# Annals of Mathematics

Published by Princeton University, Department of Mathematics

Online ISSN: 0003-486X

Published by Princeton University, Department of Mathematics

Online ISSN: 0003-486X

Publications

Article

In this paper, we reveal the formal algebraic structure underlying the intrinsic reconstitution algorithm, introduced by Singer and Shkolnisky in [9], 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.

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Conference Paper

In this paper, we study functions with low influences on product probability spaces. The analysis of Boolean functions f {-1, 1}<sup>n</sup> → {-1, 1} with low influences has become a central problem in discrete Fourier analysis. It is motivated by fundamental questions arising from the construction of probabilistically checkable proofs in theoretical computer science and from problems in the theory of social choice in economics. We prove an invariance principle for multilinear polynomials with low influences and bounded degree; it shows that under mild conditions the distribution of such polynomials is essentially invariant for all product spaces. Ours is one of the very few known non-linear invariance principles. It has the advantage that its proof is simple and that the error bounds are explicit. We also show that the assumption of bounded degree can be eliminated if the polynomials are slightly "smoothed"; this extension is essential for our applications to "noise stability "-type problems. In particular; as applications of the invariance principle we prove two conjectures: the "Majority Is Stablest" conjecture [29] from theoretical computer science, which was the original motivation for this work, and the "It Ain't Over Till It's Over" conjecture [27] from social choice theory. The "Majority Is Stablest" conjecture and its generalizations proven here, in conjunction with the "Unique Games Conjecture" and its variants, imply a number of (optimal) inapproximability results for graph problems.

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Article

We introduce spaces of exponential constructible functions in the motivic
setting for which we construct direct image functors in the absolute and
relative cases. This allows us to define a motivic Fourier transformation for
which we get various inversion statements. We define also motivic
Schwartz-Bruhat spaces on which motivic Fourier transformation induces an
isomorphism. Our motivic integrals specialize to non archimedian integrals. We
give a general transfer principle comparing identities between functions
defined by integrals over local fields of characteristic zero, resp. positive,
having the same residue field. We also prove new results about p-adic integrals
of exponential functions.

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Article

Let $M_d$ be the centered Hardy-Littlewood maximal function associated to
cubes in $\mathbb{R}^d$ with Lebesgue measure, and let $c_d$ denote the lowest
constant appearing in the weak type (1,1) inequality satisfied by $M_d$.
We show that $c_d \to \infty$ as $d\to \infty$, thus answering, for the case
of cubes, a long standing open question of E. M. Stein and J. O. Str\"{o}mberg.

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Article

We prove that for any number $r$ in $[2,3]$, there are spin (resp. non-spin
minimal) simply connected complex surfaces of general type $X$ with
$c_1^2(X)/c_2(X)$ arbitrarily close to $r$. In particular, this shows the
existence of simply connected surfaces of general type arbitrarily close to the
Bogomolov-Miyaoka-Yau line. In addition, we prove that for any $r \in [2,3]$
and any integer $q\geq 0$, there are minimal complex surfaces of general type
$X$ with $c_1^2(X)/c_2(X)$ arbitrarily close to $r$, and $\pi_1(X)$ isomorphic
to the fundamental group of a compact Riemann surface of genus $q$.

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Article

The stochastic 2D Navier-Stokes equations on the torus driven by degenerate noise are studied. We characterize the smallest closed invariant subspace for this model and show that the dynamics restricted to that subspace is ergodic. In particular, our results yield a purely geometric characterization of a class of noises for which the equation is ergodic in $\L^2_0(\TT^2)$. Unlike previous works, this class is independent of the viscosity and the strength of the noise. The two main tools of our analysis are the \textit{asymptotic strong Feller} property, introduced in this work, and an approximate integration by parts formula. The first, when combined with a weak type of irreducibility, is shown to ensure that the dynamics is ergodic. The second is used to show that the first holds under a H{\"o}rmander-type condition. This requires some interesting nonadapted stochastic analysis. Comment: 40 Pages, published version

