We introduce a new concept of solution to the KPZ equation which is shown to extend the classical Cole-Hopf solution. This notion provides a factorisation of the Cole-Hopf solution map into a “universal” measurable map from the probability space into an explicitly described auxiliary metric space, composed with a new solution map that has very good continuity properties. The advantage of such a formulation is that it essentially provides a pathwise notion of a solution, together with a very detailed approximation theory. In particular, our construction completely bypasses the Cole-Hopf transform, thus laying the groundwork for proving that the KPZ equation describes the fluctuations of systems in the KPZ universality class.

As a corollary of our construction, we obtain very detailed new regularity results about the solution, as well as its derivative with respect to the initial condition. Other byproducts of the proof include an explicit approximation to the stationary solution of the KPZ equation, a well-posedness result for the Fokker-Planck equation associated to a particle diffusing in a rough space-time dependent potential, and a new periodic homogenisation result for the heat equation with a space-time periodic potential. One ingredient in our construction is an example of a non-Gaussian rough path such that the area process of its natural approximations needs to be renormalised by a diverging term for the approximations to converge.

## Solving the KPZ equation

Pages 559-664 by
From volume 178-2

## Nonuniqueness of weak solutions to the Navier-Stokes equation

For initial datum of finite kinetic energy, Leray has proven in 1934 that there exists at least one global in time finite energy weak solution of the 3D Navier-Stokes equations. In this paper we prove that weak solutions of the 3D Navier-Stokes equations are not unique in the class of weak solutions with finite kinetic energy. Moreover, we prove that Hölder continuous dissipative weak solutions of the 3D Euler equations may be obtained as a strong vanishing viscosity limit of a sequence of finite energy weak solutions of the 3D Navier-Stokes equations.

Pages 101-144 by
From volume 189-1

## Global existence of weak solutions for compressible Navier–Stokes equations: Thermodynamically unstable pressure and anisotropic viscous stress tensor

We prove global existence of appropriate weak solutions for the compressible Navier–Stokes equations for a more general stress tensor than those previously covered by P.-L.Lions and E. Feireisl’s theory. More precisely we focus on more *general pressure laws* that are *not thermodynamically stable*; we are also able to handle some *anisotropy in the viscous stress tensor*. To give answers to these two longstanding problems, we revisit the classical compactness theory on the density by obtaining precise quantitative regularity estimates: This requires a more precise analysis of the structure of the equations combined to a novel approach to the compactness of the continuity equation. These two cases open the theory to important physical applications, for instance to describe solar events (virial pressure law), geophysical flows (eddy viscosity) or biological situations (anisotropy).

Pages 577-684 by
From volume 188-2

## Global well-posedness for the Yang-Mills equation in $4+1$ dimensions. Small energy

We consider the hyperbolic Yang-Mills equation on the Minkowski space $\mathbb{R}^{4+1}$. Our main result asserts that this problem is globally well-posed for all initial data whose energy is sufficiently small. This solves a longstanding open problem.

Pages 831-893 by
From volume 185-3

## A sharp counterexample to local existence of low regularity solutions to Einstein equations in wave coordinates

We give a sharp counterexample to local existence of low regularity solutions to Einstein equations in wave coordinates. We show that there are initial data in $H^2$ satisfying the wave coordinate condition such that there is no solution in $H^2$ to Einstein equations in wave coordinates for any positive time. This result is sharp since Klainerman-Rodnianski and Smith-Tataru proved existence for the same equations with slightly more regular initial data.

Pages 311-330 by
From volume 185-1

## Finite time singularity for the modified SQG patch equation

It is well known that the incompressible Euler equations in two dimensions have globally regular solutions. The inviscid surface quasi-geostrophic (SQG) equation has a Biot-Savart law that is one derivative less regular than in the Euler case, and the question of global regularity for its solutions is still open. We study here the patch dynamics in the half-plane for a family of active scalars that interpolates between these two equations, via a parameter $\alpha\in[0,\frac 12]$ appearing in the kernels of their Biot-Savart laws. The values $\alpha=0$ and $\alpha=\frac 12$ correspond to the 2D Euler and SQG cases, respectively. We prove global in time regularity for the 2D Euler patch model, even if the patches initially touch the boundary of the half-plane. On the other hand, for any sufficiently small $\alpha>0$, we exhibit initial data that lead to a singularity in finite time. Thus, these results show a phase transition in the behavior of solutions to these equations and provide a rigorous foundation for classifying the 2D Euler equations as critical.

Pages 909-948 by
From volume 184-3

## Decay for solutions of the wave equation on Kerr exterior spacetimes III: The full subextremalcase $|a| < M$

This paper concludes the series begun in [M. Dafermos and I. Rodnianski, Decay for solutions of the wave equation on Kerr exterior spacetimes I–II: the cases $|a| \ll M$ or axisymmetry, arXiv:1010.5132], providing the complete proof of definitive boundedness and decay results for the scalar wave equation on Kerr backgrounds in the general subextremal $|a|< M$ case without symmetry assumptions. The essential ideas of the proof (together with explicit constructions of the most difficult multiplier currents) have been announced in our survey [M. Dafermos and I. Rodnianski, The black hole stability problem for linear scalar perturbations, in *Proceedings of the 12th Marcel Grossmann Meeting on General Relativity*, T. Damour et al. (ed.), World Scientific, Singapore, 2011, pp. 132189, arXiv:1010.5137]. Our proof appeals also to the quantitative mode-stability proven in [Y. Shlapentokh-Rothman, Quantitative Mode Stability for the Wave Equation on the Kerr Spacetime, arXiv:1302.6902, to appear, *Ann. Henri Poincaré*], together with a streamlined continuity argument in the parameter $a$, appearing here for the first time. While serving as Part III of a series, this paper repeats all necessary notation so that it can be read independently of previous work.

Pages 787-913 by
From volume 183-3

## Hidden symmetries and decay for the wave equation on the Kerr spacetime

Energy and decay estimates for the wave equation on the exterior region of slowly rotating Kerr spacetimes are proved. The method used is a generalisation of the vector-field method that allows the use of higher-order symmetry operators. In particular, our method makes use of the second-order Carter operator, which is a hidden symmetry in the sense that it does not correspond to a Killing symmetry of the spacetime.

Pages 787-853 by
From volume 182-3

## Solution of Leray’s problem for stationary Navier-Stokes equations in plane and axially symmetric spatial domains

We study the nonhomogeneous boundary value problem for the Navier-Stokes equations of steady motion of a viscous incompressible fluid in arbitrary bounded multiply connected plane or axially-symmetric spatial domains. (For axially symmetric domains, data is assumed to be axially symmetric as well.) We prove that this problem has a solution under the sole necessary condition of zero total flux through the boundary. The problem was formulated by Jean Leray 80 years ago. The proof of the main result uses Bernoulli’s law for a weak solution to the Euler equations.

Pages 769-807 by
From volume 181-2

## Construction of Cauchy data of vacuum Einstein field equations evolving to black holes

We show the existence of complete, asymptotically flat Cauchy initial data for the vacuum Einstein field equations, free of trapped surfaces, whose future development must admit a trapped surface. Moreover, the datum is exactly a constant time slice in Minkowski space-time inside and exactly a constant time slice in Kerr space-time outside.

The proof makes use of the full strength of Christodoulou’s work on the dynamical formation of black holes and Corvino-Schoen’s work on the construction of initial data sets.

Pages 699-768 by
From volume 181-2

## Dispersion for the wave equation inside strictly convex domains I: the Friedlander model case

We consider a model case for a strictly convex domain $\Omega\subset\mathbb{R}^d$ of dimension $d\geq 2$ with smooth boundary $\partial\Omega\neq\emptyset$, and we describe dispersion for the wave equation with Dirichlet boundary conditions. More specifically, we obtain the optimal fixed time decay rate for the smoothed out Green function: a $t^{1/4}$ loss occurs with respect to the boundary less case, due to repeated occurrences of swallowtail type singularities in the wave front set.

** Note: To view the article, click on the URL link for the DOI number. **

Pages 323-380 by
From volume 180-1

## A correction to “Propagation of singularities for the wave equation on manifolds with corners”

We correct an error in the proof of Proposition 7.3 of the author’s paper on the propagation of singularities for the wave equation on manifolds with corners. The correction does not affect the statement of Proposition 7.3, and it does not affect any other part of the paper.

Pages 783-785 by
From volume 177-2

## Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity

For a large class of nonlinear Schrödinger equations with nonzero conditions at infinity and for any speed $c$ less than the sound velocity, we prove the existence of nontrivial finite energy traveling waves moving with speed $c$ in any space dimension $N\geq 3$. Our results are valid as well for the Gross-Pitaevskii equation and for NLS with cubic-quintic nonlinearity.

