We establish a general structure theorem for the singular part of ${\mathscr A}$-free Radon measures, where ${\mathscr A}$ is a linear PDE operator. By applying the theorem to suitably chosen differential operators ${\mathscr A}$, we obtain a simple proof of Alberti’s rank-one theorem and, for the first time, its extensions to functions of bounded deformation (BD). We also prove a structure theorem for the singular part of a finite family of normal currents. The latter result implies that the Rademacher theorem on the differentiability of Lipschitz functions can hold only for absolutely continuous measures and that every top-dimensional Ambrosio–Kirchheim metric current in $\mathbb R^d$ is a Federer–Fleming flat chain.
On the structure of ${\mathscr A}$-free measures and applications
Pages 1017-1039 by
From volume 184-3
Multiplicative functions in short intervals
We introduce a general result relating “short averages” of a multiplicative function to “long averages” which are well understood. This result has several consequences. First, for the Möbius function we show that there are cancellations in the sum of $\mu(n)$ in almost all intervals of the form $[x, x + \psi(x)]$ with $\psi(x) \rightarrow \infty$ arbitrarily slowly. This goes beyond what was previously known conditionally on the Density Hypothesis or the stronger Riemann Hypothesis. Second, we settle the long-standing conjecture on the existence of $x^{\varepsilon}$-smooth numbers in intervals of the form $[x, x + c(\varepsilon) \sqrt{x}]$, recovering unconditionally a conditional (on the Riemann Hypothesis) result of Soundararajan. Third, we show that the mean-value of $\lambda(n)\lambda(n+1)$, with $\lambda(n)$ Liouville’s function, is nontrivially bounded in absolute value by $1 – \delta$ for some $\delta > 0$. This settles an old folklore conjecture and constitutes progress towards Chowla’s conjecture. Fourth, we show that a (general) real-valued multiplicative function $f$ has a positive proportion of sign changes if and only if $f$ is negative on at least one integer and nonzero on a positive proportion of the integers. This improves on many previous works and is new already in the case of the Möbius function. We also obtain some additional results on smooth numbers in almost all intervals, and sign changes of multiplicative functions in all intervals of square-root length.\looseness=-1
Pages 1015-1056 by
From volume 183-3
Characters of odd degree
We prove the McKay conjecture on characters of odd degree. A major step in the proof is the verification of the inductive McKay condition for groups of Lie type and primes $\ell$ such that a Sylow $\ell$-subgroup or its maximal normal abelian subgroup is contained in a maximally split torus by means of a new equivariant version of Harish-Chandra induction. Specifics of characters of odd degree, namely, that most of them lie in the principal Harish-Chandra series, then allow us to deduce from it the McKay conjecture for the prime $2$, hence for characters of odd degree.
Pages 869-908 by
From volume 184-3
Anabelian geometry with étale homotopy types
Anabelian geometry with étale homotopy types generalizes in a natural way classical anabelian geometry with étale fundamental groups. We show that, both in the classical and the generalized sense, any point of a smooth variety over a field $k$ that is finitely generated over $\mathbb{Q}$ has a fundamental system of (affine) anabelian Zariski-neighborhoods. This was predicted by Grothendieck in his letter to Faltings.
Pages 817-868 by
From volume 184-3
The clique density theorem
Turán’s theorem is a cornerstone of extremal graph theory. It asserts that for any integer $r \geqslant 2$, every graph on $n$ vertices with more than ${\tfrac{r-2}{2(r-1)}\cdot n^2}$ edges contains a clique of size $r$, i.e., $r$ mutually adjacent vertices. The corresponding extremal graphs are balanced $(r-1)$-partite graphs.
The question as to how many such $r$-cliques appear at least in any $n$-vertex graph with $\gamma n^2$ edges has been intensively studied in the literature. In particular, Lov\’asz and Simonovits conjectured in the 1970’s that asymptotically the best possible lower bound is given by the complete multipartite graph with $\gamma n^2$ edges in which all but one vertex class is of the same size while the remaining one may be smaller.
Their conjecture was recently resolved for $r=3$ by Razborov and for $r=4$ by Nikiforov. In this article, we prove the conjecture for all values of $r$.
