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.

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

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

## 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

## Complex multiplication cycles and Kudla-Rapoport divisors

We study the intersections of special cycles on a unitary Shimura variety of signature $(n-1,1)$ and show that the intersection multiplicities of these cycles agree with Fourier coefficients of Eisenstein series. The results are new cases of conjectures of Kudla and suggest a Gross-Zagier theorem for unitary Shimura varieties.

Pages 1097-1171 by
From volume 176-2

## Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory

Generalizing the notion of Newton polytope, we define the Newton-Okounkov body, respectively, for semigroups of integral points, graded algebras and linear series on varieties. We prove that any semigroup in the lattice $\mathbb{Z}^n$ is asymptotically approximated by the semigroup of all the points in a sublattice and lying in a convex cone. Applying this we obtain several results. We show that for a large class of graded algebras, the Hilbert functions have polynomial growth and their growth coefficients satisfy a Brunn-Minkowski type inequality. We prove analogues of the Fujita approximation theorem for semigroups of integral points and graded algebras, which imply a generalization of this theorem for arbitrary linear series. Applications to intersection theory include a far-reaching generalization of the Kushnirenko theorem (from Newton polytope theory) and a new version of the Hodge inequality. We also give elementary proofs of the Alexandrov-Fenchel inequality in convex geometry and its analogue in algebraic geometry.

Pages 925-978 by
From volume 176-2

## Deforming three-manifolds with positive scalar curvature

In this paper we prove that the moduli space of metrics with positive scalar curvature of an orientable compact three-manifold is path-connected. The proof uses the Ricci flow with surgery, the conformal method, and the connected sum construction of Gromov and Lawson. The work of Perelman on Hamilton’s Ricci flow is fundamental. As one of the applications we prove the path-connectedness of the space of trace-free asymptotically flat solutions to the vacuum Einstein constraint equations on $\mathbb{R}^3$.

Pages 815-863 by
From volume 176-2

## Mixed Tate motives over $\mathbb{Z}$

We prove that the category of mixed Tate motives over $\mathbb{Z}$ is spanned by the motivic fundamental group of $\mathbb{P}^1$ minus three points. We prove a conjecture by M. Hoffman which states that every multiple zeta value is a $\mathbb{Q}$-linear combination of $\zeta(n_1,\ldots, n_r)$, where $n_i\in \{2,3\}$.

Pages 949-976 by
From volume 175-2

## The Weil-Petersson geodesic flow is ergodic

We prove that the geodesic flow for the Weil-Petersson metric on the moduli space of Riemann surfaces is ergodic (and in fact Bernoulli) and has finite, positive metric entropy.

Pages 835-908 by
From volume 175-2

## The Borel Conjecture for hyperbolic and CAT(0)-groups

We prove the Borel Conjecture for a class of groups containing word-hyperbolic groups and groups acting properly, isometrically and cocompactly on a finite-dimensional CAT(0)-space.

Pages 631-689 by
From volume 175-2

## Finiteness of central configurations of five bodies in the plane

We prove there are finitely many isometry classes of planar central configurations (also called relative equilibria) in the Newtonian 5-body problem, except perhaps if the 5-tuple of positive masses belongs to a given codimension 2 subvariety of the mass space.

Pages 535-588 by
From volume 176-1

## Rational points over finite fields for regular models of algebraic varieties of Hodge type $\geq 1$

Let $R$ be a discrete valuation ring of mixed characteristics $(0,p)$, with finite residue field $k$ and fraction field $K$, let $k’$ be a finite extension of $k$, and let $X$ be a regular, proper and flat $R$-scheme, with generic fibre $X_K$ and special fibre $X_k$. Assume that $X_K$ is geometrically connected and of Hodge type $\geq 1$ in positive degrees. Then we show that the number of $k’$-rational points of $X$ satisfies the congruence $|X(k’)| \equiv 1$ mod $|k’|$. We deduce such congruences from a vanishing theorem for the Witt cohomology groups $H^q(X_k, W\mathcal{O}_{X_k,\mathbb{Q}})$ for $q > 0$. In our proof of this last result, a key step is the construction of a trace morphism between the Witt cohomologies of the special fibres of two flat regular $R$-schemes $X$ and $Y$ of the same dimension, defined by a surjective projective morphism $f : Y \to X$.

