## 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

## Solution of the Littlewood-Offord problem in high dimensions

Consider the $2^n$ partial sums of arbitrary $n$ vectors of length at least one in $d$-dimensional Euclidean space. It is shown that as $n$ goes to infinity no closed ball of diameter $\Delta$ contains more than $(|\Delta| +1+o(1)) \binom{n}{|n/2|}$ out of these sums and this is best possible. For $\Delta – |\Delta|$ small an exact formula is given.

Pages 259-270 by
From volume 128-2

## On the Castelnuovo-Mumford regularity of rings of polynomial invariants

We show that when a group acts on a polynomial ring over a field the ring of invariants has Castelnuovo-Mumford regularity at most zero. As a consequence, we prove a well-known conjecture that the invariants are always generated in degrees at most $n(|G|-1)$, where $n >1$ is the number of polynomial generators and $|G|>1$ is the order of the group. We also prove some other related conjectures in invariant theory.

Pages 499-517 by
From volume 174-1

## Inverse Littlewood–Offord theorems and the condition number of random discrete matrices

Consider a random sum $\eta_1 v_1 + …+ \eta_n v_n$, where $\eta_1, \dots,\eta_n$ are independently and identically distributed (i.i.d.) random signs and $v_1, \dots,v_n$ are integers. The Littlewood-Offord problem asks to maximize concentration probabilities such as $\Bbb{P}( \eta_1 v_1 + …+ \eta_n v_n = 0)$ subject to various hypotheses on $v_1, \dots,v_n$. In this paper we develop an *inverse* Littlewood-Offord theory (somewhat in the spirit of Freiman’s inverse theory in additive combinatorics), which starts with the hypothesis that a concentration probability is large, and concludes that almost all of the $v_1, \dots,v_n$ are efficiently contained in a generalized arithmetic progression. As an application we give a new bound on the magnitude of the least singular value of a random Bernoulli matrix, which in turn provides upper tail estimates on the condition number.

Pages 595-632 by
From volume 169-2

## The stable moduli space of Riemann surfaces: Mumford’s conjecture

D. Mumford conjectured in [33] that the rational cohomology of the stable moduli space of Riemann surfaces is a polynomial algebra generated by certain classes $\kappa_i$ of dimension $2i$. For the purpose of calculating rational cohomology, one may replace the stable moduli space of Riemann surfaces by $B\Gamma_{\infty}$, where $\Gamma_\infty$ is the group of isotopy classes of automorphisms of a smooth oriented connected surface of “large” genus. Tillmann’s theorem [44] that the plus construction makes $B\Gamma_{\infty}$ into an infinite loop space led to a stable homotopy version of Mumford’s conjecture, stronger than the original [24]. We prove the stronger version, relying on Harer’s stability theorem [17], Vassiliev’s theorem concerning spaces of functions with moderate singularities [46], [45] and methods from homotopy theory.

Pages 843-941 by
From volume 165-3

## Stable minimal hypersurfaces in $\mathbb {R}^{N+1+\ell }$ with singular set an arbitrary closed $K\subset \{0\}\times \mathbb {R}^{\ell }$

With respect to a $C^{\infty }$ metric which is close to the standard Euclidean metric on $\mathbb {R}^{N+1+\ell }$, where $N\ge 7$ and $\ell \ge 1$ are given, we construct a class of embedded $(N+\ell )$-dimensional hypersurfaces (without boundary) which are minimal and strictly stable, and which have singular set equal to an arbitrary preassigned closed subset $K\subset \{0\}\times \mathbb {R}^{\ell }$. Thus the question is settled, with a strong affirmative, as to whether there can be “gaps” or even fractional dimensional parts in the singular set. Such questions, for both stable and unstable minimal submanifolds, remain open in all dimensions in the case of real analytic metrics and, in particular, for the standard Euclidean metric. \par The construction used here involves the analysis of solutions $u$ of the symmetric minimal surface equation on domains $\Omega \subset \mathbb {R}^{n}$ whose symmetric graphs (i.e., $\{(x,\xi )\in \Omega \times \mathbb {R}^{m}: |\xi |=u(x)\}$) lie on one side of a cylindrical minimal cone including, in particular, a Liouville type theorem for complete solutions (i.e., the case $\Omega =\mathbb {R}^{n}$).

Pages 1205-1234 by
From volume 197-3

## Potential automorphy over CM fields

Let $F$ be a CM number field. We prove modularity lifting theorems for regular $n$-dimensional Galois representations over $F$ without any self-duality condition. We deduce that all elliptic curves $E$ over $F$ are potentially modular, and furthermore satisfy the Sato–Tate conjecture. As an application of a different sort, we also prove the Ramanujan Conjecture for weight zero cuspidal automorphic representations for $\mathrm {GL}_2(\mathbf {A}_F)$.

Pages 897-1113 by
From volume 197-3

## Higher uniformity of bounded multiplicative functions in short intervals on average

Let $\lambda$ denote the Liouville function. We show that, as $X \rightarrow \infty$,

$$

\int^{2X}_X \sup_{\begin{smallmatrix} P(Y)\in\mathbb{R}[Y] \ \mathrm{deg}{P} \leq k\end{smallmatrix}}

\left| \sum_{x\leq n \leq x+H}

\lambda(n) e(-P(n))\right| dx=o (XH)

$$

for all fixed $k$ and $X^{\theta} \leq H \leq X$ with $0 < \theta < 1$ fixed but arbitrarily small. Previously this was only established for $k \leq 1$. We obtain this result as a special case of the corresponding statement for (non-pretentious) $1$-bounded multiplicative functions that we prove.

In fact, we are able to replace the polynomial phases $e(-P(n))$ by degree $k$ nilsequences $\overline{F}(g(n) \Gamma)$. By the inverse theory for the Gowers norms this implies the higher order asymptotic uniformity result

$$

\int_{X}^{2X} \| \lambda \|_{U^{k+1}([x,x+H])} dx = o ( X )

$$

in the same range of $H$.

We present applications of this result to patterns of various types in the Liouville sequence. Firstly, we show that the number of sign patterns of the Liouville function is superpolynomial, making progress on a conjecture of Sarnak about the Liouville sequence having positive entropy. Secondly, we obtain cancellation in averages of $\lambda$ over short polynomial progressions $(n+P_1(m),\ldots, n+P_k(m))$, which in the case of linear polynomials yields a new averaged version of Chowla’s conjecture.

We are in fact able to prove our results on polynomial phases in the wider range $H\geq \exp((\log X)^{5/8+\varepsilon})$, thus strengthening also previous work on the Fourier uniformity of the Liouville function.

Pages 739-857 by
From volume 197-2

## Special subvarieties of non-arithmetic ball quotients and Hodge theory

Let $\Gamma \subset \mathrm {PU}(1,n)$ be a lattice and $S_\Gamma $ be the associated ball quotient. We prove that, if $S_\Gamma $ contains infinitely many maximal complex totally geodesic subvarieties, then $\Gamma $ is arithmetic. We also prove an Ax–Schanuel Conjecture for $S_\Gamma $, similar to the one recently proven by Mok, Pila and Tsimerman. One of the main ingredients in the proofs is to realise $S_\Gamma $ inside a period domain for polarised integral variations of Hodge structure and interpret totally geodesic subvarieties as unlikely intersections.

Pages 159-220 by
From volume 197-1

## Rough solutions of the $3$-D compressible Euler equations

We prove the local-in-time well-posedness for the solution of the compressible Euler equations in $3$-D for the Cauchy data of the velocity, density and vorticity $(v,\varrho, \mathfrak{w}) \in H^s\times H^s\times H^{s’}$, $2<s'<s$. The result extends the sharp results of Smith-Tataru and of Wang, established in the irrotational case, i.e., $ \mathfrak{w}=0$, which is known to be optimal for $s>2$. At the opposite extreme, in the incompressible case, i.e., with a constant density, the result is known to hold for

$\mathfrak{w}\in H^s$, $s>3/2$ and fails for $s\le 3/2$. We therefore, conjecture that the optimal result should be $(v,\varrho, \mathfrak{w}) \in H^s\times H^s\times H^{s’}$, $s>2, \, s’>\frac{3}{2}$. We view our work here as an important step in proving the conjecture. The main difficulty in establishing sharp well-posedness results for general compressible Euler flow is due to the highly nontrivial interaction between the sound waves, governed by quasilinear wave equations, and vorticity which is transported by the flow. To overcome this difficulty, we separate the dispersive part of sound wave from the transported part, and gain regularity significantly by exploiting the nonlinear structure of the system and the geometric structures of the acoustical spacetime.