Pages 683-707 by
From volume 184-3
Extremal results for random discrete structures
We study thresholds for extremal properties of random discrete structures. We determine the threshold for Szemerédi’s theorem on arithmetic progressions in random subsets of the integers and its multidimensional extensions, and we determine the threshold for Turán-type problems for random graphs and hypergraphs. In particular, we verify a conjecture of Kohayakawa, Łuczak, and Rödl for Turán-type problems in random graphs. Similar results were obtained independently by Conlon and Gowers.
Pages 333-365 by
From volume 184-2
Splitting mixed Hodge structures over affine invariant manifolds
We prove that affine invariant manifolds in strata of flat surfaces are algebraic varieties. The result is deduced from a generalization of a theorem of Möller. Namely, we prove that the image of a certain twisted Abel-Jacobi map lands in the torsion of a factor of the Jacobians. This statement can be viewed as a splitting of certain mixed Hodge structures.
Pages 681-713 by
From volume 183-2
Global solutions of the Euler–Maxwell two-fluid system in 3D
The fundamental “two-fluid” model for describing plasma dynamics is given by the Euler–Maxwell system, in which compressible ion and electron fluids interact with their own self-consistent electromagnetic field. We prove global stability of a constant neutral background, in the sense that irrotational, smooth and localized perturbations of a constant background with small amplitude lead to global smooth solutions in three space dimensions for the Euler–Maxwell system. Our construction is robust in dimension 3 and applies equally well to other plasma models such as the Euler–Poisson system for two-fluids and a relativistic Euler–Maxwell system for two fluids. Our solutions appear to be the first nontrivial global smooth solutions in all of these models.
Pages 377-498 by
From volume 183-2
On the nonexistence of elements of Kervaire invariant one
We show that the Kervaire invariant one elements $\theta_{j}\in\pi_{2^{j+1}-2}S^{0}$ exist only for $j\le 6$. By Browder’s Theorem, this means that smooth framed manifolds of Kervaire invariant one exist only in dimensions $2$, $6$, $14$, $30$, $62$, and possibly $126$. Except for dimension $126$ this resolves a longstanding problem in algebraic topology.
Pages 1-262 by
From volume 184-1
Functoriality, Smith theory, and the Brauer homomorphism
If $\sigma$ is an automorphism of order $p$ of the semisimple group $\mathbf{G}$, there is a natural correspondence between $\mathrm{mod}p$ 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
Kontsevich’s graph complex, GRT, and the deformation complex of the sheaf of polyvector fields
We generalize Kontsevich’s construction of $L_{\infty}$-derivations of polyvector fields from the affine space to an arbitrary smooth algebraic variety. More precisely, we construct a map (in the homotopy category) from Kontsevich’s graph complex to the deformation complex of the sheaf of polyvector fields on a smooth algebraic variety. We show that the action of Deligne-Drinfeld elements of the Grothendieck-Teichmüller Lie algebra on the cohomology of the sheaf of polyvector fields coincides with the action of odd components of the Chern character. Using this result, we deduce that the $\hat{A}$-genus in the Calaque-Van den Bergh formula for the isomorphism between harmonic and Hochschild structures can be replaced by a generalized $\hat{A}$-genus.
Pages 855-943 by
From volume 182-3
A proof of Demailly’s strong openness conjecture
In this article, we solve the strong openness conjecture on the multiplier ideal sheaf associated to any plurisubharmonic function, which was posed by Demailly.
Pages 605-616 by
From volume 182-2
Rationality of $W$-algebras: principal nilpotent cases
We prove the rationality of all the minimal series principal $W$-algebras discovered by Frenkel, Kac and Wakimoto, thereby giving a new family of rational and $C_2$-cofinite vertex operator algebras. A key ingredient in our proof is the study of Zhu’s algebra of simple $W$-algebras via the quantized Drinfeld-Sokolov reduction. We show that the functor of taking Zhu’s algebra commutes with the reduction functor. Using this general fact we determine the maximal spectrums of the associated graded of Zhu’s algebras of vertex operator algebras associated with admissible representations of affine Kac-Moody algebras as well.
Pages 565-604 by
From volume 182-2
The good pants homology and the Ehrenpreis Conjecture
We develop the notion of the good pants homology and show that it agrees with the standard homology on closed surfaces. (Good pants are pairs of pants whose cuffs have the length nearly equal to some large number $R>0$.) Combined with our previous work on the Surface Subgroup Theorem, this yields a proof of the Ehrenpreis Conjecture.