Pages 413-508 by
From volume 176-1

## Submultiplicativity and the Hanna Neumann Conjecture

In this article, we define *submultiplicativity* of $\ell^2$-numbers in the category of $\Gamma$-complexes over a given $\Gamma$-complex $\hat{X}$, which generalizes the statement of the Strengthened Hanna Neumann Conjecture (SHNC). In the case when $\Gamma$ is a left-orderable group and $\hat{X}$ is a free $\Gamma$-complex, we prove submultiplicativity for the subcategory consisting of $\Gamma$-ordered *leafages* over $\hat{X}$ with an additional analytic assumption called the *deep-fall property*. We show that the deep-fall property is satisfied for graphs. This implies SHNC.

Pages 393-414 by
From volume 175-1

## Rational points near manifolds and metric Diophantine approximation

This work is motivated by problems on simultaneous Diophantine approximation on manifolds, namely, establishing Khintchine and Jarník type theorems for submanifolds of $\mathbb{R}^n$. These problems have attracted a lot of interest since Kleinbock and Margulis proved a related conjecture of Alan Baker and V. G. Sprindžuk. They have been settled for planar curves but remain open in higher dimensions. In this paper, Khintchine and Jarník type divergence theorems are established for arbitrary analytic nondegenerate manifolds regardless of their dimension. The key to establishing these results is the study of the distribution of rational points near manifolds — a very attractive topic in its own right. Here, for the first time, we obtain sharp lower bounds for the number of rational points near nondegenerate manifolds in dimensions $n>2$ and show that they are ubiquitous (that is uniformly distributed).

Pages 187-235 by
From volume 175-1

## The local Langlands conjecture for GSp(4)

We prove the local Langlands conjecture for $\mathrm{GSp}_4(F)$ where $F$ is a non-archimedean local field of characteristic zero.

Pages 1841-1882 by
From volume 173-3

## On a problem posed by Steve Smale

The 17th of the problems proposed by Steve Smale for the 21st century asks for the existence of a deterministic algorithm computing an approximate solution of a system of $n$ complex polynomials in $n$ unknowns in time polynomial, on the average, in the size $N$ of the input system. A partial solution to this problem was given by Carlos Beltrán and Luis Miguel Pardo who exhibited a randomized algorithm doing so. In this paper we further extend this result in several directions. Firstly, we exhibit a linear homotopy algorithm that efficiently implements a nonconstructive idea of Mike Shub. This algorithm is then used in a randomized algorithm, call it

Pages 1785-1836 by
From volume 174-3

## The conjugacy problem in ergodic theory

All common probability preserving transformations can be represented as elements of MPT, the group of measure preserving transformations of the unit interval with Lebesgue measure. This group has a natural Polish topology and the induced topology on the set of ergodic transformations is also Polish. Our main result is that the set of ergodic elements $T$ in MPT that are isomorphic to their inverse is a complete analytic set. This has as a consequence the fact that the isomorphism relation is also a complete analytic set and in particular is not Borel. This is in stark contrast to the situation of unitary operators where the spectral theorem can be used to show that conjugacy relation in the unitary group is Borel.

This result explains, perhaps, why the problem of determining whether ergodic transformations are isomorphic or not has proven to be so intractable. The construction that we use is general enough to show that the set of ergodic $T$’s with nontrivial centralizer is also complete analytic.

On the positive side we show that the isomorphism relation is Borel when restricted to the rank one transformations, which form a generic subset of MPT. It remains an open problem to find a good explicit method of checking when two rank one transformations are isomorphic.

