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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.

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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.

We establish the existence of infinitely many$polynomial$progressions in the primes; more precisely, given any integer-valued polynomials$P$_{1}, …,$P$_{$k$}∈$Z$[$m$] in one unknown$m$with$P$_{1}(0) = … =$P$_{$k$}(0) = 0, and given any$ε$> 0, we show that there are infinitely many integers$x$and$m$, with $1 \leqslant m \leqslant x^\varepsilon$ , such that$x$+$P$_{1}($m$), …,$x$+$P$_{$k$}($m$) are simultaneously prime. The arguments are based on those in [18], which treated the linear case$P$_{$j$}= ($j$− 1)$m$and$ε$= 1; the main new features are a localization of the shift parameters (and the attendant Gowers norm objects) to both coarse and fine scales, the use of PET induction to linearize the polynomial averaging, and some elementary estimates for the number of points over finite fields in certain algebraic varieties.

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$x$.