We show that in the generic case the smooth noncommutative tori associated with two$n$×$n$real skew-symmetric matrices are Morita equivalent if and only if the matrices are in the same orbit of the natural SO($n$,$n$∣$Z$) action.

We show that the scaling limit exists and is invariant under dilations and rotations. We give some tools that might be useful to show universality.

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$E$^{1}. 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$C$^{1}and$C$^{2}). 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$C$^{1}. We conclude with some remarks about the stationary measure.

In this paper we study surfaces in$R$^{3}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 ∫$σ$(∇$h$)$dx$$dy$among all Lipschitz functions$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 show that a compact Riemannian manifold with weakly pointwise 1/4-pinched sectional curvatures is either locally symmetric or diffeomorphic to a space form. More generally, we classify all compact, locally irreducible Riemannian manifolds$M$with the property that$M$×$R$^{2}has non-negative isotropic curvature.

Let$N$be a compact simply connected smooth Riemannian manifold and, for$p$∈ {2,3,...},$W$^{1,$p$}($R$^{$p$+1},$N$) be the Sobolev space of measurable maps from$R$^{$p$+1}into$N$whose gradients are in$L$^{$p$}. The restriction of$u$to almost every$p$-dimensional sphere$S$in$R$^{$p$+1}is in$W$^{1,$p$}($S$,$N$) and defines an homotopy class in π_{$p$}($N$) (White 1988). Evaluating a fixed element$z$of Hom(π_{$p$}($N$),$R$) on this homotopy class thus gives a real number Φ_{$z$,$u$}($S$). The main result of the paper is that any$W$^{1,$p$}-weakly convergent limit$u$of a sequence of smooth maps in$C$^{∞}($R$^{$p$+1},$N$), Φ_{$z$,$u$}has a$rectifiable Poincaré dual$ $ {\left( {\Gamma ,{\overrightarrow{\Gamma }} ,\theta } \right)} $ . Here Γ is a a countable union of$C$^{1}curves in$R$^{$p$+1}with Hausdorff $ {\user1{\mathcal{H}}}^{1} $ -measurable orientation $ {\overrightarrow{\Gamma }} :\Gamma \to S^{p} $ and density function$θ$: Γ→$R$. The intersection number between $ {\left( {\Gamma ,{\overrightarrow{\Gamma }} ,\theta } \right)} $ and$S$evaluates Φ_{$z$,$u$}($S$), for almost every$p$-sphere$S$. Moreover, we exhibit a non-negative integer$n$_{$z$}, depending only on homotopy operation$z$, such that $ {\int_\Gamma {{\left| \theta \right|}^{{p \mathord{\left/ {\vphantom {p {{\left( {p + n_{z} } \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {p + n_{z} } \right)}}}} d{\user1{\mathcal{H}}}^{1} < \infty } } $ even though the mass $ {\int_\Gamma {{\left| \theta \right|}d{\user1{\mathcal{H}}}^{1} } } $ may be infinite. We also provide cases of$N$,$p$and$z$for which this rational power$p$/($p$+$n$_{$z$}) is optimal. The construction of this Poincaré dual is based on 1-dimensional “bubbling” described by the notion of “scans” which was introduced in Hardt and Rivière (2003). We also describe how to generalize these results to$R$^{$m$}for any$m$⩾$p$+ 1, in which case the bubbling is described by an ($m$–$p$)-rectifiable set with orientation and density function determined by restrictions of the mappings to almost every oriented Euclidean$p$-sphere.

We consider amalgamated free product II_{1}factors$M$=$M$_{1*$B$}$M$_{2*$B$}… and use “deformation/rigidity” and “intertwining” techniques to prove that any relatively rigid von Neumann subalgebra$Q$⊂$M$can be unitarily conjugated into one of the$M$_{$i$}’s. We apply this to the case where the$M$_{$i$}’s are w-rigid II_{1}factors, with$B$equal to either$C$, to a Cartan subalgebra$A$in$M$_{$i$}, or to a regular hyperfinite II_{1}subfactor$R$in$M$_{$i$}, to obtain the following type of unique decomposition results, àla Bass–Serre: If$M$= ($N$_{1 * C}N_{2*$C$}…)^{$t$}, for some$t$> 0 and some other similar inclusions of algebras$C$⊂$N$_{$i$}then, after a permutation of indices, ($B$⊂$M$_{$i$}) is inner conjugate to ($C$⊂$N$_{$i$})^{$t$}, for all$i$. Taking$B$=$C$and $ M_{i} = {\left( {L{\left( {Z^{2} \rtimes F_{2} } \right)}} \right)}^{{t_{i} }} $ , with {$t$_{$i$}}_{$i$⩾1}=$S$a given countable subgroup of$R$_{+}^{*}, we obtain continuously many non-stably isomorphic factors$M$with fundamental group $ {\user1{\mathcal{F}}}{\left( M \right)} $ equal to$S$. For$B$=$A$, we obtain a new class of factors$M$with unique Cartan subalgebra decomposition, with a large subclass satisfying $ {\user1{\mathcal{F}}}{\left( M \right)} = {\left\{ 1 \right\}} $ and Out(M) abelian and calculable. Taking$B$=$R$, we get examples of factors with $ {\user1{\mathcal{F}}}{\left( M \right)} = {\left\{ 1 \right\}} $ , Out($M$) =$K$, for any given separable compact abelian group$K$.