Pages 1529-1586 by
From volume 173-3

## Stratifying modular representations of finite groups

We classify localising subcategories of the stable module category of a finite group that are closed under tensor product with simple (or, equivalently all) modules. One application is a proof of the telescope conjecture in this context. Others include new proofs of the tensor product theorem and of the classification of thick subcategories of the finitely generated modules which avoid the use of cyclic shifted subgroups. Along the way we establish similar classifications for differential graded modules over graded polynomial rings, and over graded exterior algebras.

Pages 1643-1684 by
From volume 174-3

## Global Schrödinger maps in dimensions $d≥ 2$: Small data in the critical Sobolev spaces

We consider the Schrödinger map initial-value problem $$\cases{ \partial_t\phi=\phi\times\Delta \phi &\text{on } \mathbb{R}^d\times\mathbb{R},\cr

\phi(0)=\phi_0, &{} }$$ where $\phi\colon\mathbb{R}^d\times\mathbb{R}\to\mathbb{S}^2\hookrightarrow \mathbb{R}^3$ is a smooth function. In all dimensions $d\geq 2$, we prove that the Schrödinger map initial-value problem admits a unique global smooth solution $\phi\in C(\mathbb{R}:H^\infty_Q)$, $Q\in\mathbb{S}^2$, provided that the data $\phi_0\in H^\infty_Q$ is smooth and satisfies the smallness condition $\|\phi_0-Q\|_{\dot{H}^{d/2}}\ll 1$. We prove also that the solution operator extends continuously to the space of data in $\dot H^{d/2}\cap \dot H^{d/2-1}_Q$ with small $\dot{H}^{d/2}$ norm.

Pages 1443-1506 by
From volume 173-3

## On the structure of the Selberg class, VII: $1\lt d\lt 2$

The Selberg class $\mathcal{S}$ is a rather general class of Dirichlet series with functional equation and Euler product and can be regarded as an axiomatic model for the global $L$-functions arising from number theory and automorphic representations. One of the main problems of the Selberg class theory is to classify the elements of $\mathcal{S}$. Such a classification is based on a real-valued invariant $d$ called degree, and the degree conjecture asserts that $d\in\mathbb{N}$ for every $L$-function in $\mathcal{S}$. The degree conjecture has been proved for $d\lt 5/3$, and in this paper we extend its validity to $d\lt 2$. The proof requires several new ingredients, in particular a rather precise description of the properties of certain nonlinear twists associated with the $L$-functions in $\mathcal{S}$.

Pages 1397-1441 by
From volume 173-3

## The link between the shape of the irrational Aubry-Mather sets and their Lyapunov exponents

We consider the irrational Aubry-Mather sets of an exact symplectic monotone $C^1$ twist map of the two-dimensional annulus, introduce for them a notion of “$C^1$-regularity” (related to the notion of Bouligand paratingent cone) and prove that

- $\bullet$ a Mather measure has zero Lyapunov exponents if and only if its support is $C^1$-regular almost everywhere;
- $\bullet$ a Mather measure has nonzero Lyapunov exponents if and only if its support is $C^1$-irregular almost everywhere;
- $\bullet$ an Aubry-Mather set is uniformly hyperbolic if and only if it is irregular everywhere;
- $\bullet$ the Aubry-Mather sets which are close to the KAM invariant curves, even if they may be $C^1$-irregular, are not “too irregular” (i.e., have small paratingent cones).

The main tools that we use in the proofs are the so-called Green bundles.