Pages 509-654 by
From volume 195-2

## On property (T) for $\mathrm {Aut}(F_n)$ and $\mathrm {SL}_n(\mathbb {Z})$

We prove that $\mathrm {Aut}(F_n)$ has Kazhdan’s property (T) for every $n \geqslant 6$. Together with a previous result of Kaluba, Nowak, and Ozawa, this gives the same statement for $n\geqslant 5$. We also provide explicit lower bounds for the Kazhdan constants of $\mathrm {SAut}(F_n)$ (with $n \geqslant 6$) and of $\mathrm {SL}_n(\mathbb {Z})$ (with $n \geqslant 3$) with respect to natural generating sets. In the latter case, these bounds improve upon previously known lower bounds whenever $n > 6$.

Pages 539-562 by
From volume 193-2

## Thresholds versus fractional expectation-thresholds

Proving a conjecture of Talagrand, a fractional version of the “expectation-threshold” conjecture of Kalai and the second author, we show that

$p_c \mathcal(F) = O(q_f (\mathcal{F}) \mathrm{log}\ell (\mathcal{F})$ for any increasing family $\mathcal{F}$ on a finite set $X$, where $p_c(\mathcal{F})$ and $q_f(\mathcal{F})$ are the threshold and “fractional expectation-threshold” of $\mathcal(F)$, and $\ell (\mathcal{F})$ is the maximum size of a minimal member of $\mathcal{F}$. This easily implies several heretofore difficult results and conjectures in probabilistic combinatorics, including thresholds for perfect hypergraph matchings (Johansson–Kahn–Vu), bounded degree spanning trees (Montgomery), and bounded degree graphs (new). We also resolve (and vastly extend) the “axial” version of the random multi-dimensional assignment problem (earlier considered by Martin–Mézard–Rivoire and Frieze–Sorkin). Our approach builds on a recent breakthrough of Alweiss, Lovett, Wu and Zhang on the Erdős–Rado “Sunflower Conjecture.”

Pages 475-495 by
From volume 194-2

## Local and global boundary rigidity and the geodesic X-ray transform in the normal gauge

In this paper we analyze the local and global boundary rigidity problem for general Riemannian manifolds with boundary $(M,g)$. We show that the boundary distance function, i.e., $d_g|_{\partial M\times \partial M}$, known near a point $p\in \partial M$ at which $\partial M$ is strictly convex, determines $g$ in a suitable neighborhood of $p$ in $M$, up to the natural diffeomorphism invariance of the problem.

We also consider the closely related lens rigidity problem which is a more natural formulation if the boundary distance is not realized by unique minimizing geodesics. The lens relation measures the point and the direction of exit from $M$ of geodesics issued from the boundary and the length of the geodesic. The lens rigidity problem is whether we can determine the metric up to isometry from the lens relation. We solve the lens rigidity problem under the assumption that there is a function on $M$ with suitable convexity properties relative to $g$. This can be considered as a complete solution of a problem formulated first by Herglotz in 1905. We also prove a semi-global results given semi-global data. This shows, for instance, that simply connected manifolds with strictly convex boundaries are lens rigid if the sectional curvature is non-positive or non-negative or if there are no focal points.

The key tool is the analysis of the geodesic X-ray transform on 2-tensors, corresponding to a metric $g$, in the normal gauge, such as normal coordinates relative to a hypersurface, where one also needs to allow weights. This is handled by refining and extending our earlier results in the solenoidal gauge.

Pages 1-95 by
From volume 194-1

## Flat Littlewood polynomials exist

We show that there exist absolute constants $\Delta > \delta > 0$ such that, for all $n \geqslant 2$, there exists a polynomial $P$ of degree\nonbreakingspace $n$, with coefficients in $\{-1,1\}$, such that \[ \delta \sqrt {n} \leqslant |P(z)| \leqslant \Delta \sqrt {n} \] for all $z\in \mathbb {C}$ with $|z|=1$. This confirms a conjecture of Littlewood from\nonbreakingspace 1966.

Pages 977-1004 by
From volume 192-3

## Lorentzian polynomials

We study the class of Lorentzian polynomials. The class contains homogeneous stable polynomials as well as volume polynomials of convex bodies and projective varieties. We prove that the Hessian of a nonzero Lorentzian polynomial has exactly one positive eigenvalue at any point on the positive orthant. This property can be seen as an analog of the Hodge–Riemann relations for Lorentzian polynomials.

Lorentzian polynomials are intimately connected to matroid theory and negative dependence properties. We show that matroids, and more generally $\mathrm {M}$-convex sets, are characterized by the Lorentzian property, and develop a theory around Lorentzian polynomials. In particular, we provide a large class of linear operators that preserve the Lorentzian property and prove that Lorentzian measures enjoy several negative dependence properties. We also prove that the class of tropicalized Lorentzian polynomials coincides with the class of $\mathrm {M}$-convex functions in the sense of discrete convex analysis. The tropical connection is used to produce Lorentzian polynomials from $\mathrm {M}$-convex functions.

We give two applications of the general theory. First, we prove that\nonbreakingspace the homogenized multivariate Tutte polynomial of a matroid is Lorentzian whenever the parameter $q$ satisfies $0\lt q \leq 1$. Consequences are proofs of the strongest Mason’s conjecture from 1972 and negative dependence properties of the random cluster model in statistical physics. Second, we prove that the multivariate characteristic polynomial of an $\mathrm {M}$-matrix is Lorentzian. This refines a result of Holtz who proved that the coefficients of the characteristic polynomial of an $\mathrm {M}$-matrix form an ultra log-concave sequence.

Pages 821-891 by
From volume 192-3

## Absolute profinite rigidity and hyperbolic geometry

We construct arithmetic Kleinian groups that are profinitely rigid in the absolute sense: each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. The Bianchi group $\mathrm {PSL}(2,\mathbb {Z}[\omega ])$ with $\omega ^2+\omega +1=0$ is rigid in this sense. Other examples include the non-uniform lattice of minimal co-volume in ${\rm {PSL}}(2,\mathbb {C})$ and the fundamental group of the Weeks manifold (the closed hyperbolic $3$-manifold of minimal volume).

The Supplemental Magma code for this paper is available at the following location:

https://doi.org/10.4007/annals.2020.192.3.1.code

Pages 679-719 by
From volume 192-3

## On the Duffin-Schaeffer conjecture

Let $\psi :\mathbb {N}\to \mathbb {R}_{\geqslant 0}$ 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 $|\alpha -a/q|\leqslant \psi (q)/q$. If $\sum _{q=1}^\infty \psi (q)\varphi (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 $|\alpha – a/q|\leqslant \psi (q)/q$, giving a refinement of Khinchin’s Theorem.

Pages 251-307 by
From volume 192-1

## Ax-Schanuel for Shimura varieties

We prove the Ax-Schanuel theorem for a general (pure) Shimura variety. A basic version of the theorem concerns the transcendence of the uniformization map from a bounded Hermitian symmetric space to a Shimura variety. We then prove a version of the theorem with derivatives in the setting of jet spaces, and finally a version in the setting of differential fields.

Our method of proof builds on previous work, combined with a new approach that uses higher-order contact conditions to place varieties yielding intersections of excessive dimension in natural algebraic families.

Pages 945-978 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

## Uniqueness of $\mathrm {K}$-polystable degenerations of Fano varieties

We prove that $\mathrm {K}$-polystable degenerations of $\mathbb {Q}$-Fano varieties are unique. Furthermore, we show that the moduli stack of $\mathrm {K}$-stable $\mathbb {Q}$-Fano varieties is separated. Together with recently proven boundedness and openness statements, the latter result yields a separated Deligne-Mumford stack parametrizing all uniformly $\mathrm {K}$-stable $\mathbb {Q}$-Fano varieties of fixed dimension and volume. The result also implies that the automorphism group of a $\mathrm {K}$-stable $\mathbb {Q}$-Fano variety is finite.