Pages 107-182 by
From volume 178-1

## The Witten equation, mirror symmetry, and quantum singularity theory

For any nondegenerate, quasi-homogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a corresponding cohomological field theory associated to the singularity. This theory is analogous to Gromov-Witten theory and generalizes the theory of $r$-spin curves, which corresponds to the simple singularity $A_{r-1}$.

We also resolve two outstanding conjectures of Witten. The first conjecture is that ADE-singularities are self-dual, and the second conjecture is that the total potential functions of ADE-singularities satisfy corresponding ADE-integrable hierarchies. Other cases of integrable hierarchies are also discussed.

Pages 1-106 by
From volume 178-1

## Klein forms and the generalized superelliptic equation

If $F(x,y) \in \mathbb{Z}[x,y]$ is an irreducible binary form of degree $k \geq 3$, then a theorem of Darmon and Granville implies that the generalized superelliptic equation $$ F(x,y)=z^l $$ has, given an integer $l \geq \mathrm{max} \{ 2, 7-k \}$, at most finitely many solutions in coprime integers $x, y$ and $z$. In this paper, for large classes of forms of degree $k=3, 4, 6$ and $12$ (including, heuristically, “most” cubic forms), we extend this to prove a like result, where the parameter $l$ is now taken to be variable. In the case of irreducible cubic forms, this provides the first examples where such a conclusion has been proven. The method of proof combines classical invariant theory, modular Galois representations, and properties of elliptic curves with isomorphic mod-$n$ Galois representations.

Pages 171-239 by
From volume 177-1

## Knots and links in steady solutions of the Euler equation

Given any possibly unbounded, locally finite link, we show that there exists a smooth diffeomorphism transforming this link into a set of stream (or vortex) lines of a vector field that solves the steady incompressible Euler equation in $\mathbb{R}^3$. Furthermore, the diffeomorphism can be chosen arbitrarily close to the identity in any $C^r$ norm.

Pages 345-367 by
From volume 175-1

## Heisenberg uniqueness pairs and the Klein-Gordon equation

A Heisenberg uniqueness pair (HUP) is a pair $(\Gamma,\Lambda)$, where $\Gamma$ is a curve in the plane and $\Lambda$ is a set in the plane, with the following property: any finite Borel measure $\mu$ in the plane supported on $\Gamma$, which is absolutely continuous with respect to arc length, and whose Fourier transform $\widehat\mu$ vanishes on $\Lambda$, must automatically be the zero measure. We prove that when $\Gamma$ is the hyperbola $x_1x_2=1$ %, and $\Lambda$ is the lattice-cross \[\Lambda=(\alpha\mathbb{Z}\times\{0\})\cup(\{0\}\times\beta\mathbb{Z}),\] where $\alpha,\beta$ are positive reals, then $(\Gamma,\Lambda)$ is an HUP if and only if $\alpha\beta\le1$; in this situation, the Fourier transform $\widehat\mu$ of the measure solves the one-dimensional Klein-Gordon equation. Phrased differently, we show that \[{\mathrm e}^{\pi{\mathrm i} \alpha n t},\,\,{\mathrm e}^{\pi{\mathrm i}\beta n/t},\qquad n\in\mathbb{Z},\] span a weak-star dense subspace in $L^\infty(\mathbb{R})$ if and only if $\alpha\beta\le1$. In order to prove this theorem, some elements of linear fractional theory and ergodic theory are needed, such as the Birkhoff Ergodic Theorem. An idea parallel to the one exploited by Makarov and Poltoratski (in the context of model subspaces) is also needed. As a consequence, we solve a problem on the density of algebras generated by two inner functions raised by Matheson and Stessin.

Pages 1507-1527 by
From volume 173-3

## The Evans-Krylov theorem for nonlocal fully nonlinear equations

We prove a regularity result for solutions of a purely integro-differential Bellman equation. This regularity is enough for the solutions to be understood in the classical sense. If we let the order of the equation approach two, we recover the theorem of Evans and Krylov about the regularity of solutions to concave uniformly elliptic partial differential equations.

Pages 1163-1187 by
From volume 174-2

## Global regularity for some classes of large solutions to the Navier-Stokes equations

In previous works by the first two authors, classes of initial data to the three-dimensional, incompressible Navier-Stokes equations were presented, generating a global smooth solution although the norm of the initial data may be chosen arbitrarily large. The main feature of the initial data considered in one of those studies is that it varies slowly in one direction, though in some sense it is “well-prepared” (its norm is large but does not depend on the slow parameter). The aim of this article is to generalize that setting to an “ill prepared” situation (the norm blows up as the small parameter goes to zero). As in those works, the proof uses the special structure of the nonlinear term of the equation.

Pages 983-1012 by
From volume 173-2

## Description of two soliton collision for the quartic gKdV equation

In this paper, we give the first description of the collision of two solitons for a nonintegrable equation in a special regime. We consider solutions of the quartic gKdV equation $\partial_t u + \partial_x (\partial_x^2 u + u^4)=0$, which behave as $t\to -\infty$ like \[ u(t,x)=Q_{c_1}(x -c_1 t) + Q_{c_2}(x-c_2 t) + \eta(t,x), \] where $Q_{c}(x-ct)$ is a soliton and $\|\eta(t)\|_{H^1} \ll \|Q_{c_2}\|_{H^1}\ll \|Q_{c_1}\|_{H^1}$.

The global behavior of $u(t)$ is given by the following stability result: for all $t\in \mathbb{R}$, $u(t,x)=Q_{ c_1(t)}(x-y_1(t)) + Q_{ c_2(t)}(x-y_2(t)) + \eta(t,x)$, where $\vert\eta(t)\|_{H^1} \ll \|Q_{c_2}\vert_{H^1}$ and $\lim_{t\to +\infty} c_1(t)=c_1^{+}$, $\lim_{t\to +\infty} c_2(t)=c_2^{+}$.

In the case where $u(t)$ is a pure $2$-soliton solution as $t\to -\infty$ (i.e. $\mathrm{lim}_{t\to -\infty} \vert\eta(t)\vert_{H^1}=0$), we obtain $ c_1^{+}>c_1, c_2^{+}\lt c_2 $ and for the residual part, $\mathrm{lim}_{t\to +\infty} \vert\eta(t)\vert_{H^1}\gt 0$. Therefore, in contrast with the integrable KdV equation (or mKdV equation), no global pure 2-soliton solution exists and the collision is inelastic. A different notion of global 2-soliton is then proposed.

Pages 757-857 by
From volume 174-2

## Functional equations for zeta functions of groups and rings

We introduce a new method to compute explicit formulae for various zeta functions associated to groups and rings. The specific form of these formulae enables us to deduce local functional equations. More precisely, we prove local functional equations for the subring zeta functions associated to rings, the subgroup, conjugacy and representation zeta functions of finitely generated, torsion-free nilpotent (or $\mathcal{T}$-)groups, and the normal zeta functions of $\mathcal{T}$-groups of class $2$. We deduce our theorems from a “blueprint result” on certain $p$-adic integrals which generalises work of Denef and others on Igusa’s local zeta function. The Malcev correspondence and a Kirillov-type theory developed by Howe are used to “linearise” the problems of counting subgroups and representations in $\mathcal{T}$-groups, respectively.

Pages 1181-1218 by
From volume 172-2

## The Euler equations as a differential inclusion

We propose a new point of view on weak solutions of the Euler equations, describing the motion of an ideal incompressible fluid in $\mathbb{R}^n$ with $n\geq 2$. We give a reformulation of the Euler equations as a differential inclusion, and in this way we obtain transparent proofs of several celebrated results of V. Scheffer and A. Shnirelman concerning the non-uniqueness of weak solutions and the existence of energy-decreasing solutions. Our results are stronger because they work in any dimension and yield bounded velocity and pressure.

Pages 1417-1436 by
From volume 170-3

## The polynomial Carleson operator

We prove affirmatively the one-dimensional case of a conjecture of Stein regarding the $L^p$-boundedness of the Polynomial Carleson operator, for $1\lt p\lt \infty$.

