds. Under weaker conditions this method proves a homological h-principle, under stronger conditions it proves a homotopical one. The three applications are as follows: a homotopical version of Vassiliev's h-principle, the contra…

We obtain a large deviation function for the stationary measures of twisted Brownian motions associated to the Lagrangians $L_λ(p,v)=\frac{1}{2}g_{p}(v,v)- λω_{p}(v)$, where $g$ is a $C^{\infty}$ Riemannian metric in a compact surface $(M,g)$ with nonpositive curvature, $ω$ is a closed 1-form such that the Aubry-Mather…

Survey of integrable billiard models and inequalities.

Geometric analysis proves weak KAM solutions constant under specific conditions.

We develop a generalization to non-Witt spaces of the intersection homology theory of Goresky-MacPherson. The second author has described the self-dual sheaves compatible with intersection homology, and the other authors have described a generalization of Cheeger's L2 de Rham cohomology. In this paper we extend both of…

We review the author's results on Mather's $β$ function : non-strict convexity of $β$ when the configuration space has dimension two, link between the size of the Aubry set and the differentiability of $β$, correlation between the rationality of the homology class and the differentiability of $β$, equality of the Mathe…

Extends cohomology to incomplete Riemannian manifolds.

In his celebrated paper "Generic projections", John Mather has shown that almost all linear projections from a submanifold of a vector space into a subspace are transverse with respect to a given modular submanifold. In this paper, an improvement of Mather's result is stated. Namely, we show that almost all linear pert…

This article is a survey of recent work of the author, together with Markus Banagl, Eric Leichtnam, Rafe Mazzeo, and Paolo Piazza, on the Hodge theory of stratified spaces. We discuss how to resolve a Thom-Mather stratified space to a manifold with corners with an iterated fibration structure and the generalization of …

Paper confirms conjecture for PL foliations of codimension 2.

In this paper, the notion of generic transversality and its characterization are given. The characterization is also a further improvement of the basic transversality result and its strengthening which was given by John Mather.

We prove a theorem on structural stability of smooth attractor-repellor endomorphisms of compact manifolds, with singularities. By attractor-repellor, we mean that the non-wandering set of the dynamics $f$ is the disjoint union of a repulsive compact subset with a hyperbolic attractor on which $f$ acts bijectively. The…

We show that the stable commutator length vanishes for certain groups defined as infinite unions of smaller groups. The argument uses a group-theoretic analogue of the Mazur swindle, and goes back to the works of Anderson, Fisher, and Mather on homeomorphism groups.

In his celebrated paper "Generic projections", John Mather has given a striking transversality theorem and its applications on generic projections. On the other hand, in this paper, two transversality theorems on generic linearly perturbed $C^r$ mappings are shown $(r\geq 1)$. Moreover, some applications of the two the…

The main result of this paper is a sufficient condition in order to have a compact Thom-Mather stratified pseudomanifold endowed with a $\hat{c}$-iterated edge metric on its regular part $q$-parabolic. Moreover, besides stratified pseudomanifolds, the $q$-parabolicity of other classes of singular spaces, such as compac…

The main result is a version of Morse inequalities for the minimum and maximum ideal boundary conditions of the de Rham complex on strata of compact Thom-Mather stratifications, endowed with adapted metrics. An adaptation of the analytic method of Witten is used in the proof, as well as certain perturbation of the harm…

For $Γ_1$-structures on 3-manifolds, we give a very simple proof of Thurston's regularization theorem, first proved in \cite{thurston}, without using Mather's homology equivalence. Moreover, in the co-orientable case, the resulting foliation can be chosen of a precise kind, namely an "open book foliation modified by su…

Proves geodesic connections on 2-torus without invariant tori.

We consider equivalence relations among smooth map germs with respect to geometry of G-structures on the target space germ. These equivalence relations are natural generalization of right-left equivalence (i.e., A-equivalence) in the sense of Thom-Mather depending on geometric structures on the target space germ. Unfor…

Study beta function for convex billiard maps, linking spectral invariants.

We introduce class A spacetimes, i.e. compact vicious spacetimes $(M,g)$ such that the Abelian cover $(\bar{M},\bar{g})$ is globally hyperbolic. We study the main properties of class A spacetimes using methods similar to the one introduced in D. Sullivan "Cycles for the dynamical study of foliated manifolds and complex…

Study characteristic classes of a specific type of determinantal varieties.

Compactness proven for isospectral Birkhoff billiard tables.

The paper classifies constraint mappings in optimization problems.

Study on positive scalar curvature on complex spaces with specific conditions.

We consider singular foliations of codimension one on 3-manifolds, in the sense defined by A. Haefliger as being Gamma_1-structures. We prove that under the obvious linear embedding condition, they are Gamma_1-homotopic to a regular foliation carried by an open book or a twisted open book. The latter concept is introdu…

For a given multicusp $f=c_{(θ_0,..., θ_i)}$ $(1\le i)$, we present a direct sum decomposition theorem of the source space of ${}_i\barωf$, where ${}_i\barωf$ is a higher version of the reduced Kodaira-Spencer-Mather map $\barωf$. As a corollary of our direct sum decomposition theorem, we show that for any $i\in \mathb…

The paper studies cohomology on incomplete manifolds and stratified spaces.

For a germ of a smooth map f and a subgroup G_V of any of the Mather groups G for which the source or target diffeomorphisms preserve some given volume form V in the source or in the target we study the G_V-moduli space of f that parameterizes the G_V-orbits inside the G-orbit of f. We find, for example, that this modu…