Pages 609-656 by
From volume 190-2

## Strong cosmic censorship in spherical symmetry for two-ended asymptotically flat initial data I. The interior of the black hole region

This is the first and main paper of a two-part series, in which we prove the $C^{2}$-formulation of the strong cosmic censorship conjecture for the Einstein–Maxwell-(real)-scalar-field system in spherical symmetry for two-ended asymptotically flat data. For this model, it is known through the works of Dafermos and Dafermos–Rodnianski that the maximal globally hyperbolic future development of any admissible two-ended asymptotically flat Cauchy initial data set possesses a non-empty Cauchy horizon, across which the spacetime is $C^{0}$-future-extendible. (In particular, the $C^{0}$-formulation of the strong cosmic censorship conjecture is false.) Nevertheless, the main conclusion of the present series of papers is that for a generic (in the sense of being open and dense relative to appropriate topologies) class of such data, the spacetime is future-inextendible with a Lorentzian metric of higher regularity (specifically, $C^{2}$).

In this paper, we prove that the solution is $C^{2}$-future-inextendible under the condition that the scalar field obeys an $L^{2}$-averaged polynomial lower bound along each of the event horizons. This, in particular, improves upon a previous result of Dafermos, which required instead a pointwise lower bound. Key to the proof are appropriate stability and instability results in the interior of the black hole region, whose proofs are in turn based on ideas from the work of Dafermos–Luk on the stability of the Kerr Cauchy horizon (without symmetry) and from our previous paper on linear instability of the Reissner–Nordström Cauchy horizon. In the second paper of the series, which concerns analysis in the exterior of the black hole region, we show that the $L^2$-averaged polynomial lower bound needed for the instability result indeed holds for a generic class of admissible two-ended asymptotically flat Cauchy initial data.

Pages 1-111 by
From volume 190-1

## Polynomial diffeomorphisms of $\bf{C}^2$: VI. Connectivity of $J$

No Abstract available for this article.

Pages 695-735 by
From volume 148-2

## The number of solutions of $\phi(x)=m$

An old conjecture of Sierpiński asserts that for every integer $k\ge 2$ , there is a number $m$ for which the equation $\phi(x) = m$ has exactly $k$ solutions. Here $\phi$ is Euler’s totient function. In 1961, Schinzel deduced this conjecture from his Hypothesis H. The purpose of this paper is to present an unconditional proof of Sierpiński’s conjecture. The proof uses many results from sieve theory, in particular the famous theorem of Chen.

Pages 283-311 by
From volume 150-1

## The algebraic hull of the Kontsevich–Zorich cocycle

We compute the algebraic hull of the Kontsevich–Zorich cocycle over any $ \mathrm {GL}^+_2(\mathbb {R}) $ invariant subvariety of the Hodge bundle, and derive from this finiteness results on such subvarieties.

Pages 281-313 by
From volume 188-1

## The Gopakumar-Vafa formula for symplectic manifolds

The Gopakumar-Vafa conjecture predicts that the Gromov-Witten invariants of a Calabi-Yau 3-fold can be canonically expressed in terms of integer invariants called BPS numbers. Using the methods of symplectic Gromov-Witten theory, we prove that the Gopakumar-Vafa conjecture holds for any symplectic Calabi-Yau 6-manifold, and hence for Calabi-Yau 3-folds. The results extend to all symplectic 6-manifolds and to the genus zero GW invariants of semipositive manifolds.

Pages 1-64 by
From volume 187-1

## The Severi bound on sections of rank two semistable bundles on a Riemann surface

Let $E$ be a semistable rank two vector bundle of degree $d$ on a Riemann surface $C$ of genus $g\ge 1$, i.e. such that the minimal degree $s$ of a tensor product of $E$ with a line bundle having a nonzero section is nonnegative. We give an analogue of Clifford’s lemma by showing that $E$ has at most $(d-s)/2+\delta$ independent sections, where $\delta$ is $2$ or $1$ according to whether the Krawtchouk polynomial $K_r(n,N)$ is zero or not at $r = (d-s)/2+1$, $n = g$, $N = 2g-s$ (the analogous bound for nonsemistable rank two bundles being stronger but easier to prove). This gives an answer to the problem posed by Severi asking for the minimal degree of a directrix of a ruled surface. In some cases, namely if $s$ has maximal value $s=g$, or if $s\ge \mathrm{gonality}(C)-2$, or if $E$ is general among those of the same Segre invariant $s$, or also if the genus is a power of two, we prove the bound holds with $\delta =1$.

The theory of Krawtchouk polynomials investigates which triples $(g,s,d)$ provide zeros of $K_r(n,N)$. Then, they generate invariants which one may expect to be associated to a Severi bundle, i.e., to a rank two semistable bundle reaching the bound $\delta =2$. According to this theory, there are only a finite number of such triples $(g,s,d)$ for each value of $d-s$, with the exception that there are infinitely many triples with $d-s = 2$ or $4$. We then find all the Severi bundles corresponding to those two exceptional values of $d-s$.

Pages 739-758 by
From volume 154-3

## The extremal function associated to intrinsic norms

Through the study of the degenerate complex Monge-Ampère equation, we establish the optimal regularity of the extremal function associated to intrinsic norms of Chern-Levine-Nirenberg and Bedford-Taylor. We prove a conjecture of Chern-Levine-Nirenberg on the extended intrinsic norms on complex manifolds and verify Bedford-Taylor’s representation formula for these norms in general.

Pages 197-211 by
From volume 156-1

## Entropy waves, the zig-zag graph product, and new constant-degree expanders

The main contribution of this work is a new type of graph product, which we call the *zig-zag product*. Taking a product of a large graph with a small graph, the resulting graph inherits (roughly) its size from the large one, its degree from the small one, and its expansion properties from both! Iteration yields simple explicit constructions of constant-degree expanders of arbitrary size, starting from one constant-size expander.

Crucial to our intuition (and simple analysis) of the properties of this graph product is the view of expanders as functions which act as “entropy wave” propagators — they transform probability distributions in which entropy is concentrated in on area to distributions where that concentration is dissipated. In these terms, the graph product affords the constructive interference of two such waves.

Subsequent work [ALW-1], MW01] relates the zig-zag product of graphs to the standard semidirect product of groups, leading to new results and constructions on expanding Cayley graphs.

Pages 157-187 by
From volume 155-1

## 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

## 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

## Homological stability for moduli spaces of high dimensional manifolds. II

We prove a homological stability theorem for moduli spaces of manifolds of dimension $2n$, for attaching handles of index at least $n$, after these manifolds have been stabilised by countably many copies of $S^n \times S^n$. Combined with previous work of the authors, we obtain an analogue of the Madsen–Weiss theorem for any simply-connected manifold of dimension $2n \geq 6$.

Pages 127-204 by
From volume 186-1

## 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

## Large gaps between consecutive prime numbers

Let $G(X)$ denote the size of the largest gap between consecutive primes below $X$. Answering a question of Erdős, we show that \[ G(X) \geq f(X) \frac{\log X \log \log X \log \log \log \log X}{(\log \log \log X)^2},\] where $f(X)$ is a function tending to infinity with $X$. Our proof combines existing arguments with a random construction covering a set of primes by arithmetic progressions. As such, we rely on recent work on the existence and distribution of long arithmetic progressions consisting entirely of primes.

Pages 935-974 by
From volume 183-3

## Large gaps between primes

We show that there exist pairs of consecutive primes less than $x$ whose difference is larger than \[ t(1+o(1))(\log{x})(\log\log{x})(\log\log\log\log{x})(\log\log\log{x})^{-2}\] for any fixed $t$. This answers a well-known question of Erdős.

Pages 915-933 by
From volume 183-3

## Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields

We prove a homological stabilization theorem for Hurwitz spaces: moduli spaces of branched covers of the complex projective line. This has the following arithmetic consequence: let $\ell > 2$ be prime and $A$ a finite abelian $\ell$-group. Then there exists $Q = Q(A)$ such that, for $q$ greater than $Q$, a positive fraction of quadratic extensions of $\mathbb{F}_q(t)$ have the $\ell$-part of their class group isomorphic to $A$.