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Article

In this paper we characterize irreducible generic representations of $\SO_{2n+1}(k)$ where $k$ is a $p$-adic field) by means of twisted local gamma factors (the Local Converse Theorem). As applications, we prove that two irreducible generic cuspidal automorphic representations of $\SO_{2n+1}({\Bbb A})$ (where ${\Bbb A}$ is the ring of adeles of a number field) are equivalent if their local components are equivalent at almost all local places (the Rigidity Theorem);and prove the Local Langlands Reciprocity Conjecture for generic supercuspidal representations of $\SO_{2n+1}(k)$.

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Article

For any given compact C^2 hypersurface \Sigma in {\bf R}^{2n} bounding a strictly convex set with nonempty interior, in this paper an invariant \varrho_n(\Sigma) is defined and satisfies \varrho_n(\Sigma)\ge [n/2]+1, where [a] denotes the greatest integer which is not greater than a\in {\bf R}. The following results are proved in this paper. There always exist at least \rho_n(\Sigma) geometrically distinct closed characteristics on \Sigma. If all the geometrically distinct closed characteristics on \Sigma are nondegenerate, then \varrho_n(\Sigma)\ge n. If the total number of geometrically distinct closed characteristics on \Sigma is finite, there exists at least an elliptic one among them, and there exist at least \varrho_n(\Sigma)-1 of them possessing irrational mean indices. If this total number is at most 2\varrho_n(\Sigma) -2, there exist at least two elliptic ones among them.

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Article

Suppose that D is a planar Jordan domain and x and y are distinct boundary
points of D. Fix \kappa \in (4,8) and let \eta\ be an SLE_\kappa process from x
to y in D. We prove that the law of the time-reversal of \eta is, up to
reparameterization, an SLE_\kappa process from y to x in D. More generally, we
prove that SLE_\kappa(\rho_1;\rho_2) processes are reversible if and only if
both \rho_i are at least \kappa/2-4, which is the critical threshold at or
below which such curves are boundary filling.
Our result supplies the missing ingredient needed to show that for all \kappa
\in (4,8) the so-called conformal loop ensembles CLE_\kappa\ are canonically
defined, with almost surely continuous loops. It also provides an interesting
way to couple two Gaussian free fields (with different boundary conditions) so
that their difference is piecewise constant and the boundaries between the
constant regions are SLE_\kappa curves.

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Article

This paper proves that for every convex body in R^n there exist 5n-4 Minkowski symmetrizations, which transform the body into an approximate Euclidean ball. This result complements the sharp c n log n upper estimate by J. Bourgain, J. Lindenstrauss and V.D. Milman, of the number of random Minkowski symmetrizations sufficient for approaching an approximate Euclidean ball.

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Article

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

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Article

For G = GL_2, PGL_2 and SL_2 we prove that the perverse filtration associated
to the Hitchin map on the cohomology of the moduli space of twisted G-Higgs
bundles on a Riemann surface C agrees with the weight filtration on the
cohomology of the twisted G character variety of C, when the cohomologies are
identified via non-Abelian Hodge theory. The proof is accomplished by means of
a study of the topology of the Hitchin map over the locus of integral spectral
curves.

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Article

We study the higher Abel-Jacobi invariant defined recently by M. Green. We first construct a counterexample to the injectivity of Green's higher Abel-Jacobi map. On the other hand, we prove that the higher Abel-Jacobi map governs Mumford's pull-back of holomorphic forms. We deduce from this that if a surface has holomorphic 2-forms, the image of the higher Abel-Jacobi map, defined on its group of zero-cycles Albanese equivalent to 0, has infinite dimensional image. Comment: 23 pages, published version

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Article

We show that the usual Hodge conjecture implies the general Hodge conjecture for certain abelian varieties of type III, and use this to deduce the general Hodge conjecture for all powers of certain 4-dimensional abelian varieties of type III. We also show the existence of a Hodge structure M such that M occurs in the cohomology of an abelian variety, but the Tate twist M(1) does not occur in the cohomology of any abelian variety, even though it is effective.