Pages 1571-1601 by
From volume 174-3

## On De Giorgi’s conjecture in dimension $N\ge 9$

A celebrated conjecture due to De Giorgi states that any bounded solution of the equation $\Delta u + (1-u^2) u = 0 \hbox{in} \mathbb{R}^N $ with $\partial_{y_N}u >0$ must be such that its level sets $\{u=\lambda\}$ are all hyperplanes, *at least* for dimension $N\le 8$. A counterexample for $N\ge 9$ has long been believed to exist. Starting from a minimal graph $\Gamma$ which is not a hyperplane, found by Bombieri, De Giorgi and Giusti in $\Bbb{R}^N$, $N\ge 9$, we prove that for any small $\alpha >0$ there is a bounded solution $u_\alpha(y)$ with $\partial_{y_N}u_\alpha >0$, which resembles $ \tanh \left ( \frac t{\sqrt{2}}\right ) $, where $t=t(y)$ denotes a choice of signed distance to the blown-up minimal graph $\Gamma_\alpha := \alpha^{-1}\Gamma$. This solution is a counterexample to De Giorgi’s conjecture for $N\ge 9$.

Pages 1485-1569 by
From volume 174-3

## Representations of Yang-Mills algebras

The aim of this article is to describe families of representations of the Yang-Mills algebras $\mathrm{YM}(n)$ ($n \in \mathbb{N}_{\geq 2}$) defined by A. Connes and M. Dubois-Violette. We first describe some irreducible finite dimensional representations. Next, we provide families of infinite dimensional representations of $\mathrm{YM}$, big enough to separate points of the algebra. In order to prove this result, we prove and use that all Weyl algebras $A_{r}(k)$ are epimorphic images of $\mathrm{YM}(n)$.

Pages 1043-1080 by
From volume 173-2

## Representation theoretic patterns in three dimensional Cryo-Electron Microscopy I: The intrinsic reconstitution algorithm

In this paper, we reveal the formal algebraic structure underlying the intrinsic reconstitution algorithm, introduced by Singer and Shkolnisky in [S], for determining three dimensional macromolecular structures from images obtained by an electron microscope. Inspecting this algebraic structure, we obtain a conceptual explanation for the admissibility (correctness) of the algorithm and a proof of its numerical stability. In addition, we explain how the various numerical observations reported in that work follow from basic representation theoretic principles.

Pages 1219-1241 by
From volume 174-2

## Livšic Theorem for matrix cocycles

We prove the Livšic Theorem for arbitrary $\mathrm{GL}(m,\Bbb{R})$ cocycles. We consider a hyperbolic dynamical system $f : X \to X$ and a Hölder continuous function $A: X \to \mathrm{GL}(m,\Bbb{R})$. We show that if $A$ has trivial periodic data, i.e. $A(f^{n-1} p) \cdots A(fp) A(p)$ $= \mathrm{Id}$ for each periodic point $p=f^n p$, then there exists a Hölder continuous function $C: X \to \mathrm{GL}(m,\Bbb{R})$ satisfying $A (x) = C(f x) C(x) ^{-1}$ for all $x \in X$. The main new ingredients in the proof are results of independent interest on relations between the periodic data, Lyapunov exponents, and uniform estimates on growth of products along orbits for an arbitrary Hölder function $A$.

Pages 1025-1042 by
From volume 173-2

## The single ring theorem

We study the empirical measure $L_{A_n}$ of the eigenvalues of nonnormal square matrices of the form $A_n=U_nT_nV_n$ with $U_n,V_n$ independent Haar distributed on the unitary group and $T_n$ real diagonal. We show that when the empirical measure of the eigenvalues of $T_n$ converges, and $T_n$ satisfies some technical conditions, $L_{A_n}$ converges towards a rotationally invariant measure $\mu$ on the complex plane whose support is a single ring. In particular, we provide a complete proof of the Feinberg-Zee single ring theorem [FZ]. We also consider the case where $U_n,V_n$ are independently Haar distributed on the orthogonal group.