Pages 729-786 by
From volume 183-3

## Combinatorial theorems in sparse random sets

We develop a new technique that allows us to show in a unified way that many well-known combinatorial theorems, including Turán’s theorem, Szemerédi’s theorem and Ramsey’s theorem, hold almost surely inside sparse random sets. For instance, we extend Turán’s theorem to the random setting by showing that for every $\epsilon > 0$ and every positive integer $t \geq 3$ there exists a constant $C$ such that, if $G$ is a random graph on $n$ vertices where each edge is chosen independently with probability at least $C n^{-2/(t+1)}$, then, with probability tending to 1 as $n$ tends to infinity, every subgraph of $G$ with at least $\left(1 – \frac{1}{t-1} + \epsilon\right) e(G)$ edges contains a copy of $K_t$. This is sharp up to the constant $C$. We also show how to prove sparse analogues of structural results, giving two main applications, a stability version of the random Turán theorem stated above and a sparse hypergraph removal lemma. Many similar results have recently been obtained independently in a different way by Schacht and by Friedgut, Rödl and Schacht.

Pages 367-454 by
From volume 184-2

## Embedded self-similar shrinkers of genus $0$

We confirm a well-known conjecture that the round sphere is the only compact, embedded self-similar shrinking solution of mean curvature flow in $\mathbb{R}^3$ with genus $0$. More generally, we show that the only properly embedded self-similar shrinkers in $\mathbb{R}^3$ with vanishing intersection form are the sphere, the cylinder, and the plane. This answers two questions posed by T. Ilmanen.

Pages 715-728 by
From volume 183-2

## 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 $\mathrm{mod}p$ 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

## Defining ${\mathbb Z}$ in ${\mathbb Q}$

We show that ${\mathbb Z}$ is definable in ${\mathbb Q}$ by a universal first-order formula in the language of rings. We also present an $\forall\exists$-formula for ${\mathbb Z}$ in ${\mathbb Q}$ with just one universal quantifier. We exhibit new diophantine subsets of ${\mathbb Q}$ like the complement of the image of the norm map under a quadratic extension, and we give an elementary proof for the fact that the set of nonsquares is diophantine.

Pages 73-93 by
From volume 183-1

## Isolation, equidistribution, and orbit closures for the $\mathrm{SL}(2,\mathbb{R})$ action on moduli space

We prove results about orbit closures and equidistribution for the $\mathrm{SL}(2,\mathbb{R})$ action on the moduli space of compact Riemann surfaces, which are analogous to the theory of unipotent flows. The proofs of the main theorems rely on the measure classification theorem of the first two authors and a certain isolation property of closed $\mathrm{SL}(2,\mathbb{R})$ invariant manifolds developed in this paper.

Pages 673-721 by
From volume 182-2

## A polynomial upper bound on Reidemeister moves

We prove that any diagram of the unknot with $c$ crossings may be reduced to the trivial diagram using at most $(236 \,c)^{11}$ Reidemeister moves.

Pages 491-564 by
From volume 182-2

## 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

## Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0

We prove an asymptotic formula for the number of $\mathrm{SL}_3(\mathbb{Z})$-equivalence classes of integral ternary cubic forms having bounded invariants. We use this result to show that the average size of the 3-Selmer group of all elliptic curves, when ordered by height, is equal to 4. This implies that the average rank of all elliptic curves, when ordered by height, is less than 1.17. Combining our counting techniques with a recent result of Dokchitser and Dokchitser, we prove that a positive proportion of all elliptic curves have rank 0. Assuming the finiteness of the Tate–Shafarevich group, we also show that a positive proportion of elliptic curves have rank 1. Finally, combining our counting results with the recent work of Skinner and Urban, we show that a positive proportion of elliptic curves have analytic rank 0; i.e., a positive proportion of elliptic curves have nonvanishing $L$-function at $s=1$. It follows that a positive proportion of all elliptic curves satisfy BSD.

Pages 587-621 by
From volume 181-2

## Small gaps between primes

We introduce a refinement of the GPY sieve method for studying prime $k$-tuples and small gaps between primes. This refinement avoids previous limitations of the method and allows us to show that for each $k$, the prime $k$-tuples conjecture holds for a positive proportion of admissible $k$-tuples. In particular, $\liminf_{n}(p_{n+m}-p_n)<\infty$ for every integer $m$. We also show that $\liminf(p_{n+1}-p_n)\le 600$ and, if we assume the Elliott-Halberstam conjecture, that $\liminf_n(p_{n+1}-p_n)\le 12$ and $\liminf_n (p_{n+2}-p_n)\le 600$.

Pages 383-413 by
From volume 181-1

## Solution of the minimum modulus problem for covering systems

We answer a question of Erdős by showing that the least modulus of a distinct covering system is at most $ 10^{16}$.

Pages 361-382 by
From volume 181-1

## Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves

We prove a theorem giving the asymptotic number of binary quartic forms having bounded invariants; this extends, to the quartic case, the classical results of Gauss and Davenport in the quadratic and cubic cases, respectively. Our techniques are quite general and may be applied to counting integral orbits in other representations of algebraic groups. We use these counting results to prove that the average rank of elliptic curves over $\Bbb{Q}$, when ordered by their heights, is bounded. In particular, we show that when elliptic curves are ordered by height, the mean size of the $2$-Selmer group is $3$. This implies that the limsup of the average rank of elliptic curves is at most $1.5$.

Pages 191-242 by
From volume 181-1

## Sets of integers with no large sum-free subset

Answering a question of P. Erdős from 1965, we show that for every $\varepsilon> 0$ there is a set $A$ of $n$ integers with the following property: every set $A’ \subset A$ with at least $\left(\frac{1}{3} + \varepsilon\right) n$ elements contains three distinct elements $x,y,z$ with $x + y = z$.

Pages 621-652 by
From volume 180-2

## On the quantitative distribution of polynomial nilsequences — erratum

This is an erratum to the paper *The quantitative behaviour of polynomial orbits on nilmanifolds* by the authors, published as Ann. of Math. (2) $\bf{175}$ (2012), no. 2, 465–540. The proof of Theorem 8.6 of that paper, which claims a distribution result for multiparameter polynomial sequences on nilmanifolds, was incorrect. We provide two fixes for this issue here. First, we deduce the “equal sides” case $N_1 = …= N_t = N$ of this result from the 1-parameter results in the paper. This is the same basic mode of argument we attempted originally, though the details are different. The equal sides case is the only one required in applications such as the proof of the inverse conjectures for the Gowers norms due to the authors and Ziegler. To remove the equal sides condition one must rerun the entire argument of our paper in the context of multiparameter polynomial sequences $g : \mathbb{Z}^t \rightarrow G$ rather than 1-parameter sequences $g : \mathbb{Z} \rightarrow G$ as is currently done: a more detailed sketch of how this may be done is available online.

Pages 1175-1183 by
From volume 179-3

## Ax-Lindemann for $\mathcal{A}_g$

We prove the Ax-Lindemann theorem for the coarse moduli space $\mathcal{A}_{g}$ of principally polarized abelian varieties of dimension $g\ge 1$. We affirm the André-Oort conjecture unconditionally for $\mathcal{A}_g$ for $g\le 6$, and under GRH for all $g$.

Pages 659-681 by
From volume 179-2

## Invariant varieties for polynomial dynamical systems

We study algebraic dynamical systems (and, more generally, $\sigma$-varieties) $\Phi:\mathbb{A}^n_\mathbb{C} \to \mathbb{A}^n_\mathbb{C}$ given by coordinatewise univariate polynomials by refining an old theorem of Ritt on compositional identities amongst polynomials. More precisely, we find a nearly canonical way to write a polynomial as a composition of “clusters” from which one may easily read off possible compositional identities. Our main result is an explicit description of the (weakly) skew-invariant varieties, that is, for a fixed field automorphism $\sigma:\mathbb{C} \to \mathbb{C}$ those algebraic varieties $X \subseteq \mathbb{A}^n_\mathbb{C}$ for which $\Phi(X) \subseteq X^\sigma$. As a special case, we show that if $f(x) \in \mathbb{C}[x]$ is a polynomial of degree at least two that is not conjugate to a monomial, Chebyshev polynomial or a negative Chebyshev polynomial, and $X \subseteq \mathbb{A}^2_\mathbb{C}$ is an irreducible curve that is invariant under the action of $(x,y) \mapsto (f(x),f(y))$ and projects dominantly in both directions, then $X$ must be the graph of a polynomial that commutes with $f$ under composition. As consequences, we deduce a variant of a conjecture of Zhang on the existence of rational points with Zariski dense forward orbits and a strong form of the dynamical Manin-Mumford conjecture for liftings of the Frobenius.