Pages 1-72 by
From volume 182-1
Periodic approximations of irrational pseudo-rotations using pseudoholomorphic curves
We prove that every $C^\infty$-smooth, area preserving diffeomorphism of the closed $2$-disk having not more than one periodic point is the uniform limit of periodic $C^\infty$-smooth diffeomorphisms. In particular, every smooth irrational pseudo-rotation can be $C^0$-approximated by integrable systems. This partially answers a long standing question of A. Katok regarding zero entropy Hamiltonian systems in low dimensions. Our approach uses pseudoholomorphic curve techniques from symplectic geometry.
Pages 1033-1086 by
From volume 181-3
Kähler–Einstein metrics with edge singularities
This article considers the existence and regularity of Kähler–Einstein metrics on a compact Kähler manifold $M$ with edge singularities with cone angle $2\pi \beta$ along a smooth divisor $D$. We prove existence of such metrics with negative, zero and some positive cases for all cone angles $2\pi \beta \leq 2\pi$. The results in the positive case parallel those in the smooth case. We also establish that solutions of this problem are polyhomogeneous, i.e., have a complete asymptotic expansion with smooth coefficients along $D$ for all $2\pi \beta < 2\pi$.
Pages 95-176 by
From volume 183-1
Properly embedded minimal planar domains
In 1997, Collin proved that any properly embedded minimal surface in $\mathbb{R}^3$ with finite topology and more than one end has finite total Gaussian curvature. Hence, by an earlier result of López and Ros, catenoids are the only nonplanar, nonsimply connected, properly embedded, minimal planar domains in $\mathbb{R}^3$ of finite topology. In 2005, Meeks and Rosenberg proved that the only simply connected, properly embedded minimal surfaces in $\mathbb{R}^3$ are planes and helicoids. Around 1860, Riemann defined a one-parameter family of periodic, infinite topology, properly embedded, minimal planar domains $\mathcal{R}_t$ in $\mathbb{R}^3$, $t \in (0,\infty)$. These surfaces are called the Riemann minimal examples, and the family $\{ \mathcal{R}_t\} _t$ has natural limits being a vertical catenoid as $t\to~0$ and a vertical helicoid as $t\to\infty$. In this paper we complete the classification of properly embedded, minimal planar domains in $\mathbb{R}^3$ by proving that the only connected examples with infinite topology are the Riemann minimal examples. We also prove that the limit ends of Riemann minimal examples are model surfaces for the limit ends of properly embedded minimal surfaces $M\subset\mathbb{R}^3$ of finite genus and infinite topology, in the sense that such an $M$ has two limit ends, each of which has a representative that is naturally asymptotic to a limit end representative of a Riemann minimal example with the same associated flux vector.
Pages 473-546 by
From volume 181-2
Random walks in Euclidean space
Fix a probability measure on the space of isometries of Euclidean space $\mathbf{R}^d$. Let $Y_0=0,Y_1,Y_2,\ldots\in\mathbf{R}^d$ be a sequence of random points such that $Y_{l+1}$ is the image of $Y_l$ under a random isometry of the previously fixed probability law, which is independent of $Y_l$. We prove a Local Limit Theorem for $Y_l$ under necessary nondegeneracy conditions. Moreover, under more restrictive but still general conditions we give a quantitative estimate which describes the behavior of the law of $Y_l$ on scales $e^{-cl^{1/4}}
Pages 243-301 by
From volume 181-1
The space of embedded minimal surfaces of fixed genus in a 3-manifold V; Fixed genus
This paper is the fifth and final in a series on embedded minimal surfaces. Following our earlier papers on disks, we prove here two main structure theorems for \itnonsimply connected embedded minimal surfaces of any given fixed genus.
The first of these asserts that any such surface without small necks can be obtained by gluing together two oppositely-oriented double spiral staircases.
The second gives a pair of pants decomposition of any such surface when there are small necks, cutting the surface along a collection of short curves. After the cutting, we are left with graphical pieces that are defined over a disk with either one or two sub-disks removed (a topological disk with two sub-disks removed is called a pair of pants).
Both of these structures occur as different extremes in the two-parameter family of minimal surfaces known as the Riemann examples.
Pages 1-153 by
From volume 181-1
The Hodge theory of Soergel bimodules
We prove Soergel’s conjecture on the characters of indecomposable
Soergel bimodules. We deduce that Kazhdan-Lusztig polynomials have positive coefficients for arbitrary Coxeter systems. Using results of Soergel one may deduce an algebraic proof of the Kazhdan-Lusztig conjecture.