Pages 1189-1217 by
From volume 174-2

## Asymptotics of characters of symmetric groups related to Stanley character formula

We prove an upper bound for characters of the symmetric groups. In particular, we show that there exists a constant $a>0$ with a property that for every Young diagram $\lambda$ with $n$ boxes, $r(\lambda)$ rows and $c(\lambda)$ columns $$ \left| \frac{\mathrm{Tr}\, \rho^{\lambda}(\pi)}{\mathrm{Tr}\, \rho^{\lambda}(e)} \right| \leq \left[a \max\left(\frac{r(\lambda)}{n},\frac{c(\lambda)}{n},\frac{|\pi|}{n} \right)\right]^{|\pi|}, $$ where $|\pi|$ is the minimal number of factors needed to write $\pi\in S_n$ as a product of transpositions. We also give uniform estimates for the error term in the Vershik-Kerov’s and Biane’s character formulas and give a new formula for free cumulants of the transition measure.

Pages 887-906 by
From volume 173-2

## Loop groups and twisted $K$-theory III

In this paper, we identify the Ad-equivariant twisted $K$-theory of a compact Lie group $G$ with the “Verlinde group” of isomorphism classes of admissible representations of its loop groups. Our identification preserves natural module structures over the representation ring $R(G)$ and a natural duality pairing. Two earlier papers in the series covered foundations of twisted equivariant $K$-theory, introduced distinguished families of Dirac operators and discussed the special case of connected groups with free $\pi_1$. Here, we recall the earlier material as needed to make the paper self-contained. Going further, we discuss the relation to semi-infinite cohomology, the fusion product of conformal field theory, the rôle of energy and a topological Peter-Weyl theorem.

Pages 947-1007 by
From volume 174-2

## A nearly-optimal method to compute the truncated theta function, its derivatives, and integrals

A poly-log time method to compute the truncated theta function, its derivatives, and integrals is presented. The method is elementary, rigorous, explicit, and suited for computer implementation. We repeatedly apply the Poisson summation formula to the truncated theta function while suitably normalizing the linear and quadratic arguments after each repetition. The method relies on the periodicity of the complex exponential, which enables the suitable normalization of the arguments, and on the self-similarity of the Gaussian, which ensures that we still obtain a truncated theta function after each application of the Poisson summation. In other words, our method relies on modular properties of the theta function. Applications to the numerical computation of the Riemann zeta function and to finding the number of solutions of Waring type Diophantine equations are discussed.

Pages 859-889 by
From volume 174-2

## 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

## Grothendieck rings of basic classical Lie superalgebras

The Grothendieck rings of finite dimensional representations of the basic classical Lie superalgebras are explicitly described in terms of the corresponding generalized root systems. We show that they can be interpreted as the subrings in the weight group rings invariant under the action of certain groupoids called super Weyl groupoids.

Pages 663-703 by
From volume 173-2

## All automorphisms of the Calkin algebra are inner

We prove that it is relatively consistent with the usual axioms of mathematics that all automorphisms of the Calkin algebra are inner. Together with a 2006 Phillips-Weaver construction of an outer automorphism using the Continuum Hypothesis, this gives a complete solution to a 1977 problem of Brown-Douglas-Fillmore. We also give a simpler and self-contained proof of the Phillips-Weaver result.

Pages 619-661 by
From volume 173-2

## On Roth’s theorem on progressions

We show that if $A \subset \{1,\dots,N\}$ contains no nontrivial three-term arithmetic progressions then $|A|=O(N/\log^{1-o(1)}N)$.

Pages 619-636 by
From volume 174-1

## The Boltzmann-Grad limit of the periodic Lorentz gas

We study the dynamics of a point particle in a periodic array of spherical scatterers and construct a stochastic process that governs the time evolution for random initial data in the limit of low scatterer density (Boltzmann-Grad limit). A generic path of the limiting process is a piecewise linear curve whose consecutive segments are generated by a Markov process with memory two.

Pages 225-298 by
From volume 174-1

## Analyticity of periodic traveling free surface water waves with vorticity

We prove that the profile of a periodic traveling wave propagating at the surface of water above a flat bed in a flow with a real analytic vorticity must be real analytic, provided the wave speed exceeds the horizontal fluid velocity throughout the flow. The real analyticity of each streamline beneath the free surface holds even if the vorticity is only Hölder continuously differentiable.