We also show that in models of ACFA$_0$, a disintegrated set defined by $\sigma(x) = f(x)$ for a polynomial $f$ has Morley rank one and is usually strongly minimal, that model theoretic algebraic closure is a locally finite closure operator on the nonalgebraic points of this set unless the skew-conjugacy class of $f$ is defined over a fixed field of a power of $\sigma$, and that nonorthogonality between two such sets is definable in families if the skew-conjugacy class of $f$ is defined over a fixed field of a power of $\sigma$.

Pages 81-177 by
From volume 179-1

## 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

## Quantum groups via Hall algebras of complexes

We describe quantum enveloping algebras of symmetric Kac-Moody Lie algebras via a finite field Hall algebra construction involving $\mathbb{Z}_2$-graded complexes of quiver representations.

Pages 739-759 by
From volume 177-2

## Disparity in Selmer ranks of quadratic twists of elliptic curves

We study the parity of $2$-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve $E$ over an arbitrary number field $K$. We prove that the fraction of twists (of a given elliptic curve over a fixed number field) having even $2$-Selmer rank exists as a stable limit over the family of twists, and we compute this fraction as an explicit product of local factors. We give an example of an elliptic curve $E$ such that as $K$ varies, these fractions are dense in $[0, 1]$. More generally, our results also apply to $p$-Selmer ranks of twists of $2$-dimensional self-dual $\mathbf{F}_p$-representations of the absolute Galois group of $K$ by characters of order $p$.

Pages 287-320 by
From volume 178-1

## Kloosterman sheaves for reductive groups

Deligne constructed a remarkable local system on $\mathbb{P}^1-\{0,\infty\}$ attached to a family of Kloosterman sums. Katz calculated its monodromy and asked whether there are Kloosterman sheaves for general reductive groups and which automorphic forms should be attached to these local systems under the Langlands correspondence.

Motivated by work of Gross and Frenkel-Gross we find an explicit family of such automorphic forms and even a simple family of automorphic sheaves in the framework of the geometric Langlands program. We use these automorphic sheaves to construct $\ell$-adic Kloosterman sheaves for any reductive group in a uniform way, and describe the local and global monodromy of these Kloosterman sheaves. In particular, they give motivic Galois representations with exceptional monodromy groups $G_2,F_4,E_7$ and $E_8$. This also gives an example of the geometric Langlands correspondence with wild ramification for any reductive group.

Pages 241-310 by
From volume 177-1

## Cover times, blanket times, and majorizing measures

We exhibit a strong connection between cover times of graphs, Gaussian processes, and Talagrand’s theory of majorizing measures. In particular, we show that the cover time of any graph $G$ is equivalent, up to universal constants, to the square of the expected maximum of the Gaussian free field on $G$, scaled by the number of edges in $G$. This allows us to resolve a number of open questions. We give a deterministic polynomial-time algorithm that computes the cover time to within an $O(1)$ factor for any graph, answering a question of Aldous and Fill (1994). We also positively resolve the blanket time conjectures of Winkler and Zuckerman (1996), showing that for any graph, the blanket and cover times are within an $O(1)$ factor. The best previous approximation factor for both these problems was $O((\log \log n)^2)$ for $n$-vertex graphs, due to Kahn, Kim, Lovász, and Vu (2000).

Pages 1409-1471 by
From volume 175-3

## Topology of Hitchin systems and Hodge theory of character varieties: the case $A_1$

For $\mathrm{G}=\mathrm{GL}_2,\mathrm{PGL}_2, \mathrm{SL}_2$ we prove that the perverse filtration associated with the Hitchin map on the rational cohomology of the moduli space of twisted $\mathrm{G}$-Higgs bundles on a compact Riemann surface $C$ agrees with the weight filtration on the rational cohomology of the twisted $\mathrm{G}$ character variety of $C$ when the cohomologies are identified via non-Abelian Hodge theory. The proof is accomplished by means of a study of the topology of the Hitchin map over the locus of integral spectral curves.

Pages 1329-1407 by
From volume 175-3

## Local entropy averages and projections of fractal measures

We show that for families of measures on Euclidean space which satisfy an ergodic-theoretic form of “self-similarity” under the operation of re-scaling, the dimension of linear images of the measure behaves in a semi-continuous way. We apply this to prove the following conjecture of Furstenberg: if $X,Y\subseteq [0,1]$ are closed and invariant, respectively, under $\times m\bmod 1$ and $\times n\bmod 1$, where $m,n$ are not powers of the same integer, then, for any $t\neq0$, \[ \dim(X+t Y)=\min\{1,\dim X+\dim Y\}.\] A similar result holds for invariant measures and gives a simple proof of the Rudolph-Johnson theorem. Our methods also apply to many other classes of conformal fractals and measures. As another application, we extend and unify results of Peres, Shmerkin and Nazarov, and of Moreira, concerning projections of products of self-similar measures and Gibbs measures on regular Cantor sets. We show that under natural irreducibility assumptions on the maps in the IFS, the image measure has the maximal possible dimension under any linear projection other than the coordinate projections. We also present applications to Bernoulli convolutions and to the images of fractal measures under differentiable maps.

Pages 1001-1059 by
From volume 175-3

## Thom polynomials of Morin singularities

We prove a formula for Thom polynomials of $A_d$ singularities in any codimension. We use a combination of the test-curve model of Porteous, and the localization methods in equivariant cohomology. Our formulas are independent of the codimension, and are computationally effective up to $d=6$.

Pages 567-629 by
From volume 175-2

## The existence of an abelian variety over $\overline{\mathbb{Q}}$ isogenous to no Jacobian

We prove the existence of an abelian variety $A$ of dimension $g$ over $\overline{\mathbb{Q}}$ that is not isogenous to any Jacobian, subject to the necessary condition $g\!>\!3$. Recently, C. Chai and F. Oort gave such a proof assuming the André-Oort conjecture. We modify their proof by constructing a special sequence of CM points for which we can avoid any unproven hypotheses. We make use of various techniques from the recent work of Klingler-Yafaev et al.

Pages 637-650 by
From volume 176-1

## O-minimality and the André-Oort conjecture for $\mathbb{C}^{n}$

We give an *unconditional* proof of the André-Oort conjecture for arbitrary products of modular curves. We establish two generalizations. The first includes the Manin-Mumford conjecture for arbitrary products of elliptic curves defined over $\bar{\mathbb{Q}}$ as well as Lang’s conjecture for torsion points in powers of the multiplicative group. The second includes the Manin-Mumford conjecture for abelian varieties defined over $\bar{\mathbb{Q}}$. Our approach uses the theory of o-minimal structures, a part of Model Theory, and follows a strategy proposed by Zannier and implemented in three recent papers: a new proof of the Manin-Mumford conjecture by Pila-Zannier; a proof of a special (but new) case of Pink’s relative Manin-Mumford conjecture by Masser-Zannier; and new proofs of certain known results of André-Oort-Manin-Mumford type by Pila.

Pages 1779-1840 by
From volume 173-3

## Weyl group multiple Dirichlet series, Eisenstein series and crystal bases

We show that the Whittaker coefficients of Borel Eisenstein series on the metaplectic covers of ${\rm GL}_{r+1}$ can be described as multiple Dirichlet series in $r$ complex variables, whose coefficients are computed by attaching a number-theoretic quantity (a product of Gauss sums) to each vertex in a crystal graph. These Gauss sums depend on “string data” previously introduced in work of Lusztig, Berenstein and Zelevinsky, and Littelmann. These data are the lengths of segments in a path from the given vertex to the vertex of lowest weight, depending on a factorization of the long Weyl group element into simple reflections. The coefficients may also be described as sums over strict Gelfand-Tsetlin patterns. The description is uniform in the degree of the metaplectic cover.