Our proof relies on two new ideas: i) we develop a framework for *higher-order wave-packet analysis* that is consistent with the time-frequency analysis of the (generalized) Carleson operator, and ii) we introduce a *local analysis* adapted to the concepts of mass and counting function, which yields a new tile discretization of the time-frequency plane that has the major consequence of *eliminating the exceptional sets* from the analysis of the Carleson operator. As a further consequence, we are able to deliver the full $L^p$ boundedness range and prove directly–without interpolation techniques–the strong $L^2$ bound for the (generalized) Carleson operator, answering a question raised by C. Fefferman.

by
From To appear in forthcoming issues

## On the Duffin-Schaeffer conjecture

Let $\psi:\mathbb{N}\to\mathbb{R}_{\ge0}$ be an arbitrary function from the positive integers to the non-negative reals. Consider the set $\mathcal{A}$ of real numbers $\alpha$ for which there are infinitely many reduced fractions $a/q$ such that $|\alpha-a/q|\le \psi(q)/q$. If $\sum_{q=1}^\infty \psi(q)\phi(q)/q=\infty$, we show that $\mathcal{A}$ has full Lebesgue measure. This answers a question of Duffin and Schaeffer. As a corollary, we also establish a conjecture due to Catlin regarding non-reduced solutions to the inequality $|\alpha- a/q|\le \psi(q)/q$, giving a refinement of Khinchin’s Theorem.

by
From To appear in forthcoming issues

## Uniform Manin-Mumford for a family of genus 2 curves

We introduce a general strategy for proving quantitative and uniform bounds on the number of common points of height zero for a pair of inequivalent height functions on $\mathbb {P}^1(\overline {\mathbb {Q}}).$ We apply this strategy to prove a conjecture of Bogomolov, Fu, and Tschinkel asserting uniform bounds on the number of common torsion points of elliptic curves in the case of two Legendre curves over $\mathbb {C}$. As a consequence, we obtain two uniform bounds for a two-dimensional family of genus 2 curves: a uniform Manin-Mumford bound for the family over $\mathbb {C}$, and a uniform Bogomolov bound for the family over $\overline {\mathbb {Q}}.$

Pages 949-1001 by
From volume 191-3

## A converse to a theorem of Gross, Zagier, and Kolyvagin

Let $E$ be a semistable elliptic curve over $\mathbb {Q}$. We prove that if $E$ has non-split multiplicative reduction at at least one odd prime or split multiplicative reduction at at least two odd primes, then $$\mathrm{rank}_\mathbb{Z} E(\mathbb {Q}) =1 \,\, \text {and} \,\, \#Ш(E)<\infty \Rightarrow \mathrm {ord}_{s=1}L(E,s)=1. $$ We also prove the corresponding result for the abelian variety associated with a weight 2 newform $f$ of trivial character. These, and other related results, are consequences of our main theorem, which establishes criteria for $f$ and $H^1_f(\mathbb {Q},V)$, where $V$ is the $p$-adic Galois representation associated with $f$, that ensure that $\mathrm {ord}_{s=1}L(f,s)=1$. The main theorem is proved using the Iwasawa theory of $V$ over an imaginary quadratic field to show that the $p$-adic logarithm of a suitable Heegner point is non-zero.

Pages 329-354 by
From volume 191-2

## Integrability of Liouville theory: proof of the DOZZ formula

Dorn and Otto (1994) and independently Zamolodchikov and Zamolodchikov (1996) proposed a remarkable explicit expression, the so-called DOZZ formula, for the three point structure constants of Liouville Conformal Field Theory (LCFT), which is expected to describe the scaling limit of large planar maps properly embedded into the Riemann sphere. In this paper we give a proof of the DOZZ formula based on a rigorous probabilistic construction of LCFT in terms of Gaussian Multiplicative Chaos given earlier by F. David and the authors. This result is a fundamental step in the path to prove integrability of LCFT, i.e., to mathematically justify the methods of Conformal Bootstrap used by physicists. From the purely probabilistic point of view, our proof constitutes the first nontrivial rigorous integrability result on Gaussian Multiplicative Chaos measures.

Pages 81-166 by
From volume 191-1

## An application of Cartan’s equivalence method to Hirschowitz’s conjecture on the formal principle

A conjecture of Hirschowitz’s predicts that a globally generated vector bundle $W$ on a compact complex manifold $A$ satisfies the formal principle; i.e., the formal neighborhood of its zero section determines the germ of neighborhoods in the underlying complex manifold of the vector bundle $W$. By applying Cartan’s equivalence method to a suitable differential system on the universal family of the Douady space of the complex manifold, we prove that this conjecture is true if $A$ is a Fano manifold, or if the global sections of $W$ separate points of $A$. Our method shows more generally that for any unobstructed compact submanifold $A$ in a complex manifold, if the normal bundle is globally generated and its sections separate points of $A$, then a sufficiently general deformation of $A$ satisfies the formal principle. In particular, a sufficiently general smooth free rational curve on a complex manifold satisfies the formal principle.

Pages 979-1000 by
From volume 189-3

## Explicit Chabauty–Kim for the split Cartan modular curve of level 13

We extend the explicit quadratic Chabauty methods developed in previous work by the first two authors to the case of non-hyperelliptic curves. This results in a method to compute a finite set of $p$-adic points, containing the rational points, on a curve of genus $g \ge 2$ over the rationals whose Jacobian has Mordell–Weil rank $g$ and Picard number greater than one, and which satisfies some additional conditions. This is then applied to determine the rational points of the modular curve $X_{\mathrm { s}}(13)$, completing the classification of non-CM elliptic curves over $\mathbf {Q} $ with split Cartan level structure due to Bilu–Parent and Bilu–Parent–Rebolledo.

Pages 885-944 by
From volume 189-3

## Sharp $L^2$ estimates of the Schrödinger maximal function in higher dimensions

We show that, for $n\geq 3$, $\lim_{t \to 0} e^{it\Delta}f(x) = f(x)$ holds almost everywhere for all $f \in H^s (\mathbb{R}^n)$ provided that $s>\frac{n}{2(n+1)}$. Due to a counterexample by Bourgain, up to the endpoint, this result is sharp and fully resolves a problem raised by Carleson. Our main theorem is a fractal $L^2$ restriction estimate, which also gives improved results on the size of the divergence set of the Schrödinger solutions, the Falconer distance set problem and the spherical average Fourier decay rates of fractal measures. The key ingredients of the proof include multilinear Kakeya estimates, decoupling and induction on scales.

Pages 837-861 by
From volume 189-3

## On the $K$-theory of pullbacks

To any pullback square of ring spectra we associate a new ring spectrum and use it to describe the failure of excision in algebraic $K$-theory. The construction of this new ring spectrum is categorical and hence allows us to determine the failure of excision for any localizing invariant in place of $K$-theory.

As immediate consequences we obtain an improved version of Suslin’s excision result in $K$-theory, generalizations of results of Geisser and Hesselholt on torsion in (bi)relative $K$-groups, and a generalized version of pro-excision for $K$-theory. Furthermore, we show that any truncating invariant satisfies excision, nilinvariance, and cdh-descent. Examples of truncating invariants include the fibre of the cyclotomic trace, the fibre of the rational Goodwillie–Jones Chern character, periodic cyclic homology in characteristic zero, and homotopy $K$-theory.

Various of the results we obtain have been known previously, though most of them in weaker forms and with less direct proofs.

Pages 877-930 by
From volume 190-3

## An asymptotic formula for integer points on Markoff-Hurwitz varieties

We establish an asymptotic formula for the number of integer solutions to the Markoff-Hurwitz equation \[ x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}=ax_{1}x_{2}\cdots x_{n}+k. \] When $n\geq 4$, the previous best result is by Baragar (1998) that gives an exponential rate of growth with exponent $\beta $ that is not in general an integer when $n\geq 4$. We give a new interpretation of this exponent of growth in terms of the unique parameter for which there exists a certain conformal measure on projective space.

Pages 751-809 by
From volume 190-3

## Macdonald-positive specializations of the algebra of symmetric functions: Proof of the Kerov conjecture

We prove the classification of homomorphisms from the algebra of symmetric functions to $\mathbb {R}$ with non-negative values on Macdonald symmetric functions $P_{\lambda }$, which was conjectured by S. V. Kerov in 1992.

Pages 277-316 by
From volume 189-1

## Cichoń’s maximum

Assuming four strongly compact cardinals, it is consistent that all entries in Cichoń’s diagram (apart from $\mathrm {add}(\mathcal {M})$ and $\mathrm {cof}(\mathcal {M})$, whose values are determined by the others) are pairwise different; more specifically,

$\aleph _1 < \mathrm{add}(\mathcal {N})$ $\lt \mathrm{cov}(\mathcal {N}) < \mathfrak {b} < \mathrm{non}(\mathcal {M}) < \mathrm{cov}(\mathcal {M}) < \mathfrak{d} < \mathrm{non}(\mathcal{N}) < \mathrm{cof}(\mathcal{N}) < 2^{\aleph _0}$.

Pages 113-143 by
From volume 190-1

## The first stable homotopy groups of motivic spheres

We compute the $1$-line of stable homotopy groups of motivic spheres over fields of characteristic not two in terms of hermitian and Milnor $K$-groups. This is achieved by solving questions about convergence and differentials in the slice spectral sequence.

Pages 1-74 by
From volume 189-1

## Erratum for “Geometric Langlands duality and representations of algebraic groups over commutative rings”

No Abstract available for this article.

