Let M be a smooth manifold and V a Euclidean space. Let $ \overline{{{\text{Emb}}}} $(M, V) be the homotopy fiber of the map Emb(M, V) → Imm(M, V). This paper is about the rational homology of $ \overline{{{\text{Emb}}}} $(M, V). We study it by applying embedding calculus and orthogonal calculus to the bifunctor (M, V)↦ HQ ∧ $ \overline{{{\text{Emb}}}} $(M, V)₊. Our main theorem states that if $ \dim V \geqslant 2{\text{ED}}{\left( M \right)} + 1 $(where ED(M) is the embedding dimension of M), the Taylor tower in the sense of orthogonal calculus (henceforward called “the orthogonal tower”) of this functor splits as a product of its layers. Equivalently, the rational homology spectral sequence associated with the tower collapses at E¹. In the case of knot embeddings, this spectral sequence coincides with the Vassiliev spectral sequence. The main ingredients in the proof are embedding calculus and Kontsevich's theorem on the formality of the little balls operad. We write explicit formulas for the layers in the orthogonal tower of the functor $ HQ \wedge \overline{{{\text{Emb}}}} {\left( {M,V} \right)}_{ + }. $The formulas show, in particular, that the (rational) homotopy type of the layers of the orthogonal tower is determined by the (rational) homotopy type of M. This, together with our rational splitting theorem, implies that, under the above assumption on codimension, rational homology equivalences of manifolds induce isomorphisms between the rational homology groups of $ \overline{{{\text{Emb}}}} $(–,V).

Using probabilistic methods, we prove new rigidity results for groups and pseudo-groups of diffeomorphisms of one dimensional manifolds with intermediate regularity class (i.e. between C¹ and C²). In particular, we show some generalizations of Denjoy theorem and the classical Kopell lemma for abelian groups. These techniques are also applied to the study of codimension-1 foliations. For instance, we obtain several generalized versions of Sacksteder theorem in class C¹. We conclude with some remarks about the stationary measure.

Par des méthodes de nature probabiliste, nous démontrons des nouveaux résultats de rigidité pour des groupes et des pseudo-groupes de difféomorphismes de variétés unidimensionnelles dont la classe de différentiabilité est intermédiaire (i.e. entre C¹ et C²). En particulier, nous prouvons des généralisations du théorème de Denjoy et d'un lemme classique de Kopell pour des groupes abéliens. Ensuite, nous appliquons les techniques introduites à l'étude des feuilletages de codimension 1 dont la régularité transverse est intermédiaire. Nous obtenons notamment des versions généralisées du théorème de Sacksteder en classe C¹. Nous finissons par quelques remarques à propos de la mesure stationnaire.

In this paper we study surfaces in R³ that arise as limit shapes in random surface models related to planar dimers. These limit shapes are surface tension minimizers, that is, they minimize a functional of the form ∫σ(∇h) dxdy among all Lipschitz functions h taking given values on the boundary of the domain. The surface tension σ has singularities and is not strictly convex, which leads to formation of facets and edges in the limit shapes.

We find a change of variables that reduces the Euler–Lagrange equation for the variational problem to the complex inviscid Burgers equation (complex Hopf equation). The equation can thus be solved in terms of an arbitrary holomorphic function, which is somewhat similar in spirit to Weierstrass parametrization of minimal surfaces. We further show that for a natural dense set of boundary conditions, the holomorphic function in question is, in fact, algebraic. The tools of algebraic geometry can thus be brought in to study the minimizers and, especially, the formation of their singularities. This is illustrated by several explicitly computed examples.