Pages 1081-1120 by
From volume 173-2

## Asymptotics of Toeplitz, Hankel, and Toeplitz+Hankel determinants with Fisher-Hartwig singularities

We study the asymptotics in $n$ for $n$-dimensional Toeplitz determinants whose symbols possess Fisher-Hartwig singularities on a smooth background. We prove the general nondegenerate asymptotic behavior as conjectured by Basor and Tracy. We also obtain asymptotics of Hankel determinants on a finite interval as well as determinants of Toeplitz+Hankel type. Our analysis is based on a study of the related system of orthogonal polynomials on the unit circle using the Riemann-Hilbert approach.

Pages 1243-1299 by
From volume 174-2

## Distribution of periodic torus orbits and Duke’s theorem for cubic fields

We study periodic torus orbits on spaces of lattices. Using the action of the group of adelic points of the underlying tori, we define a natural equivalence relation on these orbits, and show that the equivalence classes become uniformly distributed. This is a cubic analogue of Duke’s theorem about the distribution of closed geodesics on the modular surface: suitably interpreted, the ideal classes of a cubic totally real field are equidistributed in the modular $5$-fold $\mathrm{SL}_3(\mathbb{Z}) \backslash \mathrm{SL}_3(\mathbb{R}) / \mathrm{SO}_3$. In particular, this proves (a stronger form of) the folklore conjecture that the collection of maximal compact flats in $\mathrm{SL}_3(\mathbb{Z}) \backslash \mathrm{SL}_3(\mathbb{R}) / \mathrm{SO}_3$ of volume $\leq V$ becomes equidistributed as $V \rightarrow \infty$.

The proof combines subconvexity estimates, measure classification, and local harmonic analysis.

Pages 815-885 by
From volume 173-2

## Stable homology of automorphism groups of free groups

Homology of the group $\operatorname{Aut}(F_n)$ of automorphisms of a free group on $n$ generators is known to be independent of $n$ in a certain stable range. Using tools from homotopy theory, we prove that in this range it agrees with homology of symmetric groups. In particular we confirm the conjecture that stable rational homology of $\operatorname{Aut}(F_n)$ vanishes.

Pages 705-768 by
From volume 173-2

## Small subspaces of $L_p$

We prove that if $X$ is a subspace of $L_p$ $(2\lt p\lt \infty)$, then either $X$ embeds isomorphically into $\ell_p \oplus \ell_2$ or $X$ contains a subspace $Y,$ which is isomorphic to $\ell_p(\ell_2)$. We also give an intrinsic characterization of when $X$ embeds into $\ell_p \oplus \ell_2$ in terms of weakly null trees in $X$ or, equivalently, in terms of the “infinite asymptotic game” played in $X$. This solves problems concerning small subspaces of $L_p$ originating in the 1970’s. The techniques used were developed over several decades, the most recent being that of weakly null trees developed in the 2000’s.

Pages 169-209 by
From volume 173-1

## Densities for rough differential equations under Hörmander’s condition

We consider stochastic differential equations $dY=V\left( Y\right) dX$ driven by a multidimensional Gaussian process $X$ in the rough path sense [T. Lyons, *Rev. Mat. Iberoamericana* **14**, (1998), 215–310]. Using Malliavin Calculus we show that $Y_{t}$ admits a density for $t\in (0,T]$ provided (i) the vector fields $V=\left( V_{1},\dots,V_{d}\right) $ satisfy Hörmander’s condition and (ii) the Gaussian driving signal $X$ satisfies certain conditions. Examples of driving signals include fractional Brownian motion with Hurst parameter $H>1/4$, the Brownian bridge returning to zero after time $T$ and the Ornstein-Uhlenbeck process.

Pages 2115-2141 by
From volume 171-3

## Quantum unique ergodicity for ${\rm SL}_2(\mathbb{Z})\backslash\mathbb{H}$

We eliminate the possibility of “escape of mass” for Hecke-Maass forms of large eigenvalue for the modular group. Combined with the work of Lindenstrauss, this establishes the Quantum Unique Ergodicity conjecture of Rudnick and Sarnak for Hecke-Maass forms on the modular surface ${\rm SL}_2(\mathbb{Z})\backslash \mathbb{H}$.

Pages 1529-1538 by
From volume 172-2

## Mass equidistribution for Hecke eigenforms

We prove a conjecture of Rudnick and Sarnak on the mass equidistribution of Hecke eigenforms. This builds upon independent work of the authors.

Pages 1517-1528 by
From volume 172-2

## Weak subconvexity for central values of $L$-functions

We describe a general method to obtain weak subconvexity bounds for many classes of $L$-functions. We give several examples of our bound, and our work has applications to a conjecture of Rudnick and Sarnak for the mass equidistribution of Hecke eigenforms

Pages 1469-1498 by
From volume 172-2

## Global solutions of shock reflection by large-angle wedges for potential flow

When a plane shock hits a wedge head on, it experiences a reflection-diffraction process and then a self-similar reflected shock moves outward as the original shock moves forward in time. Experimental, computational, and asymptotic analysis has shown that various patterns of shock reflection may occur, including regular and Mach reflection. However, most of the fundamental issues for shock reflection have not been understood, including the global structure, stability, and transition of the different patterns of shock reflection. Therefore, it is essential to establish the global existence and structural stability of solutions of shock reflection in order to understand fully the phenomena of shock reflection. On the other hand, there has been no rigorous mathematical result on the global existence and structural stability of shock reflection, including the case of potential flow which is widely used in aerodynamics. Such problems involve several challenging difficulties in the analysis of nonlinear partial differential equations such as mixed equations of elliptic-hyperbolic type, free boundary problems, and corner singularity where an elliptic degenerate curve meets a free boundary. In this paper we develop a rigorous mathematical approach to overcome these difficulties involved and establish a global theory of existence and stability for shock reflection by large-angle wedges for potential flow. The techniques and ideas developed here will be useful for other nonlinear problems involving similar difficulties.

Pages 1067-1182 by
From volume 171-2

## Sparse equidistribution problems, period bounds and subconvexity

We introduce a “geometric” method to bound periods of automorphic forms. The key features of this method are the use of equidistribution results in place of mean value theorems, and the systematic use of mixing and the spectral gap. Applications are given to equidistribution of sparse subsets of horocycles and to equidistribution of CM points; to subconvexity of the triple product period in the level aspect over number fields, which implies subconvexity for certain standard and Rankin-Selberg $L$-functions; and to bounding Fourier coefficients of automorphic forms.

Pages 989-1094 by
From volume 172-2

## A family of Calabi-Yau varieties and potential automorphy

We prove potential modularity theorems for $l$-adic representations of any dimension. From these results we deduce the Sato-Tate conjecture for all elliptic curves with nonintegral $j$-invariant defined over a totally real field.

Pages 779-813 by
From volume 171-2

## Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate

Consider a system of $N$ bosons in three dimensions interacting via a repulsive short range pair potential $N^2V(N(x_i-x_j))$, where $\mathbf{x}=(x_1, \ldots, x_N)$ denotes the positions of the particles. Let $H_N$ denote the Hamiltonian of the system and let $\psi_{N,t}$ be the solution to the Schrödinger equation. Suppose that the initial data $\psi_{N,0}$ satisfies the energy condition \[ \langle \psi_{N,0}, H_N^k \psi_{N,0} \rangle \leq C^k N^k \; \] for $k=1,2,\ldots\; $. We also assume that the $k$-particle density matrices of the initial state are asymptotically factorized as $N\to\infty$. We prove that the $k$-particle density matrices of $\psi_{N,t}$ are also asymptotically factorized and the one particle orbital wave function solves the Gross-Pitaevskii equation, a cubic nonlinear Schrödinger equation with the coupling constant given by the scattering length of the potential $V$. We also prove the same conclusion if the energy condition holds only for $k=1$ but the factorization of $\psi_{N,0}$ is assumed in a stronger sense.