Pages 1017-1018 by
From volume 188-3

## A proof of Onsager’s conjecture

For any $\alpha < 1/3$, we construct weak solutions to the $3D$ incompressible Euler equations in the class $C_tC_x^\alpha $ that have nonempty, compact support in time on ${\mathbb R} \times {\mathbb T}^3$ and therefore fail to conserve the total kinetic energy. This result, together with the proof of energy conservation for $\alpha > 1/3$ due to [Eyink] and [Constantin, E, Titi], solves Onsager’s conjecture that the exponent $\alpha = 1/3$ marks the threshold for conservation of energy for weak solutions in the class $L_t^\infty C_x^\alpha $. The previous best results were solutions in the class $C_tC_x^\alpha $ for $\alpha < 1/5$, due to [Isett], and in the class $L_t^1 C_x^\alpha $ for $\alpha < 1/3$ due to [Buckmaster, De Lellis, Székelyhidi], both based on the method of convex integration developed for the incompressible Euler equations by [De Lellis, Székelyhidi]. The present proof combines the method of convex integration and a new “Gluing Approximation” technique. The convex integration part of the proof relies on the “Mikado flows” introduced by [Daneri, Székelyhidi] and the framework of estimates developed in the author’s previous work.

Pages 871-963 by
From volume 188-3

## Convergence of Ricci flows with bounded scalar curvature

In this paper we prove convergence and compactness results for Ricci flows with bounded scalar curvature and entropy. More specifically, we show that Ricci flows with bounded scalar curvature converge smoothly away from a singular set of codimension $\geq 4$. We also establish a general form of the Hamilton-Tian Conjecture, which is even true in the Riemannian case.

These results are based on a compactness theorem for Ricci flows with bounded scalar curvature, which states that any sequence of such Ricci flows converges, after passing to a subsequence, to a metric space that is smooth away from a set of codimension $\geq 4$. In the course of the proof, we will also establish $L^{p < 2}$-curvature bounds on time-slices of such flows.

Pages 753-831 by
From volume 188-3

## Semipositivity theorems for moduli problems

We prove some semipositivity theorems for singular varieties coming from graded polarizable admissible variations of mixed Hodge structure. As an application, we obtain that the moduli functor of stable varieties is semipositive in the sense of Kollár. This completes Kollár’s projectivity criterion for the moduli spaces of higher-dimensional stable varieties.

Pages 639-665 by
From volume 187-3

## Hodge theory for combinatorial geometries

We prove the hard Lefschetz theorem and the Hodge-Riemann relations for a commutative ring associated to an arbitrary matroid M. We use the Hodge-Riemann relations to resolve a conjecture of Heron, Rota, and Welsh that postulates the log-concavity of the coefficients of the characteristic polynomial of $\mathrm {M}$. We furthermore conclude that the $f$-vector of the independence complex of a matroid forms a log-concave sequence, proving a conjecture of Mason and Welsh for general matroids.

Pages 381-452 by
From volume 188-2

## Hyperbolic triangles without embedded eigenvalues

We consider the Neumann Laplacian acting on square-integrable functions on a triangle in the hyperbolic plane that has one cusp. We show that the generic such triangle has no eigenvalues embedded in its continuous spectrum. To prove this result we study the behavior of the real-analytic eigenvalue branches of a degenerating family of triangles. In particular, we use a careful analysis of spectral projections near the crossings of these eigenvalue branches with the eigenvalue branches of a model operator.

Pages 301-377 by
From volume 187-2

## Bilinear forms with Kloosterman sums and applications

We prove nontrivial bounds for general bilinear forms in hyper-Kloosterman sums when the sizes of both variables may be below the range controlled by Fourier-analytic methods (Pólya-Vinogradov range). We then derive applications to the second moment of cusp forms twisted by characters modulo primes, and to the distribution in arithmetic progressions to large moduli of certain Eisenstein-Hecke coefficients on $\mathrm{GL}_3$. Our main tools are new bounds for certain complete sums in three variables over finite fields, proved using methods from algebraic geometry, especially $\ell$-adic cohomology and the Riemann Hypothesis.

Pages 413-500 by
From volume 186-2

## Borel circle squaring

We give a completely constructive solution to Tarski’s circle squaring problem. More generally, we prove a Borel version of an equidecomposition theorem due to Laczkovich. If $k \geq 1$ and $A, B \subset \mathbb{R}^k$ are bounded Borel sets with the same positive Lebesgue measure whose boundaries have upper Minkowski dimension less than $k$, then $A$ and $B$ are equidecomposable by translations using Borel pieces. This answers a question of Wagon. Our proof uses ideas from the study of flows in graphs, and a recent result of Gao, Jackson, Krohne, and Seward on special types of witnesses to the hyperfiniteness of free Borel actions of $\mathbb{Z}^d$.

Pages 581-605 by
From volume 186-2

## A sharp Schrödinger maximal estimate in $\mathbb {R}^2$

We show that $\lim_{t \to 0} e^{it\Delta}f(x) = f(x)$ almost everywhere for all $f \in H^s (\mathbb{R}^2)$ provided that $s>1/3$. This result is sharp up to the endpoint. The proof uses polynomial partitioning and decoupling.

Pages 607-640 by
From volume 186-2

## Monochromatic sums and products in $\mathbb N$

An old question in Ramsey theory asks whether any finite coloring of the natural numbers admits a monochromatic pair $\{x+y,xy\}$. We answer this question affirmatively in a strong sense by exhibiting a large new class of nonlinear patterns that can be found in a single cell of any finite partition of $\mathbb N$. Our proof involves a correspondence principle that transfers the problem into the language of topological dynamics. As a corollary of our main theorem we obtain partition regularity for new types of equations, such as $x^2-y^2=z$ and $x^2+2y^2-3z^2=w$.

Pages 1069-1090 by
From volume 185-3

## Cubic curves and totally geodesic subvarieties of moduli space

In this paper we present the first example of a primitive, totally geodesic subvariety $F \subset \mathcal{M}_{g,n}$ with $\mathrm{dim}(F)>1$. The variety we consider is a surface $F \subset \mathcal{M}_{1,3}$ defined using the projective geometry of plane cubic curves. We also obtain a new series of Teichmüller curves in $\mathcal{M}_4$, and new $\mathrm{SL}_2(\mathbb{R})$-invariant varieties in the moduli spaces of quadratic differentials and holomorphic 1-forms.

Pages 957-990 by
From volume 185-3

## Higher ramification and the local Langlands correspondence

Let $F$ be a non-Archimedean locally compact field. We show that the local Langlands correspondence over $F$ has a property generalizing the higher ramification theorem of local class field theory. If $\pi$ is an irreducible cuspidal representation of a general linear group ${\rm GL}_n(F)$ and $\sigma$ the corresponding irreducible representation of the Weil group $\mathcal{W}_F$ of $F$, the restriction of $\sigma$ to a ramification subgroup of $\mathcal{W}_F$ is determined by a truncation of the simple character $\theta_\pi$ contained in $\pi$, and conversely. Numerical aspects of the relation are governed by an Herbrand-like function $\Psi_\varTheta$ depending on the endo-class $\varTheta$ of $\theta_\pi$. We give a method for calculating $\Psi_\varTheta$ directly from $\varTheta$. Consequently, the ramification-theoretic structure of $\sigma$ can be predicted from the simple character $\theta_\pi$ alone.

Pages 919-955 by
From volume 185-3

## On the growth of $L^2$-invariants for sequences of lattices in Lie groups

We study the asymptotic behaviour of Betti numbers, twisted torsion and other spectral invariants of sequences of locally symmetric spaces. Our main results are uniform versions of the DeGeorge–Wallach Theorem, of a theorem of Delorme and various other limit multiplicity theorems.

A basic idea is to adapt the notion of Benjamini–Schramm convergence (BS-convergence), originally introduced for sequences of finite graphs of bounded degree, to sequences of Riemannian manifolds, and analyze the possible limits. We show that BS-convergence of locally symmetric spaces $\Gamma\backslash G/K$ implies convergence, in an appropriate sense, of the normalized relative Plancherel measures associated to $L^2 (\Gamma\backslash G)$. This then yields convergence of normalized multiplicities of unitary representations, Betti numbers and other spectral invariants. On the other hand, when the corresponding Lie group $G$ is simple and of real rank at least two, we prove that there is only one possible BS-limit; i.e., when the volume tends to infinity, locally symmetric spaces always BS-converge to their universal cover $G/K$. This leads to various general uniform results.

When restricting to arbitrary sequences of congruence covers of a fixed arithmetic manifold we prove a strong quantitative version of BS-convergence, which in turn implies upper estimates on the rate of convergence of normalized Betti numbers in the spirit of Sarnak–Xue.

An important role in our approach is played by the notion of Invariant Random Subgroups. For higher rank simple Lie groups $G$, we exploit rigidity theory and, in particular, the Nevo–Stück–Zimmer theorem and Kazhdan`s property (T), to obtain a complete understanding of the space of IRS’s of $G$.