We verify a conjecture of Rognes by establishing a localization cofiber sequence of spectra $K(\mathbb{Z})\to K(ku)\to K(KU) \to\Sigma K(\mathbb{Z})$ for the algebraic$K$-theory of topological$K$-theory. We deduce the existence of this sequence as a consequence of a dévissage theorem identifying the$K$-theory of the Waldhausen category of finitely generated finite stage Postnikov towers of modules over a connective $A_\infty$ ring spectrum$R$with the Quillen$K$-theory of the abelian category of finitely generated $\pi_{0}R$ -modules.

We establish the intrinsic Harnack inequality for non-negative solutions of a class of degenerate, quasilinear, parabolic equations, including equations of the$p$-Laplacian and porous medium type. It is shown that the classical Harnack estimate, while failing for degenerate parabolic equations, it continues to hold in a space-time geometry intrinsic to the degeneracy. The proof uses only measure-theoretical arguments, it reproduces the classical Moser theory, for non-degenerate equations, and it is novel even in that context. Hölder estimates are derived as a consequence of the Harnack inequality. The results solve a long standing problem in the theory of degenerate parabolic equations.

We consider random Schrödinger equations on$R$^{$d$}for$d$≽ 3 with a homogeneous Anderson–Poisson type random potential. Denote by λ the coupling constant and $\psi_t$ the solution with initial data $\psi_0$ . The space and time variables scale as $ x\sim\lambda ^{{ - 2 - \varkappa/2}} {\text{ and }}t\sim\lambda ^{{ - 2 - \varkappa}} {\text{ with }}0 < \varkappa < \varkappa_{0} {\left( d \right)} $ . We prove that, in the limit λ → 0, the expectation of the Wigner distribution of $\psi_t$ converges weakly to the solution of a heat equation in the space variable$x$for arbitrary$L$^{2}initial data.

We give a concrete and surprisingly simple characterization of compact sets $ K \subset \mathbb{R}^{{2 \times 2}} $ for which families of approximate solutions to the inclusion problem$Du$∈$K$are compact. In particular our condition is algebraic and can be tested algorithmically. We also prove that the quasiconvex hull of compact sets of 2 × 2 matrices can be localized. This is false for compact sets in higher dimensions in general.

We count the number$S$($x$) of quadruples $ {\left( {x_{1} ,x_{2} ,x_{3} ,x_{4} } \right)} \in \mathbb{Z}^{4} $ for which $$ p = x^{2}_{1} + x^{2}_{2} + x^{2}_{3} + x^{2}_{4} \leqslant x $$ is a prime number and satisfying the determinant condition:$x$_{1}$x$_{4}−$x$_{2}$x$_{3}= 1. By means of the sieve, one shows easily the upper bound$S$($x$) ≪$x$/log$x$. Under a hypothesis about prime numbers, which is stronger than the Bombieri–Vinogradov theorem but is weaker than the Elliott–Halberstam conjecture, we prove that this order is correct, that is$S$($x$) ≫$x$/log$x$.

We prove that the chordal contour lines of the discrete Gaussian free field converge to forms of SLE(4). Specifically, there is a constant λ > 0 such that when$h$is an interpolation of the discrete Gaussian free field on a Jordan domain—with boundary values −λ on one boundary arc and λ on the complementary arc—the zero level line of$h$joining the endpoints of these arcs converges to SLE(4) as the domain grows larger. If instead the boundary values are −$a$< 0 on the first arc and b > 0 on the complementary arc, then the convergence is to SLE(4;$a$/λ - 1,$b$/λ - 1), a variant of SLE(4).

We calculate the small quantum orbifold cohomology of arbitrary weighted projective spaces. We generalize Givental’s heuristic argument, which relates small quantum cohomology to$S$^{1}-equivariant Floer cohomology of loop space, to weighted projective spaces and use this to conjecture an explicit formula for the small$J$-function, a generating function for certain genus-zero Gromov–Witten invariants. We prove this conjecture using a method due to Bertram. This provides the first non-trivial example of a family of orbifolds of arbitrary dimension for which the small quantum orbifold cohomology is known. In addition we obtain formulas for the small$J$-functions of weighted projective complete intersections satisfying a combinatorial condition; this condition naturally singles out the class of orbifolds with terminal singularities.

The embedded cobordism category under study in this paper generalizes the category of conformal surfaces, introduced by G. Segal in [S2] in order to formalize the concept of field theories. Our main result identifies the homotopy type of the classifying space of the embedded$d$-dimensional cobordism category for all$d$. For$d$= 2, our results lead to a new proof of the generalized Mumford conjecture, somewhat different in spirit from the original one, presented in [MW].

We give a necessary and sufficient condition on the cost function so that the map solution of Monge’s optimal transportation problem is continuous for arbitrary smooth positive data. This condition was first introduced by Ma, Trudinger and Wang [24], [30] for a priori estimates of the corresponding Monge–Ampère equation. It is expressed by a socalled$cost-sectional curvature$being non-negative. We show that when the cost function is the squared distance of a Riemannian manifold, the cost-sectional curvature yields the sectional curvature. As a consequence, if the manifold does not have non-negative sectional curvature everywhere, the optimal transport map$cannot be continuous$for arbitrary smooth positive data. The non-negativity of the cost-sectional curvature is shown to be equivalent to the connectedness of the contact set between any cost-convex function (the proper generalization of a convex function) and any of its supporting functions. When the cost-sectional curvature is uniformly positive, we obtain that optimal maps are continuous or Hölder continuous under quite weak assumptions on the data, compared to what is needed in the Euclidean case. This case includes the quadratic cost on the round sphere.