Pages 291-370 by
From volume 172-1

## A rigid irregular connection on the projective line

In this paper we construct a connection $\nabla$ on the trivial $G$-bundle on $\mathbb{P}^1$ for any simple complex algebraic group $G$, which is regular outside of the points $0$ and $\infty$, has a regular singularity at the point $0$, with principal unipotent monodromy, and has an irregular singularity at the point $\infty$, with slope $1/h$, the reciprocal of the Coxeter number of $G$. The connection $\nabla$, which admits the structure of an oper in the sense of Beilinson and Drinfeld, appears to be the characteristic $0$ counterpart of a hypothetical family of $\ell$-adic representations, which should parametrize a specific automorphic representation under the global Langlands correspondence. These $\ell$-adic representations, and their characteristic $0$ counterparts, have been constructed in some cases by Deligne and Katz. Our connection is constructed uniformly for any simple algebraic group, and characterized using the formalism of opers. It provides an example of the geometric Langlands correspondence with wild ramification. We compute the de Rham cohomology of our connection with values in a representation $V$ of $G$, and describe the differential Galois group of $\nabla$ as a subgroup of $G$.

Pages 1469-1512 by
From volume 170-3

## Subgroups of direct products of limit groups

SelaIf $\Gamma_1,\dots,\Gamma_n$ are limit groups and $S\subset\Gamma_1\times\dots\times\Gamma_n$ is of type ${\rm FP}_n(\mathbb Q)$ then $S$ contains a subgroup of finite index that is itself a direct product of at most $n$ limit groups. This answers a question of Sela.

Pages 1447-1467 by
From volume 170-3

## Localization of $\mathfrak g$-modules on the affine Grassmannian

We consider the category of modules over the affine Kac-Moody algebra $\widehat{\mathfrak g}$ of critical level with regular central character. In our previous paper we conjectured that this category is equivalent to the category of Hecke eigen-D-modules on the affine Grassmannian $G(\!(t)\!)/G[\mskip-2mu[t]\mskip-2mu]$. This conjecture was motivated by our proposal for a local geometric Langlands correspondence. In this paper we prove this conjecture for the corresponding $I^0$ equivariant categories, where $I^0$ is the radical of the Iwahori subgroup of $G(\!(t)\!)$. Our result may be viewed as an affine analogue of the equivalence of categories of ${\mathfrak g}$-modules and D-modules on the flag variety $G/B$, due to Beilinson-Bernstein and Brylinski-Kashiwara.

Pages 1339-1381 by
From volume 170-3

## Moduli of finite flat group schemes, and modularity

We prove that, under some mild conditions, a two dimensional $p$-adic Galois representation which is residually modular and potentially Barsotti-Tate at $p$ is modular. This provides a more conceptual way of establishing the Shimura-Taniyama-Weil conjecture, especially for elliptic curves which acquire good reduction over a wildly ramified extension of $\mathbb Q_3$. The main ingredient is a new technique for analyzing flat deformation rings. It involves resolving them by spaces which parametrize finite flat group scheme models of Galois representations.

Pages 1085-1180 by
From volume 170-3

## Moments of the Riemann zeta function

Assuming the Riemann hypothesis, we obtain an upper bound for the moments of the Riemann zeta function on the critical line. Our bound is nearly as sharp as the conjectured asymptotic formulae for these moments. The method extends to moments in other families of $L$-functions.

Pages 981-993 by
From volume 170-2

## The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz

We prove the B. and M. Shapiro conjecture that if the Wronskian of a set of polynomials has real roots only, then the complex span of this set of polynomials has a basis consisting of polynomials with real coefficients. This, in particular, implies the following result:

If all ramification points of a parametrized rational curve $\phi:\Bbb{C}\mathbb P^1 \to \Bbb{C}\mathbb P^r$ lie on a circle in the Riemann sphere $\Bbb{C}\mathbb P^1$, then $\phi$ maps this circle into a suitable real subspace $\mathbb R\mathbb P^r \subset \Bbb{C}\mathbb P^r$.

The proof is based on the Bethe ansatz method in the Gaudin model. The key observation is that a symmetric linear operator on a Euclidean space has real spectrum.

In Appendix A, we discuss properties of differential operators associated with Bethe vectors in the Gaudin model. In particular, we prove a statement, which may be useful in complex algebraic geometry; it claims that certain Schubert cycles in a Grassmannian intersect transversally if the spectrum of the corresponding Gaudin Hamiltonians is simple.

In Appendix B, we formulate a conjecture on reality of orbits of critical points of master functions and prove this conjecture for master functions associated with Lie algebras of types $A_r$, $ B_r$ and $ C_r$.

Pages 863-881 by
From volume 170-2

## An embedded genus-one helicoid

There exists a properly embedded minimal surface of genus one with one end. The end is asymptotic to the end of the helicoid. This genus one helicoid is constructed as the limit of a continuous one-parameter family of screw-motion invariant minimal surfaces—also asymptotic to the helicoid—that have genus equal to one in the quotient.

Pages 347-448 by
From volume 169-2

## On the classification of isoparametric hypersurfaces with four distinct curvatures in spheres

In this paper we give a new proof for the classification result in [3]. We show that isoparametric hypersurfaces with four distinct principal curvatures in spheres are of Clifford type provided that the multiplicities $m_1, m_2$ of the principal curvatures satisfy $m_2 \geq 2m_1 – 1$. This inequality is satisfied for all but five possible pairs $(m_1, m_2)$ with $m_1 \leq m_2$. Our proof implies that for $(m_1, m_2) \neq (1,1)$ the Clifford system may be chosen in such a way that the associated quadratic forms vanish on the higher-dimensional of the two focal manifolds. For the remaining five possible pairs $(m_1, m_2)$ with $m_1 \leq m_2$ (see [13], [1], and [15]) this stronger form of our result is incorrect: for the three pairs $(3,4)$, $(6,9)$, and $(7,8)$ there are examples of Clifford type such that the associated quadratic forms necessarily vanish on the lower-dimensional of the two focal manifolds, and for the two pairs $(2,2)$ and $(4,5)$ there exist homogeneous examples that are not of Clifford type; cf. [5, 4.3, 4.4].

Pages 1011-1024 by
From volume 168-3

## Propagation of singularities for the wave equation on manifolds with corners

In this paper we describe the propagation of ${\mathcal C}^{\infty}$ and Sobolev singularities for the wave equation on ${\mathcal C}^{\infty}$ manifolds with corners $M$ equipped with a Riemannian metric $g$. That is, for $X=M\times\mathbb{R}_t$, $P=D_t^2-\Delta_M$, and $u\in H^1_{\mathrm{loc}}(X)$ solving $Pu=0$ with homogeneous Dirichlet or Neumann boundary conditions, we show that $\mathrm{WF}_{b}(u)$ is a union of maximally extended generalized broken bicharacteristics. This result is a ${\mathcal{C}}^{\infty}$ counterpart of Lebeau’s results for the propagation of analytic singularities on real analytic manifolds with appropriately stratified boundary, [11]. Our methods rely on b-microlocal positive commutator estimates, thus providing a new proof for the propagation of singularities at hyperbolic points even if $M$ has a smooth boundary (and no corners).

Pages 749-812 by
From volume 168-3

## The distribution of integers with a divisor in a given interval

We determine the order of magnitude of $H(x,y,z)$, the number of integers $n\le x$ having a divisor in $(y,z]$, for all $x,y$ and $z$. We also study $H_r(x,y,z)$, the number of integers $n\le x$ having exactly $r$ divisors in $(y,z]$. When $r=1$ we establish the order of magnitude of $H_1(x,y,z)$ for all $x,y,z$ satisfying $z\le x^{1/2-\varepsilon}$. For every $r\ge 2$, $C>1$ and $\varepsilon>0$, we determine the order of magnitude of $H_r(x,y,z)$ uniformly for $y$ large and $y+y/(\log y)^{\log 4 -1 – \varepsilon} \le z \le \min(y^{C},x^{1/2-\varepsilon})$. As a consequence of these bounds, we settle a 1960 conjecture of Erdős and some conjectures of Tenenbaum. One key element of the proofs is a new result on the distribution of uniform order statistics.

Pages 367-433 by
From volume 168-2

## Derived equivalences for symmetric groups and $\mathfrak{s}\mathfrak{l}_2$-categorification

We define and study ${\mathfrak{sl}}_2$-categorifications on abelian categories. We show in particular that there is a self-derived (even homotopy) equivalence categorifying the adjoint action of the simple reflection. We construct categorifications for blocks of symmetric groups and deduce that two blocks are splendidly Rickard equivalent whenever they have isomorphic defect groups and we show that this implies Broué’s abelian defect group conjecture for symmetric groups. We give similar results for general linear groups over finite fields. The constructions extend to cyclotomic Hecke algebras. We also construct categorifications for category $\mathcal{O}$ of $\mathfrak{gl}_n(\mathbf{C})$ and for rational representations of general linear groups over $\bar{\mathbf{F}}_p$, where we deduce that two blocks corresponding to weights with the same stabilizer under the dot action of the affine Weyl group have equivalent derived (and homotopy) categories, as conjectured by Rickard.