Pages 711-790 by
From volume 185-3

## On the stability threshold for the 3D Couette flow in Sobolev regularity

We study Sobolev regularity disturbances to the periodic, plane Couette flow in the 3D incompressible Navier-Stokes equations at high Reynolds number \textbfRe. Our goal is to estimate how the stability threshold scales in $\textbf{Re}$: the largest the initial perturbation can be while still resulting in a solution that does not transition away from Couette flow. In this work we prove that initial data that satisfies $\|u_{\rm in}\|_{H^\sigma} \leq \delta\textbf{Re}^{-3/2}$ for any $\sigma > 9/2$ and some $\delta = \delta(\sigma) > 0$ depending only on $\sigma$ is global in time, remains within $O(\textbf{Re}^{-1/2})$ of the Couette flow in $L^2$ for all time, and converges to the class of “2.5-dimensional” streamwise-independent solutions referred to as *streaks* for times $t \gtrsim \textbf{Re}^{1/3}$. Numerical experiments performed by Reddy et. al. with “rough” initial data estimated a threshold of $\sim \textbf{Re}^{-31/20}$, which shows very close agreement with our estimate.

Pages 541-608 by
From volume 185-2

## On large subsets of $\mathbb{F}_q^n$ with no three-term arithmetic progression

In this note, we show that the method of Croot, Lev, and Pach can be used to bound the size of a subset of $\mathbb{F}_q^n$ with no three terms in arithmetic progression by $c^n$ with $c < q$. For $q=3$, the problem of finding the largest subset of $\mathbb{F}_3^n$ with no three terms in arithmetic progression is called the *cap set problem*. Previously the best known upper bound for the affine cap problem, due to Bateman and Katz, was on order $n^{-1-\epsilon} 3^n$.

Pages 339-343 by
From volume 185-1

## Progression-free sets in $\mathbb Z_4^n$ are exponentially small

We show that for an integer $n\ge 1$, any subset $A\subseteq \mathbb{Z}_4^n$ free of three-term arithmetic progressions has size $|A|\le 4^{\gamma n}$, with an absolute constant $\gamma\approx 0.926$.

Pages 331-337 by
From volume 185-1

## Rectifiable-Reifenberg and the regularity of stationary and minimizing harmonic maps

In this paper we study the regularity of stationary and minimizing harmonic maps $f:B_2(p)\subseteq M\to N$ between Riemannian manifolds. If $S^k(f)\equiv\{x\in M: \text{ no tangent map at $x$ is }k+1\text{-symmetric}\}$ is the $k^{\rm th}$-stratum of the singular set of $f$, then it is well known that $\dim S^k\leq k$, however little else about the structure of $S^k(f)$ is understood in any generality. Our first result is for a general stationary harmonic map, where we prove that $S^k(f)$ is $k$-rectifiable. In fact, we prove for $k$-a.e. point $x\in S^k(f)$ that there exists a unique $k$-plane $V^k\subseteq T_xM$ such that *every* tangent map at $x$ is $k$-symmetric with respect to $V$.

In the case of minimizing harmonic maps we go further and prove that the singular set $S(f)$, which is well known to satisfy $\dim S(f)\leq n-3$, is in fact $n-3$-rectifiable with uniformly *finite* $n-3$-measure. An effective version of this allows us to prove that $|\nabla f|$ has estimates in $L^3_{\rm weak}$, an estimate that is sharp as $|\nabla f|$ may not live in $L^3$. More generally, we show that the regularity scale $r_f$ also has $L^3_{\rm weak}$ estimates.

The above results are in fact just applications of a new class of estimates we prove on the *quantitative* stratifications $S^k_{\epsilon,r}(f)$ and $S^k_{\epsilon}(f)\equiv S^k_{\epsilon,0}(f)$. Roughly, $S^k_{\epsilon}\subseteq M$ is the collection of points $x\in M$ for which no ball $B_r(x)$ is $\epsilon$-close to being $k+1$-symmetric. We show that $S^k_\epsilon$ is $k$-rectifiable and satisfies the Minkowski estimate $\mathrm{Vol}(B_r\,S_\epsilon^k)\leq C r^{n-k}$.

The proofs require a new $L^2$-subspace approximation theorem for stationary harmonic maps, as well as new $W^{1,p}$-Reifenberg and rectifiable-Reifenberg type theorems. These results are generalizations of the classical Reifenberg and give checkable criteria to determine when a set is $k$-rectifiable with uniform measure estimates. The new Reifenberg type theorems may be of some independent interest. The $L^2$-subspace approximation theorem we prove is then used to help break down the quantitative stratifications into pieces that satisfy these criteria.

Pages 131-227 by
From volume 185-1

## On the structure of ${\mathscr A}$-free measures and applications

We establish a general structure theorem for the singular part of ${\mathscr A}$-free Radon measures, where ${\mathscr A}$ is a linear PDE operator. By applying the theorem to suitably chosen differential operators ${\mathscr A}$, we obtain a simple proof of Alberti’s rank-one theorem and, for the first time, its extensions to functions of bounded deformation (BD). We also prove a structure theorem for the singular part of a finite family of normal currents. The latter result implies that the Rademacher theorem on the differentiability of Lipschitz functions can hold only for absolutely continuous measures and that every top-dimensional Ambrosio–Kirchheim metric current in $\mathbb R^d$ is a Federer–Fleming flat chain.

Pages 1017-1039 by
From volume 184-3

## Multiplicative functions in short intervals

We introduce a general result relating “short averages” of a multiplicative function to “long averages” which are well understood. This result has several consequences. First, for the Möbius function we show that there are cancellations in the sum of $\mu(n)$ in almost all intervals of the form $[x, x + \psi(x)]$ with $\psi(x) \rightarrow \infty$ arbitrarily slowly. This goes beyond what was previously known conditionally on the Density Hypothesis or the stronger Riemann Hypothesis. Second, we settle the long-standing conjecture on the existence of $x^{\varepsilon}$-smooth numbers in intervals of the form $[x, x + c(\varepsilon) \sqrt{x}]$, recovering unconditionally a conditional (on the Riemann Hypothesis) result of Soundararajan. Third, we show that the mean-value of $\lambda(n)\lambda(n+1)$, with $\lambda(n)$ Liouville’s function, is nontrivially bounded in absolute value by $1 – \delta$ for some $\delta > 0$. This settles an old folklore conjecture and constitutes progress towards Chowla’s conjecture. Fourth, we show that a (general) real-valued multiplicative function $f$ has a positive proportion of sign changes if and only if $f$ is negative on at least one integer and nonzero on a positive proportion of the integers. This improves on many previous works and is new already in the case of the Möbius function. We also obtain some additional results on smooth numbers in almost all intervals, and sign changes of multiplicative functions in all intervals of square-root length.\looseness=-1

Pages 1015-1056 by
From volume 183-3

## Characters of odd degree

We prove the McKay conjecture on characters of odd degree. A major step in the proof is the verification of the inductive McKay condition for groups of Lie type and primes $\ell$ such that a Sylow $\ell$-subgroup or its maximal normal abelian subgroup is contained in a maximally split torus by means of a new equivariant version of Harish-Chandra induction. Specifics of characters of odd degree, namely, that most of them lie in the principal Harish-Chandra series, then allow us to deduce from it the McKay conjecture for the prime $2$, hence for characters of odd degree.

Pages 869-908 by
From volume 184-3

## Anabelian geometry with étale homotopy types

Anabelian geometry with étale homotopy types generalizes in a natural way classical anabelian geometry with étale fundamental groups. We show that, both in the classical and the generalized sense, any point of a smooth variety over a field $k$ that is finitely generated over $\mathbb{Q}$ has a fundamental system of (affine) anabelian Zariski-neighborhoods. This was predicted by Grothendieck in his letter to Faltings.

Pages 817-868 by
From volume 184-3

## The clique density theorem

Turán’s theorem is a cornerstone of extremal graph theory. It asserts that for any integer $r \geqslant 2$, every graph on $n$ vertices with more than ${\tfrac{r-2}{2(r-1)}\cdot n^2}$ edges contains a clique of size $r$, i.e., $r$ mutually adjacent vertices. The corresponding extremal graphs are balanced $(r-1)$-partite graphs.

The question as to how many such $r$-cliques appear at least in any $n$-vertex graph with $\gamma n^2$ edges has been intensively studied in the literature. In particular, Lov\’asz and Simonovits conjectured in the 1970’s that asymptotically the best possible lower bound is given by the complete multipartite graph with $\gamma n^2$ edges in which all but one vertex class is of the same size while the remaining one may be smaller.

Their conjecture was recently resolved for $r=3$ by Razborov and for $r=4$ by Nikiforov. In this article, we prove the conjecture for all values of $r$.

Pages 683-707 by
From volume 184-3

## Extremal results for random discrete structures

We study thresholds for extremal properties of random discrete structures. We determine the threshold for Szemerédi’s theorem on arithmetic progressions in random subsets of the integers and its multidimensional extensions, and we determine the threshold for Turán-type problems for random graphs and hypergraphs. In particular, we verify a conjecture of Kohayakawa, Łuczak, and Rödl for Turán-type problems in random graphs. Similar results were obtained independently by Conlon and Gowers.