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Article

We prove that for a large class of subvarieties of abelian varieties over global function fields, the Brauer-Manin condition on adelic points cuts out exactly the rational points. This result is obtained from more general results concerning the intersection of the adelic points of a subvariety with the adelic closure of the group of rational points of the abelian variety.

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Article

Let (A,\lambda) be a principally polarized abelian variety defined over a global field k, and let \Sha(A) be its Shafarevich-Tate group. Let \Sha(A)_\nd denote the quotient of \Sha(A) by its maximal divisible subgroup. Cassels and Tate constructed a nondegenerate pairing \Sha(A)_\nd \times \Sha(A)_\nd \rightarrow \Q/\Z. If A is an elliptic curve, then by a result of Cassels the pairing is alternating. But in general it is only antisymmetric. Using some new but equivalent definitions of the pairing, we derive general criteria deciding whether it is alternating and whether there exists some alternating nondegenerate pairing on \Sha(A)_\nd. These criteria are expressed in terms of an element c \in \Sha(A)_\nd that is canonically associated to the polarization \lambda. In the case that A is the Jacobian of some curve, a down-to-earth version of the result allows us to determine effectively whether \#\Sha(A) (if finite) is a square or twice a square. We then apply this to prove that a positive proportion (in some precise sense) of all hyperelliptic curves of even genus g \ge 2 over \Q have a Jacobian with nonsquare \#\Sha (if finite). For example, it appears that this density is about 13% for curves of genus 2. The proof makes use of a general result relating global and local densities; this result can be applied in other situations. Comment: 41 pages, published version

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Article

We prove that minimal area-preserving flows locally given by a smooth Hamiltonian on a closed surface of any genus are typically (in the measure-theoretical sense) not mixing. The result is obtained by considering special flows over interval exchange transformations under roof functions with symmetric logarithmic singularities and proving absence of mixing for a full measure set of interval exchange transformations. 1 Definitions and Main Results 1.1 Flows given by multi-valued Hamiltonians Let us consider the following natural construction of area-preserving flows on surfaces. On a closed, compact, orientable surface of genus g ≥ 1 with a fixed smooth area form, consider a smooth closed differential 1-form ω. Since ω is closed, it is locally given by dH for some real-valued function H. The flow {ϕt}t∈R determined by ω is the associated Hamiltonian flow, which is given by local solutions of

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Article

I explore some consequences of a groundbreaking result of Breimesser and Pearson on the absolutely continuous spectrum of one-dimensional Schr"odinger operators. These include an Oracle Theorem that predicts the potential and rather general results on the approach to certain limit potentials. In particular, we prove a Denisov-Rakhmanov type theorem for the general finite gap case. The main theme is the following: It is extremely difficult to produce absolutely continuous spectrum in one space dimension and thus its existence has strong implications. Comment: (slightly) revised version

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Chapter

In real Hilbert space H there is a finitely additive measure n on the ring of sets defined by finitely many linear conditions, which is analogous to the normal distribution in the finite-dimensional case. This has been examined from various points of view in Gelfand and Vilenkin [11], Gross [12], Segal [19], as well as by many earlier authors. Now, if the Hilbert space is “completed” in any of number of ways, this “cylinder set measure” extends to an actual Borel measure on the completed space. For example, if we define | x |T
= P Tx P, where T is a Hilbert-Schmidt operator with trivial nullspace, then the completion of H with respect to the norm | • |T
is such a completion. A more subtle example is the following. Let H be realized explicity as L
2(0, 1). Define a norm on H by \(|f|={\rm sup}_{0\leqq t \leqq 1}|\left|\int^{{t}}_{0}f\left(s\right)ds\right|.\) Then the completion of H with respect to this is isomorphic to the continuous functions on [0, 1] with suo norm and vanishing at zero, via the map which sends f to the function g whose value at t is \(\int^{{t}}_{0}f\left(s\right)ds, 0\leqq t \leqq 1.\) The finitely additive measure n is then realized as Wiener measure. Other natural norms give rise to realizations of n as Wiener measure on the Hölder–continuous function, exponent α < α < 1/2. This sort of phenomenon has been investigated abstractly by L. Gross in [14]. He shows there (Theorem 4) that if H is completed with respect to any of his measurable seminorms, as defined in Gross [13], then n gives rise to a country additive Borel measure on the Banach space obtained frome H by means of the seminorm.