Pages 559-568 by
From volume 173-1

## A reciprocity map and the two-variable $p$-adic $L$-function

For primes $p \ge 5$, we propose a conjecture that relates the values of cup products in the Galois cohomology of the maximal unramified outside $p$ extension of a cyclotomic field on cyclotomic $p$-units to the values of $p$-adic $L$-functions of cuspidal eigenforms that satisfy mod $p$ congruences with Eisenstein series. Passing up the cyclotomic and Hida towers, we construct an isomorphism of certain spaces that allows us to compare the value of a reciprocity map on a particular norm compatible system of $p$-units to what is essentially the two-variable $p$-adic $L$-function of Mazur and Kitagawa.

Pages 251-300 by
From volume 173-1

## Counting arithmetic lattices and surfaces

We give estimates on the number $\operatorname{AL}_H(x)$ of conjugacy classes of arithmetic lattices $\Gamma$ of covolume at most $x$ in a simple Lie group $H$. In particular, we obtain a first concrete estimate on the number of arithmetic $3$-manifolds of volume at most $x$. Our main result is for the classical case $H=\operatorname{PSL}(2,\mathbb{R})$ where we show that \[ \lim_{x\to\infty}\frac{\log \operatorname{AL}_H(x)}{x\log x}=\frac{1}{2\pi}. \] The proofs use several different techniques: geometric (bounding the number of generators of $\Gamma$ as a function of its covolume), number theoretic (bounding the number of maximal such $\Gamma$) and sharp estimates on the character values of the symmetric groups (to bound the subgroup growth of $\Gamma$).

Pages 2197-2221 by
From volume 172-3

## Measure equivalence rigidity of the mapping class group

We show that the mapping class group of a compact orientable surface with higher complexity satisfies the following rigidity in the sense of measure equivalence: If the mapping class group is measure equivalent to a discrete group, then they are commensurable up to finite kernels. Moreover, we describe all locally compact second countable groups containing a lattice isomorphic to the mapping class group. We obtain similar results for finite direct products of mapping class groups.

Pages 1851-1901 by
From volume 171-3

## The Atiyah-Singer index formula for subelliptic operators on contact manifolds. Part II

We present a new solution to the index problem for hypoelliptic operators in the Heisenberg calculus on contact manifolds, by constructing the appropriate topological $K$-theory cocycle for such operators. Its Chern character gives a cohomology class to which the Atiyah-Singer index formula can be applied. Such a $K$-cocycle has already been constructed by Boutet de Monvel for Toeplitz operators, and, more recently, by Melrose and Epstein for the class of Hermite operators. Our construction applies to general hypoelliptic pseudodifferential operators in the Heisenberg calculus. As in the Hermite Index Formula of Melrose and Epstein, our construction gives a vector bundle automorphism of the symmetric tensors of the contact hyperplane bundle. This automorphism is constructed directly from the invertible Heisenberg symbol of the operator, and is easily computed in the case of differential operators.

Pages 1683-1706 by
From volume 171-3

## The Atiyah-Singer index formula for subelliptic operators on contact manifolds. Part I

The Atiyah-Singer index theorem gives a topological formula for the index of an elliptic differential operator. The topological index depends on a cohomology class that is constructed from the principal symbol of the operator. On contact manifolds, the important Fredholm operators are not elliptic, but hypoelliptic. Their symbolic calculus is noncommutative, and is closely related to analysis on the Heisenberg group. For a hypoelliptic differential operator in the Heisenberg calculus on a contact manifold we construct a symbol class in the $K$-theory of a noncommutative $C^*$-algebra that is associated to the algebra of symbols. There is a canonical map from this analytic $K$-theory group to the ordinary cohomology of the manifold, which gives a de Rham class to which the Atiyah-Singer formula can be applied. We prove that the index formula holds for these hypoelliptic operators. Our methods derive from Connes’ tangent groupoid proof of the index theorem.