Pages 333-365 by
From volume 184-2

## Splitting mixed Hodge structures over affine invariant manifolds

We prove that affine invariant manifolds in strata of flat surfaces are algebraic varieties. The result is deduced from a generalization of a theorem of Möller. Namely, we prove that the image of a certain twisted Abel-Jacobi map lands in the torsion of a factor of the Jacobians. This statement can be viewed as a splitting of certain mixed Hodge structures.

Pages 681-713 by
From volume 183-2

## Global solutions of the Euler–Maxwell two-fluid system in 3D

The fundamental “two-fluid” model for describing plasma dynamics is given by the Euler–Maxwell system, in which compressible ion and electron fluids interact with their own self-consistent electromagnetic field. We prove global stability of a constant neutral background, in the sense that irrotational, smooth and localized perturbations of a constant background with small amplitude lead to global smooth solutions in three space dimensions for the Euler–Maxwell system. Our construction is robust in dimension 3 and applies equally well to other plasma models such as the Euler–Poisson system for two-fluids and a relativistic Euler–Maxwell system for two fluids. Our solutions appear to be the first nontrivial global smooth solutions in all of these models.

Pages 377-498 by
From volume 183-2

## On the nonexistence of elements of Kervaire invariant one

We show that the Kervaire invariant one elements $\theta_{j}\in\pi_{2^{j+1}-2}S^{0}$ exist only for $j\le 6$. By Browder’s Theorem, this means that smooth framed manifolds of Kervaire invariant one exist only in dimensions $2$, $6$, $14$, $30$, $62$, and possibly $126$. Except for dimension $126$ this resolves a longstanding problem in algebraic topology.

Pages 1-262 by
From volume 184-1

## Functoriality, Smith theory, and the Brauer homomorphism

If $\sigma$ is an automorphism of order $p$ of the semisimple group $\mathbf{G}$, there is a natural correspondence between \rm mod $p$ cohomological automorphic forms on $\mathbf{G}$ and $\mathbf{G}^{\sigma}$. We describe this correspondence in the global and local settings.

Pages 177-228 by
From volume 183-1

## Kontsevich’s graph complex, GRT, and the deformation complex of the sheaf of polyvector fields

We generalize Kontsevich’s construction of $L_{\infty}$-derivations of polyvector fields from the affine space to an arbitrary smooth algebraic variety. More precisely, we construct a map (in the homotopy category) from Kontsevich’s graph complex to the deformation complex of the sheaf of polyvector fields on a smooth algebraic variety. We show that the action of Deligne-Drinfeld elements of the Grothendieck-Teichmüller Lie algebra on the cohomology of the sheaf of polyvector fields coincides with the action of odd components of the Chern character. Using this result, we deduce that the $\hat{A}$-genus in the Calaque-Van den Bergh formula for the isomorphism between harmonic and Hochschild structures can be replaced by a generalized $\hat{A}$-genus.

Pages 855-943 by
From volume 182-3

## A proof of Demailly’s strong openness conjecture

In this article, we solve the strong openness conjecture on the multiplier ideal sheaf associated to any plurisubharmonic function, which was posed by Demailly.

Pages 605-616 by
From volume 182-2

## Rationality of $W$-algebras: principal nilpotent cases

We prove the rationality of all the minimal series principal $W$-algebras discovered by Frenkel, Kac and Wakimoto, thereby giving a new family of rational and $C_2$-cofinite vertex operator algebras. A key ingredient in our proof is the study of Zhu’s algebra of simple $W$-algebras via the quantized Drinfeld-Sokolov reduction. We show that the functor of taking Zhu’s algebra commutes with the reduction functor. Using this general fact we determine the maximal spectrums of the associated graded of Zhu’s algebras of vertex operator algebras associated with admissible representations of affine Kac-Moody algebras as well.

Pages 565-604 by
From volume 182-2

## The good pants homology and the Ehrenpreis Conjecture

We develop the notion of the good pants homology and show that it agrees with the standard homology on closed surfaces. (Good pants are pairs of pants whose cuffs have the length nearly equal to some large number $R>0$.) Combined with our previous work on the Surface Subgroup Theorem, this yields a proof of the Ehrenpreis Conjecture.

Pages 1-72 by
From volume 182-1

## Periodic approximations of irrational pseudo-rotations using pseudoholomorphic curves

We prove that every $C^\infty$-smooth, area preserving diffeomorphism of the closed $2$-disk having not more than one periodic point is the uniform limit of periodic $C^\infty$-smooth diffeomorphisms. In particular, every smooth irrational pseudo-rotation can be $C^0$-approximated by integrable systems. This partially answers a long standing question of A. Katok regarding zero entropy Hamiltonian systems in low dimensions. Our approach uses pseudoholomorphic curve techniques from symplectic geometry.

Pages 1033-1086 by
From volume 181-3

## Kähler–Einstein metrics with edge singularities

This article considers the existence and regularity of Kähler–Einstein metrics on a compact Kähler manifold $M$ with edge singularities with cone angle $2\pi \beta$ along a smooth divisor $D$. We prove existence of such metrics with negative, zero and some positive cases for all cone angles $2\pi \beta \leq 2\pi$. The results in the positive case parallel those in the smooth case. We also establish that solutions of this problem are polyhomogeneous, i.e., have a complete asymptotic expansion with smooth coefficients along $D$ for all $2\pi \beta < 2\pi$.

Pages 95-176 by
From volume 183-1

## Properly embedded minimal planar domains

In 1997, Collin proved that any properly embedded minimal surface in $\mathbb{R}^3$ with finite topology and more than one end has finite total Gaussian curvature. Hence, by an earlier result of López and Ros, catenoids are the only nonplanar, nonsimply connected, properly embedded, minimal planar domains in $\mathbb{R}^3$ of finite topology. In 2005, Meeks and Rosenberg proved that the only simply connected, properly embedded minimal surfaces in $\mathbb{R}^3$ are planes and helicoids. Around 1860, Riemann defined a one-parameter family of periodic, infinite topology, properly embedded, minimal planar domains $\mathcal{R}_t$ in $\mathbb{R}^3$, $t \in (0,\infty)$. These surfaces are called the *Riemann minimal examples*, and the family $\{ \mathcal{R}_t\} _t$ has natural limits being a vertical catenoid as $t\to~0$ and a vertical helicoid as $t\to\infty$. In this paper we complete the classification of properly embedded, minimal planar domains in $\mathbb{R}^3$ by proving that the only connected examples with infinite topology are the Riemann minimal examples. We also prove that the limit ends of Riemann minimal examples are model surfaces for the limit ends of properly embedded minimal surfaces $M\subset\mathbb{R}^3$ of finite genus and infinite topology, in the sense that such an $M$ has two limit ends, each of which has a representative that is naturally asymptotic to a limit end representative of a Riemann minimal example with the same associated flux vector.

Pages 473-546 by
From volume 181-2

## Random walks in Euclidean space

Fix a probability measure on the space of isometries of Euclidean space $\mathbf{R}^d$. Let $Y_0=0,Y_1,Y_2,\ldots\in\mathbf{R}^d$ be a sequence of random points such that $Y_{l+1}$ is the image of $Y_l$ under a random isometry of the previously fixed probability law, which is independent of $Y_l$. We prove a Local Limit Theorem for $Y_l$ under necessary nondegeneracy conditions. Moreover, under more restrictive but still general conditions we give a quantitative estimate which describes the behavior of the law of $Y_l$ on scales $e^{-cl^{1/4}}

Pages 243-301 by
From volume 181-1

## The space of embedded minimal surfaces of fixed genus in a 3-manifold V; Fixed genus

This paper is the fifth and final in a series on embedded minimal surfaces. Following our earlier papers on disks, we prove here two main structure theorems for \itnonsimply connected embedded minimal surfaces of any given fixed genus.

The first of these asserts that any such surface *without* small necks can be obtained by gluing together two oppositely-oriented double spiral staircases.

The second gives a pair of pants decomposition of any such surface when there *are* small necks, cutting the surface along a collection of short curves. After the cutting, we are left with graphical pieces that are defined over a disk with either one or two sub-disks removed (a topological disk with two sub-disks removed is called a pair of pants).

Both of these structures occur as different extremes in the two-parameter family of minimal surfaces known as the Riemann examples.

Pages 1-153 by
From volume 181-1

## The Hodge theory of Soergel bimodules

We prove Soergel’s conjecture on the characters of indecomposable

Soergel bimodules. We deduce that Kazhdan-Lusztig polynomials have positive coefficients for arbitrary Coxeter systems. Using results of Soergel one may deduce an algebraic proof of the Kazhdan-Lusztig conjecture.

** Note: To view the article, click on the URL link for the DOI number. **

Pages 1089-1136 by
From volume 180-3

## Fourier transform and the global Gan–Gross–Prasad conjecture for unitary groups

By the relative trace formula approach of Jacquet–Rallis, we prove the global Gan–Gross–Prasad conjecture for unitary groups under some local restrictions for the automorphic representations.