Pages 245-298 by
From volume 167-1

## Finding large Selmer rank via an arithmetic theory of local constants

We obtain lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields.

Suppose $K/k$ is a quadratic extension of number fields, $E$ is an elliptic curve defined over $k$, and $p$ is an odd prime. Let $\mathcal{K}^-$ denote the maximal abelian $p$-extension of $K$ that is unramified at all primes where $E$ has bad reduction and that is Galois over $k$ with dihedral Galois group (i.e., the generator $c$ of $\mathrm{Gal}(K/k)$ acts on $\mathrm{Gal}(\mathcal{K}^-/K)$ by inversion). We prove (under mild hypotheses on $p$) that if the $\mathbf{Z}_p$-rank of the pro-$p$ Selmer group $\mathcal{S}_p(E/K)$ is odd, then $\mathrm{rank}_{\mathbf{Z}_p} \mathcal{S}_p(E/F) \ge [F:K]$ for every finite extension $F$ of $K$ in $\mathcal{K}^-$.

Pages 579-612 by
From volume 166-2

## An uncertainty principle for arithmetic sequences

Analytic number theorists usually seek to show that sequences which appear naturally in arithmetic are “well-distributed” in some appropriate sense. In various discrepancy problems, combinatorics researchers have analyzed limitations to equidistribution, as have Fourier analysts when working with the “uncertainty principle”. In this article we find that these ideas have a natural setting in the analysis of distributions of sequences in analytic number theory, formulating a general principle, and giving several examples.

Pages 593-635 by
From volume 165-2

## Weyl group multiple Dirichlet series III: Eisenstein series and twisted unstable $A_r$

*Weyl group multiple Dirichlet series* were associated with a root system $\Phi$ and a number field $F$ containing the $n$-th roots of unity by Brubaker, Bump, Chinta, Friedberg and Hoffstein [3] and Brubaker, Bump and Friedberg [4] provided $n$ is sufficiently large; their coefficients involve $n$-th order Gauss sums. The case where $n$ is small is harder, and is addressed in this paper when $\Phi = A_r$. “Twisted” Dirichet series are considered, which contain the series of [4] as a special case. These series are not Euler products, but due to the twisted multiplicativity of their coefficients, they are determined by their $p$-parts. The $p$-part is given as a sum of products of Gauss sums, parametrized by strict Gelfand-Tsetlin patterns. It is conjectured that these multiple Dirichlet series are Whittaker coefficients of Eisenstein series on the $n$-fold metaplectic cover of $\mathrm{GL}_{r + 1}$, and this is proved if $r = 2$ or $n = 1$. The equivalence of our definition with that of Chinta [11] when $n = 2$ and $r \leqslant 5$ is also established.

Pages 293-316 by
From volume 166-1

## Stability of mixing and rapid mixing for hyperbolic flows

We obtain general results on the stability of mixing and rapid mixing (superpolynomial decay of correlations) for hyperbolic flows. Amongst $C^r$ Axiom A flows, $r\ge2$, we show that there is a $C^2$-open, $C^r$-dense set of flows for which each nontrivial hyperbolic basic set is rapid mixing. This is the first general result on the stability of rapid mixing (or even mixing) for Axiom A flows that holds in a $C^r$, as opposed to Hölder, topology.

Pages 269-291 by
From volume 166-1

## On finitely generated profinite groups, II: products in quasisimple groups

We prove two results. (1) There is an absolute constant $D$ such that for any finite quasisimple group $S$, given $2D$ arbitrary automorphisms of $S$, every element of $S$ is equal to a product of $D$ ‘twisted commutators’ defined by the given automorphisms.

(2) Given a natural number $q$, there exist $C=C(q)$ and $M=M(q)$ such that: if $S$ is a finite quasisimple group with $\left| S/\mathrm{Z}(S)\right| > C$, $\beta_{j}$ $ (j=1,\ldots,M)$ are any automorphisms of $S$, and $q_{j}$ $ (j=1,\ldots,M)$ are any divisors of $q$, then there exist inner automorphisms $\alpha_{j}$ of $S$ such that $S=\prod_{1}^{M}[S,(\alpha _{j}\beta_{j})^{q_{j}}]$.

These results, which rely on the classification of finite simple groups, are needed to complete the proofs of the main theorems of Part I.

Pages 239-273 by
From volume 165-1

## On finitely generated profinite groups, I: strong completeness and uniform bounds

We prove that in every finitely generated profinite group, every subgroup of finite index is open; this implies that the topology on such groups is determined by the algebraic structure. This is deduced from the main result about finite groups: let $w$ be a `locally finite’ group word and $d\in\mathbb{N}$. Then there exists $f=f(w,d)$ such that in every $d$-generator finite group $G$, every element of the verbal subgroup $w(G)$ is equal to a product of $f$ $w$-values.

An analogous theorem is proved for commutators; this implies that in every finitely generated profinite group, each term of the lower central series is closed.

The proofs rely on some properties of the finite simple groups, to be established in Part II.

Pages 171-238 by
From volume 165-1

## Quantum Riemann–Roch, Lefschetz and Serre

Given a holomorphic vector bundle $E$ over a compact Kähler manifold $X$, one defines twisted Gromov-Witten invariants of $X$ to be intersection numbers in moduli spaces of stable maps $f:\Sigma \to X$ with the cap product of the virtual fundamental class and a chosen multiplicative invertible characteristic class of the virtual vector bundle $H^0(\Sigma,f^* E) \ominus H^1(\Sigma,f^* E)$. Using the formalism of quantized quadratic Hamiltonians [25], we express the descendant potential for the twisted theory in terms of that for $X$. This result (Theorem $1$) is a consequence of Mumford’s Grothendieck-Riemann-Roch theorem applied to the universal family over the moduli space of stable maps. It determines all twisted Gromov-Witten invariants, of all genera, in terms of untwisted invariants.

When $E$ is concave and the $\mathbb{C}^\times$-equivariant inverse Euler class is chosen as the characteristic class, the twisted invariants of $X$ give Gromov-Witten invariants of the total space of $E$. “Nonlinear Serre duality” [21], [23] expresses Gromov-Witten invariants of $E$ in terms of those of the super-manifold $\Pi E$: it relates Gromov-Witten invariants of $X$ twisted by the inverse Euler class and $E$ to Gromov-Witten invariants of $X$ twisted by the Euler class and $E^*$. We derive from Theorem 1 nonlinear Serre duality in a very general form (Corollary 2).

When the bundle $E$ is convex and a submanifold $Y\subset X$ is defined by a global section of $E$, the genus-zero Gromov-Witten invariants of $\Pi E$ coincide with those of $Y$. We establish a “quantum Lefschetz hyperplane section principle” (Theorem $2$) expressing genus-zero Gromov-Witten invariants of a complete intersection $Y$ in terms of those of $X$. This extends earlier results [4], [9], [18], [29], [33] and yields most of the known mirror formulas for toric complete intersections.

Pages 15-53 by
From volume 165-1

## Isoparametric hypersurfaces with four principal curvatures

Let $M$ be an isoparametric hypersurface in the sphere $S^n$ with four distinct principal curvatures. Münzner showed that the four principal curvatures can have at most two distinct multiplicities $m_1, m_2$, and Stolz showed that the pair $(m_1,m_2)$ must either be $(2,2)$, $(4,5)$, or be equal to the multiplicities of an isoparametric hypersurface of FKM-type, constructed by Ferus, Karcher and Münzner from orthogonal representations of Clifford algebras. In this paper, we prove that if the multiplicities satisfy $m_2 \geq 2m_1 – 1$, then the isoparametric hypersurface $M$ must be of FKM-type. Together with known results of Takagi for the case $m_1 = 1$, and Ozeki and Takeuchi for $m_1 = 2$, this handles all possible pairs of multiplicities except for four cases, for which the classification problem remains open.

Pages 1-76 by
From volume 166-1