Pages 1647-1681 by
From volume 171-3

## Subconvexity bounds for triple $L$-functions and representation theory

We describe a new method to estimate the trilinear period on automorphic representations of $\operatorname{PGL}_2(\mathbb{R})$. Such a period gives rise to a special value of the triple $L$-function. We prove a bound for the triple period which amounts to a subconvexity bound for the corresponding special value of the triple $L$-function. Our method is based on the study of the analytic structure of the corresponding unique trilinear functional on unitary representations of $\operatorname{PGL}_2(\mathbb{R})$.

Pages 1679-1718 by
From volume 172-3

## Divergent square averages

In this paper we answer a question of J. Bourgain which was motivated by questions A. Bellow and H. Furstenberg. We show that the sequence $\{ n^{2}\}_{n=1}^{\infty}$ is $L^{1}$-universally bad. This implies that it is not true that given a dynamical system $(X ,\Sigma, \mu, T)$ and $f\in L^{1}(\mu)$, the ergodic means \[ \lim_{N\to \infty}\frac{1}N\sum _{n=1}^{N}f(T^{n^{2}}(x)) \] converge almost surely.

Pages 1479-1530 by
From volume 171-3

## The global stability of Minkowski space-time in harmonic gauge

We give a new proof of the global stability of Minkowski space originally established in the vacuum case by Christodoulou and Klainerman. The new approach, which relies on the classical harmonic gauge, shows that the Einstein-vacuum and the Einstein-scalar field equations with asymptotically flat initial data satisfying a global smallness condition produce global (causally geodesically complete) solutions asymptotically convergent to the Minkowski space-time.

Pages 1401-1477 by
From volume 171-3

## A classification of $\operatorname{SL}(n)$ invariant valuations

A classification of upper semicontinuous and $\operatorname{SL}(n)$ invariant valuations on the space of $n$-dimensional convex bodies is established. As a consequence, complete characterizations of centro-affine and $L_p$ affine surface areas are obtained. The proofs make use of a new $\operatorname{SL}(n)$ shaping process for convex bodies.

Pages 1219-1267 by
From volume 172-2

## 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

## Random conformal snowflakes

In many problems of classical analysis extremal configurations appear to exhibit complicated fractal structures, making it hard to describe them and to attack such problems. This is particularly true for questions related to the *multifractal analysis* of *harmonic measure*. We argue that, searching for extremals in such problems, one should work with random fractals rather than deterministic ones. We introduce a new class of fractals: *random conformal snowflakes*, and investigate their properties, developing tools to estimate spectra and showing that extremals can be found in this class. As an application we significantly improve known estimates from below on the extremal behavior of harmonic measure, showing how to construct a rather simple snowflake, which has a spectrum quite close to the conjectured extremal value.

Pages 597-615 by
From volume 172-1

## Dyson’s ranks and Maass forms

Motivated by work of Ramanujan, Freeman Dyson defined the rank of an integer partition to be its largest part minus its number of parts. If $N(m,n)$ denotes the number of partitions of $n$ with rank $m$, then it turns out that \[ R(w;q):=1+\!\sum_{n=1}^{\infty}\sum_{m=-\infty}^{\infty} \!\!\! N(m,n)w^mq^n \! =\! 1+\!\sum_{n=1}^{\infty}\!\frac{q^{n^2}} {\prod_{j=1}^{n}(1\!-\!(w\!+\!w^{-1})q^j\!+ q^{2j})}. \] We show that if $\zeta\neq 1$ is a root of unity, then $R(\zeta;q)$ is essentially the holomorphic part of a weight $1/2$ weak Maass form on a subgroup of $\operatorname{SL}_2(\mathbb Z)$. For integers $0\leq r\lt t$, we use this result to determine the modularity of the generating function for $N(r,t;n)$, the number of partitions of $n$ whose rank is congruent to $r\pmod t$. We extend the modularity above to construct an infinite family of vector valued weight $1/2$ forms for the full modular group $\operatorname{SL}_2(\mathbb Z)$, a result which is of independent interest.