Pages 1089-1136 by
From volume 180-3
Fourier transform and the global Gan–Gross–Prasad conjecture for unitary groups
By the relative trace formula approach of Jacquet–Rallis, we prove the global Gan–Gross–Prasad conjecture for unitary groups under some local restrictions for the automorphic representations.
Pages 971-1049 by
From volume 180-3
The geometry of the moduli space of odd spin curves
The spin moduli space $\overline{\mathcal{S}}_g$ is the parameter space of theta characteristics (spin structures) on stable curves of genus $g$. It has two connected components, $\overline{\mathcal{S}}_g^-$ and $\overline{\mathcal{S}}_g^+$, depending on the parity of the spin structure. We establish a complete birational classification by Kodaira dimension of the odd component $\overline{\mathcal{S}}_g^-$ of the spin moduli space. We show that $\overline{\mathcal{S}}_g^-$ is uniruled for $g<12$ and even unirational for $g\leq 8$. In this range, introducing the concept of cluster for the Mukai variety whose one-dimensional linear sections are general canonical curves of genus $g$, we construct new birational models of $\overline{\mathcal{S}}_g^-$. These we then use to explicitly describe the birational structure of $\overline{\mathcal{S}}_g^-$. For instance, $\overline{\mathcal{S}}_8^-$ is birational to a locally trivial $\textbf{P}^7$-bundle over the moduli space of elliptic curves with seven pairs of marked points. For $g\geq 12$, we prove that $\overline{\mathcal{S}}_g^-$ is a variety of general type. In genus $12$, this requires the construction of a counterexample to the Slope Conjecture on effective divisors on the moduli space of stable curves of genus $12$.
Pages 927-970 by
From volume 180-3
On self-similar sets with overlaps and inverse theorems for entropy
We study the dimension of self-similar sets and measures on the line. We show that if the dimension is less than the generic bound of $\mathrm{min}\{1,s\}$, where $s$ is the similarity dimension, then there are superexponentially close cylinders at all small enough scales. This is a step towards the conjecture that such a dimension drop implies exact overlaps and confirms it when the generating similarities have algebraic coefficients. As applications we prove Furstenberg’s conjecture on projections of the one-dimensional Sierpinski gasket and achieve some progress on the Bernoulli convolutions problem and, more generally, on problems about parametric families of self-similar measures. The key tool is an inverse theorem on the structure of pairs of probability measures whose mean entropy at scale $2^{-n}$ has only a small amount of growth under convolution.
Pages 773-822 by
From volume 180-2
ACC for log canonical thresholds
We show that log canonical thresholds satisfy the \rm ACC.
Pages 523-571 by
From volume 180-2
Stable logarithmic maps to Deligne–Faltings pairs I
We introduce a new compactification of the space of relative stable maps. This approach uses logarithmic geoemetry in the sense of Kato-Fontaine-Illusie without taking expansions of the target. The underlying structures of the stable logarithmic maps are stable in the usual sense.
Pages 455-521 by
From volume 180-2
Bounded gaps between primes
It is proved that $$ \liminf_{n\to\infty}(p_{n+1}-p_n)<7\times 10^7, $$ where $p_n$ is the $n$-th prime.
Our method is a refinement of the recent work of Goldston, Pintz and Yıldırım on the small gaps between consecutive primes. A major ingredient of the proof is a stronger version of the Bombieri-Vinogradov theorem that is applicable when the moduli are free from large prime divisors only, but it is adequate for our purpose.
Pages 1121-1174 by
From volume 179-3
Kodaira dimension and zeros of holomorphic one-forms
We show that every holomorphic one-form on a smooth complex projective variety of general type must vanish at some point. The proof uses generic vanishing theory for Hodge modules on abelian varieties.
