We study the energy-critical focusing non-linear wave equation, with data in the energy space, in dimensions 3, 4 and 5. We prove that for Cauchy data of energy smaller than the one of the static solution$W$which gives the best constant in the Sobolev embedding, the following alternative holds. If the initial data has smaller norm in the homogeneous Sobolev space$H$^{1}than the one of$W$, then we have global well-posedness and scattering. If the norm is larger than the one of$W$, then we have break-down in finite time.

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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 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 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 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 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$K$.

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 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.

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.

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.

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).

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.

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 prove$L$^{$p$}boundedness of certain non-translation-invariant discrete maximal Radon transforms and discrete singular Radon transforms. We also prove maximal, pointwise, and$L$^{$p$}ergodic theorems for certain families of non-commuting operators.

A$generalized polynomial$is a real-valued function which is obtained from conventional polynomials by the use of the operations of addition, multiplication, and taking the integer part; a$generalized polynomial mapping$is a vector-valued mapping whose coordinates are generalized polynomials. We show that any bounded generalized polynomial mapping$u$:$Z$^{$d$}→$R$^{$l$}has a representation$u$($n$) =$f$($ϕ$($n$)$x$),$n$∈$Z$^{$d$}, where$f$is a piecewise polynomial function on a compact nilmanifold$X$,$x$∈$X$, and$ϕ$is an ergodic$Z$^{$d$}-action by translations on$X$. This fact is used to show that the sequence$u$($n$),$n$∈$Z$^{$d$}, is well distributed on a piecewise polynomial surface $\mathcal{S}\subset\mathbf{R}^{l}$ (with respect to the Borel measure on $\mathcal{S}$ that is the image of the Lebesgue measure under the piecewise polynomial function defining $\mathcal{S}$ ). As corollaries we also obtain a von Neumann-type ergodic theorem along generalized polynomials and a result on Diophantine approximations extending the work of van der Corput and of Furstenberg–Weiss.

We use microlocal and paradifferential techniques to obtain$L$^{8}norm bounds for spectral clusters associated with elliptic second-order operators on two-dimensional manifolds with boundary. The result leads to optimal$L$^{$q$}bounds, in the range 2⩽q⩽∞, for$L$^{2}- normalized spectral clusters on bounded domains in the plane and, more generally, for two-dimensional compact manifolds with boundary. We also establish new sharp$L$^{$q$}estimates in higher dimensions for a range of exponents q̅_{n}⩽$q$⩽∞.

We study property ($T$) and the fixed-point property for actions on$L$^{$p$}and other Banach spaces. We show that property ($T$) holds when$L$^{2}is replaced by$L$^{$p$}(and even a subspace/quotient of$L$^{$p$}), and that in fact it is independent of 1≤$p$<∞. We show that the fixed-point property for$L$^{$p$}follows from property ($T$) when 1<$p$< 2+$ε$. For simple Lie groups and their lattices, we prove that the fixed-point property for$L$^{$p$}holds for any 1<$p$<∞ if and only if the rank is at least two. Finally, we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive spaces.

We prove the Zorich–Kontsevich conjecture that the non-trivial Lyapunov exponents of the Teichmüller ow on (any connected component of a stratum of) the moduli space of Abelian differentials on compact Riemann surfaces are all distinct. By previous work of Zorich and Kontsevich, this implies the existence of the complete asymptotic Lagrangian flag describing the behavior in homology of the vertical foliation in a typical translation surface.

Let$χ$be an analytic vector field defined in a real-analytic manifold of dimension three. We prove that all the singularities of$χ$can be made elementary by a finite number of blowing-ups in the ambient space.

We prove Pólya’s conjecture of 1943: For a real entire function of order greater than 2 with finitely many non-real zeros, the number of non-real zeros of the$n$th derivative tends to infinity, as $$n\to\infty$$ . We use the saddle point method and potential theory, combined with the theory of analytic functions with positive imaginary part in the upper half-plane.

We show that closed, locally symmetric spaces of non-compact type have positive simplicial volume. This gives a positive answer to a question that was first raised by (Gromov in Inst Hautes Études Sci Publ Math 56:5,1982).

In this paper, we investigate the topology of a class of non-Kähler compact complex manifolds generalizing that of Hopf and Calabi-Eckmann manifolds. These manifolds are diffeomorphic to special systems of real quadrics$C$^{$n$}which are invariant with respect to the natural action of the real torus ($S$^{1})^{$n$}onto$C$^{$n$}. The quotient space is a simple convex polytope. The problem reduces thus to the study of the topology of certain real algebraic sets and can be handled using combinatorial results on convex polytopes. We prove that the homology groups of these compact complex manifolds can have arbitrary amount of torsion so that their topology is extremely rich. We also resolve an associated wall-crossing problem by introducing holomorphic equivariant elementary surgeries related to some transformations of the simple convex polytope. Finally, as a nice consequence, we obtain that affine non-Kähler compact complex manifolds can have arbitrary amount of torsion in their homology groups, contrasting with the Kähler situation.

We answer Hubbard's question on determining the Thurston equivalence class of “twisted rabbits”, i.e. composita of the “rabbit” polynomial with$n$th powers of the Dehn twists about its ears. The answer is expressed in terms of the 4-adic expansion of$n$. We also answer the equivalent question for the other two families of degree-2 topological polynomials with three post-critical points. In the process, we rephrase the questions in group-theoretical language, in terms of wreath recursions.