** Note: To view the article, click on the URL link for the DOI number. **

Pages 971-1049 by
From volume 180-3

## The geometry of the moduli space of odd spin curves

The spin moduli space $\overline{\mathcal{S}}_g$ is the parameter space of theta characteristics (spin structures) on stable curves of genus $g$. It has two connected components, $\overline{\mathcal{S}}_g^-$ and $\overline{\mathcal{S}}_g^+$, depending on the parity of the spin structure. We establish a complete birational classification by Kodaira dimension of the odd component $\overline{\mathcal{S}}_g^-$ of the spin moduli space. We show that $\overline{\mathcal{S}}_g^-$ is uniruled for $g<12$ and even unirational for $g\leq 8$. In this range, introducing the concept of cluster for the Mukai variety whose one-dimensional linear sections are general canonical curves of genus $g$, we construct new birational models of $\overline{\mathcal{S}}_g^-$. These we then use to explicitly describe the birational structure of $\overline{\mathcal{S}}_g^-$. For instance, $\overline{\mathcal{S}}_8^-$ is birational to a locally trivial $\textbf{P}^7$-bundle over the moduli space of elliptic curves with seven pairs of marked points. For $g\geq 12$, we prove that $\overline{\mathcal{S}}_g^-$ is a variety of general type. In genus $12$, this requires the construction of a counterexample to the Slope Conjecture on effective divisors on the moduli space of stable curves of genus $12$.

** Note: To view the article, click on the URL link for the DOI number. **

Pages 927-970 by
From volume 180-3

## On self-similar sets with overlaps and inverse theorems for entropy

We study the dimension of self-similar sets and measures on the line. We show that if the dimension is less than the generic bound of $\mathrm{min}\{1,s\}$, where $s$ is the similarity dimension, then there are superexponentially close cylinders at all small enough scales. This is a step towards the conjecture that such a dimension drop implies exact overlaps and confirms it when the generating similarities have algebraic coefficients. As applications we prove Furstenberg’s conjecture on projections of the one-dimensional Sierpinski gasket and achieve some progress on the Bernoulli convolutions problem and, more generally, on problems about parametric families of self-similar measures. The key tool is an inverse theorem on the structure of pairs of probability measures whose mean entropy at scale $2^{-n}$ has only a small amount of growth under convolution.

** Note: To view the article, click on the URL link for the DOI number. **

Pages 773-822 by
From volume 180-2

## ACC for log canonical thresholds

We show that log canonical thresholds satisfy the \rm ACC.

** Note: To view the article, click on the URL link for the DOI number. **

Pages 523-571 by
From volume 180-2

## Stable logarithmic maps to Deligne–Faltings pairs I

We introduce a new compactification of the space of relative stable maps. This approach uses logarithmic geoemetry in the sense of Kato-Fontaine-Illusie without taking expansions of the target. The underlying structures of the stable logarithmic maps are stable in the usual sense.

** Note: To view the article, click on the URL link for the DOI number. **

Pages 455-521 by
From volume 180-2

## Bounded gaps between primes

It is proved that $$ \liminf_{n\to\infty}(p_{n+1}-p_n)<7\times 10^7, $$ where $p_n$ is the $n$-th prime.

Our method is a refinement of the recent work of Goldston, Pintz and Yıldırım on the small gaps between consecutive primes. A major ingredient of the proof is a stronger version of the Bombieri-Vinogradov theorem that is applicable when the moduli are free from large prime divisors only, but it is adequate for our purpose.

** Note: To view the article, click on the URL link for the DOI number. **

Pages 1121-1174 by
From volume 179-3

## Kodaira dimension and zeros of holomorphic one-forms

We show that every holomorphic one-form on a smooth complex projective variety of general type must vanish at some point. The proof uses generic vanishing theory for Hodge modules on abelian varieties.

** Note: To view the article, click on the URL link for the DOI number. **

Pages 1109-1120 by
From volume 179-3

## A general regularity theory for stable codimension 1 integral varifolds

We give a necessary and sufficient geometric structural condition, which we call the $\alpha$-Structural Hypothesis, for a stable codimension 1 integral varifold on a smooth Riemannian manifold to correspond to an embedded smooth hypersurface away from a small set of generally unavoidable singularities. The $\alpha$-Structural Hypothesis says that no point of the support of the varifold has a neighborhood in which the support is the union of three or more embedded $C^{1, \alpha}$ hypersurfaces-with-boundary meeting (only) along their common boundary. We establish that whenever a stable integral $n$-varifold on a smooth $(n+1)$-dimensional Riemannian manifold satisfies the $\alpha$-Structural Hypothesis for some $\alpha \in (0, 1/2)$, its singular set is empty if $n \leq 6$, discrete if $n =7$ and has Hausdorff dimension $\leq n-7$ if $n \geq 8$; in view of well-known examples, this is the best possible general dimension estimate on the singular set of a varifold satisfying our hypotheses. We also establish compactness of mass-bounded subsets of the class of stable codimension 1 integral varifolds satisfying the $\alpha$-Structural Hypothesis for some $\alpha \in (0, 1/2)$. The $\alpha$-Structural Hypothesis on an $n$-varifold for any $\alpha \in (0, 1/2)$ is readily implied by either of the following two hypotheses: (i) the varifold corresponds to an absolutely area minimizing rectifiable current with no boundary, (ii) the singular set of the varifold has vanishing $(n-1)$-dimensional Hausdorff measure. Thus, our theory subsumes the well-known regularity theory for codimension 1 area minimizing rectifiable currents and settles the long standing question as to which weakest size hypothesis on the singular set of a stable minimal hypersurface guarantees the validity of the above regularity conclusions.

An optimal strong maximum principle for stationary codimension 1 integral varifolds follows from our regularity and compactness theorems.

** Note: To view the article, click on the URL link for the DOI number. **

Pages 843-1007 by
From volume 179-3

## Calabi flow, geodesic rays, and uniqueness of constant scalar curvature Kähler metrics

We prove that constant scalar curvature Kähler metric “adjacent” to a fixed Kähler class is unique up to isomorphism. The proof is based on the study of a fourth order evolution equation, namely, the *Calabi flow*, from a new geometric perspective, and on the geometry of the space of Kähler metrics.

** Note: To view the article, click on the URL link for the DOI number. **

Pages 407-454 by
From volume 180-2

## The Oort Conjecture on lifting covers of curves

We show that the conjecture of Oort on lifting covers of curves is true. The main ingredients in the proof are a deformation argument in characteristic $p$ and (a special case of) a very recent result by Obus–Wewers. A kind of boundedness result is given as well.

** Note: To view the article, click on the URL link for the DOI number. **

Pages 285-322 by
From volume 180-1

## Limit theorems for translation flows

The aim of this paper is to obtain an asymptotic expansion for ergodic integrals of translation flows on flat surfaces of higher genus (Theorem 1) and to give a limit theorem for these flows (Theorem 2).

** Note: To view the article, click on the URL link for the DOI number. **

Pages 431-499 by
From volume 179-2

## Log minimal model program for the moduli space of stable curves: the first flip

We give a geometric invariant theory (GIT) construction of the log canonical model $\bar M_g(\alpha)$ of the pairs $(\bar M_g, \alpha \delta)$ for $\alpha \in (7/10 – \epsilon, 7/10]$ for small $\epsilon \in \mathbb Q_+$. We show that $\bar M_g(7/10)$ is isomorphic to the GIT quotient of the Chow variety of bicanonical curves; $\bar M_g(7/10-\epsilon)$ is isomorphic to the GIT quotient of the asymptotically-linearized Hilbert scheme of bicanonical curves. In each case, we completely classify the (semi)stable curves and their orbit closures. Chow semistable curves have ordinary cusps and tacnodes as singularities but do not admit elliptic tails. Hilbert semistable curves satisfy further conditions; e.g., they do not contain elliptic chains. We show that there is a small contraction $\Psi: \bar M_g(7/10+\epsilon) \to \bar M_g(7/10)$ that contracts the locus of elliptic bridges. Moreover, by using the GIT interpretation of the log canonical models, we construct a small contraction $\Psi^+ : \bar M_g(7/10-\epsilon) \to \bar M_g(7/10)$ that is the Mori flip of $\Psi$.

Pages 911-968 by
From volume 177-3

## Diophantine geometry over groups VIII: Stability

This paper is the eighth in a sequence on the structure of sets of solutions to systems of equations in free and hyperbolic groups, projections of such sets (Diophantine sets), and the structure of definable sets over free and hyperbolic groups. In this eighth paper we use a modification of the sieve procedure, which was used in proving quantifier elimination in the theory of a free group, to prove that free and torsion-free (Gromov) hyperbolic groups are stable.