Pages 1109-1120 by
From volume 179-3
A general regularity theory for stable codimension 1 integral varifolds
We give a necessary and sufficient geometric structural condition, which we call the $\alpha$-Structural Hypothesis, for a stable codimension 1 integral varifold on a smooth Riemannian manifold to correspond to an embedded smooth hypersurface away from a small set of generally unavoidable singularities. The $\alpha$-Structural Hypothesis says that no point of the support of the varifold has a neighborhood in which the support is the union of three or more embedded $C^{1, \alpha}$ hypersurfaces-with-boundary meeting (only) along their common boundary. We establish that whenever a stable integral $n$-varifold on a smooth $(n+1)$-dimensional Riemannian manifold satisfies the $\alpha$-Structural Hypothesis for some $\alpha \in (0, 1/2)$, its singular set is empty if $n \leq 6$, discrete if $n =7$ and has Hausdorff dimension $\leq n-7$ if $n \geq 8$; in view of well-known examples, this is the best possible general dimension estimate on the singular set of a varifold satisfying our hypotheses. We also establish compactness of mass-bounded subsets of the class of stable codimension 1 integral varifolds satisfying the $\alpha$-Structural Hypothesis for some $\alpha \in (0, 1/2)$. The $\alpha$-Structural Hypothesis on an $n$-varifold for any $\alpha \in (0, 1/2)$ is readily implied by either of the following two hypotheses: (i) the varifold corresponds to an absolutely area minimizing rectifiable current with no boundary, (ii) the singular set of the varifold has vanishing $(n-1)$-dimensional Hausdorff measure. Thus, our theory subsumes the well-known regularity theory for codimension 1 area minimizing rectifiable currents and settles the long standing question as to which weakest size hypothesis on the singular set of a stable minimal hypersurface guarantees the validity of the above regularity conclusions.
An optimal strong maximum principle for stationary codimension 1 integral varifolds follows from our regularity and compactness theorems.
Pages 843-1007 by
From volume 179-3
Calabi flow, geodesic rays, and uniqueness of constant scalar curvature Kähler metrics
We prove that constant scalar curvature Kähler metric “adjacent” to a fixed Kähler class is unique up to isomorphism. The proof is based on the study of a fourth order evolution equation, namely, the Calabi flow, from a new geometric perspective, and on the geometry of the space of Kähler metrics.
Pages 407-454 by
From volume 180-2
The Oort Conjecture on lifting covers of curves
We show that the conjecture of Oort on lifting covers of curves is true. The main ingredients in the proof are a deformation argument in characteristic $p$ and (a special case of) a very recent result by Obus–Wewers. A kind of boundedness result is given as well.
Pages 285-322 by
From volume 180-1
Limit theorems for translation flows
The aim of this paper is to obtain an asymptotic expansion for ergodic integrals of translation flows on flat surfaces of higher genus (Theorem 1) and to give a limit theorem for these flows (Theorem 2).
Pages 431-499 by
From volume 179-2
Log minimal model program for the moduli space of stable curves: the first flip
We give a geometric invariant theory (GIT) construction of the log canonical model $\bar M_g(\alpha)$ of the pairs $(\bar M_g, \alpha \delta)$ for $\alpha \in (7/10 – \epsilon, 7/10]$ for small $\epsilon \in \mathbb Q_+$. We show that $\bar M_g(7/10)$ is isomorphic to the GIT quotient of the Chow variety of bicanonical curves; $\bar M_g(7/10-\epsilon)$ is isomorphic to the GIT quotient of the asymptotically-linearized Hilbert scheme of bicanonical curves. In each case, we completely classify the (semi)stable curves and their orbit closures. Chow semistable curves have ordinary cusps and tacnodes as singularities but do not admit elliptic tails. Hilbert semistable curves satisfy further conditions; e.g., they do not contain elliptic chains. We show that there is a small contraction $\Psi: \bar M_g(7/10+\epsilon) \to \bar M_g(7/10)$ that contracts the locus of elliptic bridges. Moreover, by using the GIT interpretation of the log canonical models, we construct a small contraction $\Psi^+ : \bar M_g(7/10-\epsilon) \to \bar M_g(7/10)$ that is the Mori flip of $\Psi$.
Pages 911-968 by
From volume 177-3
Diophantine geometry over groups VIII: Stability
This paper is the eighth in a sequence on the structure of sets of solutions to systems of equations in free and hyperbolic groups, projections of such sets (Diophantine sets), and the structure of definable sets over free and hyperbolic groups. In this eighth paper we use a modification of the sieve procedure, which was used in proving quantifier elimination in the theory of a free group, to prove that free and torsion-free (Gromov) hyperbolic groups are stable.
