The Whitney embedding theorem gives an upper bound on the smallest embedding dimension of a manifold. If a data set lies on a manifold, a random projection into this reduced dimension will retain the manifold structure. Here we present an algorithm to find a projection that distorts the data as little as possible.

The article considers distributed gradient flow (DGF) for multiagent nonconvex optimization. DGF is a continuous-time approximation of descent that often easier to study than its discrete-time counterpart. has two main contributions. First, the optimization nonsmooth, objective functions. It shown converges critical points in this setting. then problem avoiding saddle points. if agents’ functio...

Strata in manifold stratified spaces are shown to have neighborhoods that are teardrops of manifold stratified approximate fibrations (under dimension and compactness assumptions). This is the best possible version of the tubular neighborhood theorem for strata in the topological setting. Applications are given to replacement of singularities, to the structure of neighborhoods of points in mani...

We prove entropy rigidity for finite volume strictly convex projective manifolds in dimensions $$\ge 3$$ , generalizing the work of [1] to setting. The theorem uses techniques Besson, Courtois, and Gallot’s theorem. It implies uniform lower bounds on any manifold .

Let X be a projective K3 surface defined over C and H an ample divisor on X. For a coherent sheaf E on X, v(E) := ch(E) √ tdX ∈ H∗(X,Z) is the Mukai vector of E, where tdX is the Todd class of X. We denote the moduli space of stable sheaves E of v(E) = v by MH(v). If v is primitive and H is general (i.e. H does not lie on walls [Y3]), then MH(v) is a smooth projective scheme. In [Mu1], Mukai sh...

In this paper, we have defined recurrence and symmetry of different kinds in Hsu-structure manifold. Some theorem establishing relationship between recurrent H- HSU-Structure manifold involving equivalent conditions with respect to projective, conformal, conharmonic concircular curvature tensors has been discussed parameter also studied.

We prove a relative version of Kontsevich’s formality theorem. This theorem involves a manifold M and a submanifold C and reduces to Kontsevich’s theorem if C=M. It states that the DGLA of multivector fields on an infinitesimal neighbourhood of C is L∞-quasiisomorphic to the DGLA of multidifferential operators acting on sections of the exterior algebra of the conormal bundle. Applications to th...

We establish a S1-equivariant index theorem for Dirac operators on Z/kmanifolds. As an application, we generalize the Atiyah-Hirzebruch vanishing theorem for S1-actions on closed spin manifolds to the case of Z/k-manifolds. Résumé français On établit un théorème d’indice S-équivariant pour les opérateurs de Dirac sur des Z/k variétés. On donne une application de ce résultat, qui généralise le t...

Thurston’s hyperbolic Dehn surgery theorem is one of the most important results in 3-manifold theory, and it has stimulated an enormous amount of research. IfM is a compact orientable hyperbolic 3-manifold with boundary a single torus, then the theorem asserts that, for all but finitely many slopes s on ∂M , the manifold M(s) obtained by Dehn filling along s also admits a hyperbolic structure. ...