Pages 787-868 by
From volume 177-3

## Small eigenvalues of the Laplacian for algebraic measures in moduli space, and mixing properties of the Teichmüller flow

We consider the $\mathrm{SL}(2, \mathbb{R})$ action on moduli spaces of quadratic differentials. If $\mu$ is an $\mathrm{SL}(2, \mathbb{R})$-invariant probability measure, crucial information about the associated representation on $L^2(\mu)$ (and, in particular, fine asymptotics for decay of correlations of the diagonal action, the Teichmüller flow) is encoded in the part of the spectrum of the corresponding foliated hyperbolic Laplacian that lies in $\!(0,1/4)\!$ (which controls the contribution of the complementary series). Here we prove that the essential spectrum of an invariant algebraic measure is contained in $[1/4,\infty)$; i.e., for every $\delta\!>\!0$, there are only finitely many eigenvalues (counted with multiplicity) in $(0,1/4\!-\!\delta)$. In particular, all algebraic invariant measures have a spectral gap.

Pages 385-442 by
From volume 178-2

## The universal relation between scaling exponents in first-passage percolation

It has been conjectured in numerous physics papers that in ordinary first-passage percolation on integer lattices, the fluctuation exponent $\chi$ and the wandering exponent $\xi$ are related through the universal relation $\chi=2\xi -1$, irrespective of the dimension. This is sometimes called the KPZ relation between the two exponents. This article gives a rigorous proof of this conjecture assuming that the exponents exist in a certain sense.

Pages 663-697 by
From volume 177-2

## A further improvement of the Quantitative Subspace Theorem

In 2002, Evertse and Schlickewei obtained a quantitative version of the so-called Absolute Parametric Subspace Theorem. This result deals with a parametrized class of twisted heights. One of the consequences of this result is a quantitative version of the Absolute Subspace Theorem, giving an explicit upper bound for the number of subspaces containing the solutions of the Diophantine inequality under consideration.

In the present paper, we further improve Evertse’s and Schlickewei’s quantitative version of the Absolute Parametric Subspace Theorem and deduce an improved quantitative version of the Absolute Subspace Theorem. We combine ideas from the proof of Evertse and Schlickewei (which is basically a substantial refinement of Schmidt’s proof of his Subspace Theorem from 1972), with ideas from Faltings’ and Wüstholz’ proof of the Subspace Theorem. A new feature is an “interval result,” which gives more precise information on the distribution of the heights of the solutions of the system of inequalities considered in the Subspace Theorem.

Pages 513-590 by
From volume 177-2

## A class of superrigid group von Neumann algebras

We prove that for any group $G$ in a fairly large class of generalized wreath product groups, the associated von Neumann algebra $\mathrm{L} G$ completely “remembers” the group $G$. More precisely, if $\mathrm{L} G$ is isomorphic to the von Neumann algebra $\mathrm{L} \Lambda$ of an arbitrary countable group $\Lambda$, then $\Lambda$ must be isomorphic to $G$. This represents the first superrigidity result pertaining to group von Neumann algebras.

Pages 231-286 by
From volume 178-1

## Descente par éclatements en $K$-théorie invariante par homotopie

Ces notes donnent une preuve de la représentabilité de la $K$-théorie invariante par homotopie dans la catégorie homotopique stable des schémas (résultat annoncé par Voevodsky). On en déduit, grâce au théorème de changement de base propre en théorie de l’homotopie stable des schémas, un théorème de descente par éclatements en $K$-théorie invariante par homotopie.

These notes give a proof of the representability of homotopy invariant $K$-theory in the stable homotopy category of schemes (which was announced by Voevodsky). One deduces from the proper base change theorem in stable homotopy theory of schemes a descent by blow-ups theorem for homotopy invariant $K$-theory.

Pages 425-448 by
From volume 177-2

## Higher finiteness properties of reductive arithmetic groups in positive characteristic: The Rank Theorem

We show that the finiteness length of an $S$-arithmetic subgroup $\Gamma$ in a noncommutative isotropic absolutely almost simple group $\mathcal{G}$ over a global function field is one less than the sum of the local ranks of $\mathcal{G}$ taken over the places in $S$. This determines the finiteness properties for $S$-arithmetic subgroups in isotropic reductive groups, confirming the conjectured finiteness properties for this class of groups.

Our main tool is Behr–Harder reduction theory which we recast in terms of the metric structure of euclidean buildings.

Pages 311-366 by
From volume 177-1

## Norm convergence of nilpotent ergodic averages

We show that multiple polynomial ergodic averages arising from nilpotent groups of measure preserving transformations of a probability space always converge in the $L^2$ norm.

Pages 1667-1688 by
From volume 175-3

## The second fundamental theorem of invariant theory for the orthogonal group

Let $V=\mathbb{C}^n$ be endowed with an orthogonal form and $G=\mathrm{O}(V)$ be the corresponding orthogonal group. Brauer showed in 1937 that there is a surjective homomorphism $\nu:B_r(n)\to\mathrm{End}_G(V^{\otimes r})$, where $B_r(n)$ is the $r$-string Brauer algebra with parameter $n$. However the kernel of $\nu$ has remained elusive. In this paper we show that, in analogy with the case of $\mathrm{GL}(V)$, for $r\geq n+1$, $\nu$ has a kernel which is generated by a single idempotent element $E$, and we give a simple explicit formula for $E$. Using the theory of cellular algebras, we show how $E$ may be used to determine the multiplicities of the irreducible representations of $\mathrm{O}(V)$ in $V^{\otimes r}$. We also show how our results extend to the case where $\mathbb{C}$ is replaced by an appropriate field of positive characteristic, and we comment on quantum analogues of our results.

Pages 2031-2054 by
From volume 176-3

## The sharp weighted bound for general Calderón–Zygmund operators

For a general Calderón–Zygmund operator $T$ on $\Bbb{R}^N$, it is shown that $$ \Vert{Tf}\Vert{L^2(w)}\leq C(T)\cdot\sup_Q\Big(∫_Q w\cdot ∫_Q w^{-1}\Big)\cdot\Vert{f}\Vert{L^2(w)}$$ for all Muckenhoupt weights $w\in A_2$. This optimal estimate was known as the $A_2$ conjecture. A recent result of Pérez–Treil–Volberg reduced the problem to a testing condition on indicator functions, which is verified in this paper.

The proof consists of the following elements: (i) a variant of the Nazarov–Treil–Volberg method of random dyadic systems with just one random system and completely without “bad” parts; (ii) a resulting representation of a general Calderón–Zygmund operator as an average of “dyadic shifts;” and (iii) improvements of the Lacey–Petermichl–Reguera estimates for these dyadic shifts, which allow summing up the series in the obtained representation.

Pages 1473-1506 by
From volume 175-3

## Operator monotone functions and Löwner functions of several variables

We prove generalizations of Löwner’s results on matrix monotone functions to several variables. We give a characterization of when a function of $d$ variables is locally monotone on $d$-tuples of commuting self-adjoint $n$-by-$n$ matrices. We prove a generalization to several variables of Nevanlinna’s theorem describing analytic functions that map the upper half-plane to itself and satisfy a growth condition. We use this to characterize all rational functions of two variables that are operator monotone.

Pages 1783-1826 by
From volume 176-3

## Linearization of generalized interval exchange maps

A standard interval exchange map is a one-to-one map of the interval that is locally a translation except at finitely many singularities. We define for such maps, in terms of the Rauzy-Veech continuous fraction algorithm, a diophantine arithmetical condition called restricted Roth type, which is almost surely satisfied in parameter space. Let $T_0$ be a standard interval exchange map of restricted Roth type, and let $r$ be an integer $\geq 2$. We prove that, amongst $C^{r+3}$ deformations of $T_0$ that are $C^{r+3}$ tangent to $T_0$ at the singularities, those that are conjugated to $T_0$ by a $C^r$-diffeomorphism close to the identity form a $C^1$-submanifold of codimension $(g-1)(2r+1) +s$. Here, $g$ is the genus and $s$ is the number of marked points of the translation surface obtained by suspension of $T_0$. Both $g$ and $s$ can be computed from the combinatorics of $T_0$.

Pages 1583-1646 by
From volume 176-3

## Immersing almost geodesic surfaces in a closed hyperbolic three manifold

Let $\mathbf{M}^3$ be a closed hyperbolic three manifold. We construct closed surfaces that map by immersions into $\mathbf{M}^3$ so that for each, one the corresponding mapping on the universal covering spaces is an embedding, or, in other words, the corresponding induced mapping on fundamental groups is an injection.

Pages 1127-1190 by
From volume 175-3

## Linear Shafarevich conjecture

In this paper we settle affirmatively Shafarevich’s uniformization conjecture for varieties with linear fundamental groups. We prove the strongest to date uniformization result — the universal covering space of a complex projective manifold with a linear fundamental group is holomorphically convex. The proof is based on both known and newly developed techniques in non-abelian Hodge theory.

Pages 1545-1581 by
From volume 176-3