Pages 787-868 by
From volume 177-3
Small eigenvalues of the Laplacian for algebraic measures in moduli space, and mixing properties of the Teichmüller flow
We consider the $\mathrm{SL}(2, \mathbb{R})$ action on moduli spaces of quadratic differentials. If $\mu$ is an $\mathrm{SL}(2, \mathbb{R})$-invariant probability measure, crucial information about the associated representation on $L^2(\mu)$ (and, in particular, fine asymptotics for decay of correlations of the diagonal action, the Teichmüller flow) is encoded in the part of the spectrum of the corresponding foliated hyperbolic Laplacian that lies in $\!(0,1/4)\!$ (which controls the contribution of the complementary series). Here we prove that the essential spectrum of an invariant algebraic measure is contained in $[1/4,\infty)$; i.e., for every $\delta\!>\!0$, there are only finitely many eigenvalues (counted with multiplicity) in $(0,1/4\!-\!\delta)$. In particular, all algebraic invariant measures have a spectral gap.
Pages 385-442 by
From volume 178-2
The universal relation between scaling exponents in first-passage percolation
It has been conjectured in numerous physics papers that in ordinary first-passage percolation on integer lattices, the fluctuation exponent $\chi$ and the wandering exponent $\xi$ are related through the universal relation $\chi=2\xi -1$, irrespective of the dimension. This is sometimes called the KPZ relation between the two exponents. This article gives a rigorous proof of this conjecture assuming that the exponents exist in a certain sense.
Pages 663-697 by
From volume 177-2
A further improvement of the Quantitative Subspace Theorem
In 2002, Evertse and Schlickewei obtained a quantitative version of the so-called Absolute Parametric Subspace Theorem. This result deals with a parametrized class of twisted heights. One of the consequences of this result is a quantitative version of the Absolute Subspace Theorem, giving an explicit upper bound for the number of subspaces containing the solutions of the Diophantine inequality under consideration.
In the present paper, we further improve Evertse’s and Schlickewei’s quantitative version of the Absolute Parametric Subspace Theorem and deduce an improved quantitative version of the Absolute Subspace Theorem. We combine ideas from the proof of Evertse and Schlickewei (which is basically a substantial refinement of Schmidt’s proof of his Subspace Theorem from 1972), with ideas from Faltings’ and Wüstholz’ proof of the Subspace Theorem. A new feature is an “interval result,” which gives more precise information on the distribution of the heights of the solutions of the system of inequalities considered in the Subspace Theorem.
Pages 513-590 by
From volume 177-2
A class of superrigid group von Neumann algebras
We prove that for any group $G$ in a fairly large class of generalized wreath product groups, the associated von Neumann algebra $\mathrm{L} G$ completely “remembers” the group $G$. More precisely, if $\mathrm{L} G$ is isomorphic to the von Neumann algebra $\mathrm{L} \Lambda$ of an arbitrary countable group $\Lambda$, then $\Lambda$ must be isomorphic to $G$. This represents the first superrigidity result pertaining to group von Neumann algebras.
Pages 231-286 by
From volume 178-1
Descente par éclatements en $K$-théorie invariante par homotopie
Ces notes donnent une preuve de la représentabilité de la $K$-théorie invariante par homotopie dans la catégorie homotopique stable des schémas (résultat annoncé par Voevodsky). On en déduit, grâce au théorème de changement de base propre en théorie de l’homotopie stable des schémas, un théorème de descente par éclatements en $K$-théorie invariante par homotopie.
These notes give a proof of the representability of homotopy invariant $K$-theory in the stable homotopy category of schemes (which was announced by Voevodsky). One deduces from the proper base change theorem in stable homotopy theory of schemes a descent by blow-ups theorem for homotopy invariant $K$-theory.
Pages 425-448 by
From volume 177-2
Higher finiteness properties of reductive arithmetic groups in positive characteristic: The Rank Theorem
We show that the finiteness length of an $S$-arithmetic subgroup $\Gamma$ in a noncommutative isotropic absolutely almost simple group $\mathcal{G}$ over a global function field is one less than the sum of the local ranks of $\mathcal{G}$ taken over the places in $S$. This determines the finiteness properties for $S$-arithmetic subgroups in isotropic reductive groups, confirming the conjectured finiteness properties for this class of groups.
Our main tool is Behr–Harder reduction theory which we recast in terms of the metric structure of euclidean buildings.
Pages 311-366 by
From volume 177-1
Norm convergence of nilpotent ergodic averages
We show that multiple polynomial ergodic averages arising from nilpotent groups of measure preserving transformations of a probability space always converge in the $L^2$ norm.