Pages 419-449 by
From volume 171-1

## 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

## Donaldson-Thomas type invariants via microlocal geometry

We prove that Donaldson-Thomas type invariants are equal to weighted Euler characteristics of their moduli spaces. In particular, such invariants depend only on the scheme structure of the moduli space, not the symmetric obstruction theory used to define them. We also introduce new invariants generalizing Donaldson-Thomas type invariants to moduli problems with open moduli space. These are useful for computing Donaldson-Thomas type invariants over stratifications.

Pages 1307-1338 by
From volume 170-3

## Strong cosmic censorship in $T^{3}$-Gowdy spacetimes

Einstein’s vacuum equations can be viewed as an initial value problem, and given initial data there is one part of spacetime, the so-called maximal globally hyperbolic development (MGHD), which is uniquely determined up to isometry. Unfortunately, it is sometimes possible to extend the spacetime beyond the MGHD in inequivalent ways. Consequently, the initial data do not uniquely determine the spacetime, and in this sense the theory is not deterministic. It is then natural to make the strong cosmic censorship conjecture, which states that for generic initial data, the MGHD is inextendible. Since it is unrealistic to hope to prove this conjecture in all generality, it is natural to make the same conjecture within a class of spacetimes satisfying some symmetry condition. Here, we prove strong cosmic censorship in the class of $T^{3}$-Gowdy spacetimes. In a previous paper, we introduced a set $\mathcal{G}_{i,c}$ of smooth initial data and proved that it is open in the $C^{1}\times C^{0}$-topology. The solutions corresponding to initial data in $\mathcal{G}_{i,c}$ have the following properties. First, the MGHD is $C^{2}$-inextendible. Second, following a causal geodesic in a given time direction, it is either complete, or a curvature invariant, the Kretschmann scalar, is unbounded along it (in fact the Kretschmann scalar is unbounded along any causal curve that ends on the singularity). The purpose of the present paper is to prove that $\mathcal{G}_{i,c}$ is dense in the $C^{\infty}$-topology.

Pages 1181-1240 by
From volume 170-3

## A new upper bound for diagonal Ramsey numbers

We prove a new upper bound for diagonal two-colour Ramsey numbers, showing that there exists a constant $C$ such that \[r(k+1, k+1) \leq k^{- C {\log k}/{\log \log k}} \textstyle \binom{2k}{k}.\]

Pages 941-960 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

## Generalizations of Siegel’s and Picard’s theorems

We prove new theorems that are higher-dimensional generalizations of the classical theorems of Siegel on integral points on affine curves and of Picard on holomorphic maps from $\mathbb{C}$ to affine curves. These include results on integral points over varying number fields of bounded degree and results on Kobayashi hyperbolicity. We give a number of new conjectures describing, from our point of view, how we expect Siegel’s and Picard’s theorems to optimally generalize to higher dimensions.

Pages 609-655 by
From volume 170-2

## Cubic structures, equivariant Euler characteristics and lattices of modular forms

We use the theory of cubic structures to give a fixed point Riemann-Roch formula for the equivariant Euler characteristics of coherent sheaves on projective flat schemes over $\mathbb{Z}$ with a tame action of a finite abelian group. This formula supports a conjecture concerning the extent to which such equivariant Euler characteristics may be determined from the restriction of the sheaf to an infinitesimal neighborhood of the fixed point locus. Our results are applied to study the module structure of modular forms having Fourier coefficients in a ring of algebraic integers, as well as the action of diamond Hecke operators on the Mordell-Weil groups and Tate-Shafarevich groups of Jacobians of modular curves.

Pages 561-608 by
From volume 170-2