Pages 1667-1688 by
From volume 175-3
The second fundamental theorem of invariant theory for the orthogonal group
Let $V=\mathbb{C}^n$ be endowed with an orthogonal form and $G=\mathrm{O}(V)$ be the corresponding orthogonal group. Brauer showed in 1937 that there is a surjective homomorphism $\nu:B_r(n)\to\mathrm{End}_G(V^{\otimes r})$, where $B_r(n)$ is the $r$-string Brauer algebra with parameter $n$. However the kernel of $\nu$ has remained elusive. In this paper we show that, in analogy with the case of $\mathrm{GL}(V)$, for $r\geq n+1$, $\nu$ has a kernel which is generated by a single idempotent element $E$, and we give a simple explicit formula for $E$. Using the theory of cellular algebras, we show how $E$ may be used to determine the multiplicities of the irreducible representations of $\mathrm{O}(V)$ in $V^{\otimes r}$. We also show how our results extend to the case where $\mathbb{C}$ is replaced by an appropriate field of positive characteristic, and we comment on quantum analogues of our results.
Pages 2031-2054 by
From volume 176-3
The sharp weighted bound for general Calderón–Zygmund operators
For a general Calderón–Zygmund operator $T$ on $\Bbb{R}^N$, it is shown that $$ \Vert{Tf}\Vert{L^2(w)}\leq C(T)\cdot\sup_Q\Big(∫_Q w\cdot ∫_Q w^{-1}\Big)\cdot\Vert{f}\Vert{L^2(w)} \end{equation*} for all Muckenhoupt weights $w\in A_2$. This optimal estimate was known as the $A_2$ conjecture. A recent result of Pérez–Treil–Volberg reduced the problem to a testing condition on indicator functions, which is verified in this paper.
The proof consists of the following elements: (i) a variant of the Nazarov–Treil–Volberg method of random dyadic systems with just one random system and completely without “bad” parts; (ii) a resulting representation of a general Calderón–Zygmund operator as an average of “dyadic shifts;” and (iii) improvements of the Lacey–Petermichl–Reguera estimates for these dyadic shifts, which allow summing up the series in the obtained representation.
Pages 1473-1506 by
From volume 175-3
Operator monotone functions and Löwner functions of several variables
We prove generalizations of Löwner’s results on matrix monotone functions to several variables. We give a characterization of when a function of $d$ variables is locally monotone on $d$-tuples of commuting self-adjoint $n$-by-$n$ matrices. We prove a generalization to several variables of Nevanlinna’s theorem describing analytic functions that map the upper half-plane to itself and satisfy a growth condition. We use this to characterize all rational functions of two variables that are operator monotone.
Pages 1783-1826 by
From volume 176-3
Linearization of generalized interval exchange maps
A standard interval exchange map is a one-to-one map of the interval that is locally a translation except at finitely many singularities. We define for such maps, in terms of the Rauzy-Veech continuous fraction algorithm, a diophantine arithmetical condition called restricted Roth type, which is almost surely satisfied in parameter space. Let $T_0$ be a standard interval exchange map of restricted Roth type, and let $r$ be an integer $\geq 2$. We prove that, amongst $C^{r+3}$ deformations of $T_0$ that are $C^{r+3}$ tangent to $T_0$ at the singularities, those that are conjugated to $T_0$ by a $C^r$-diffeomorphism close to the identity form a $C^1$-submanifold of codimension $(g-1)(2r+1) +s$. Here, $g$ is the genus and $s$ is the number of marked points of the translation surface obtained by suspension of $T_0$. Both $g$ and $s$ can be computed from the combinatorics of $T_0$.
Pages 1583-1646 by
From volume 176-3
Immersing almost geodesic surfaces in a closed hyperbolic three manifold
Let $\mathbf{M}^3$ be a closed hyperbolic three manifold. We construct closed surfaces that map by immersions into $\mathbf{M}^3$ so that for each, one the corresponding mapping on the universal covering spaces is an embedding, or, in other words, the corresponding induced mapping on fundamental groups is an injection.
Pages 1127-1190 by
From volume 175-3
Linear Shafarevich conjecture
In this paper we settle affirmatively Shafarevich’s uniformization conjecture for varieties with linear fundamental groups. We prove the strongest to date uniformization result — the universal covering space of a complex projective manifold with a linear fundamental group is holomorphically convex. The proof is based on both known and newly developed techniques in non-abelian Hodge theory.
Pages 1545-1581 by
From volume 176-3
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