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Article

We study the ac-conductivity in linear response theory in the general
framework of ergodic magnetic Schr\"odinger operators. For the Anderson model,
if the Fermi energy lies in the localization regime, we prove that the
ac-conductivity is bounded by $C \nu^2 (\log \frac 1 \nu)^{d+2}$ at small
frequencies $\nu$. This is to be compared to Mott's formula, which predicts the
leading term to be $C \nu^2 (\log \frac 1 \nu)^{d+1}$.

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Article

We show that log canonical thresholds satisfy the ACC

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Article

We introduce a class of metric spaces which we call "bolic". They include hyperbolic spaces, simply conneccted complete manifolds of nonpositive curvature, euclidean buildings, etc. We prove the Novikov conjecture on higher signatures for any discrete group which admits a proper isometric action on a "bolic", weakly geodesic metric space of bounded geometry.

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Article

We prove results about orbit closures and equidistribution for the SL(2,R)
action on the moduli space of compact Riemann surfaces, which are analogous to
the theory of unipotent flows. The proofs of the main theorems rely on the
measure classification theorem of [EMi2] and a certain isolation property of
closed SL(2,R) invariant manifolds developed in this paper.

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Article

I prove the existence, and describe the structure, of moduli space of pairs $(p,\Theta)$ consisting of a projective variety $P$ with semiabelian group action and an ample Cartier divisor on it satisfying a few simple conditions. Every connected component of this moduli space is proper. A component containing a projective toric variety is described by a configuration of several polytopes, the main one of which is the secondary polytope. On the other hand, the component containing a principally polarized abelian variety provides a moduli compactification of $A_g$. The main irreducible component of this compactification is described by an "infinite periodic" analog of the secondary polytope and coincides with the toroidal compactification of $A_g$ for the second Voronoi decomposition.

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Article

In 1964, Weil gave a criterion for local rigidity of a homomorphism from a finitely generated group Γ to a finite dimensional Lie group G in terms of cohomology of Γ with coefficients in the Lie algebra of G. Here we generalize Weil’s result to a class of homomorphisms into certain infinite dimensional Lie groups, namely groups of diffeomorphism compact manifolds. This gives a criterion for local rigidity of group actions which implies local rigidity of: (1) all isometric actions of groups with property (T), (2) all isometric actions of irreducible lattices in products of simple Lie groups and certain more general locally compact groups and (3) a certain class of isometric actions of a certain class of cocompact lattices in SU(1, n). 1. A cohomological criterion for local rigidity and applications In 1964, André Weil showed that a homomorphism π from a finitely generated group Γ to a Lie group G is locally rigid whenever H 1 (Γ, g) = 0. Here π is locally rigid if any nearby homomorphism is conjugate to π by a small element of G, g is the Lie algebra of G, and Γ acts on g by the composition of π and the adjoint representation of G. Weil’s proof also applies to G an algebraic group over a local field of characteristic zero, but his use of the implicit function theorem forced G to be finite dimensional. Here we prove the following generalization of Weil’s theorem to some cases where G is an infinite dimensional Lie group. Theorem 1.1. Let Γ be a finitely presented group, (M, g) a compact Riemannian manifold and π: Γ → Isom(M, g)⊂Diff ∞ (M) a homomorphism. If H 1 (Γ, Vect ∞ (M)) = 0, the homomorphism π is locally rigid

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Article

For \Gamma a countable amenable group consider those actions of \Gamma as measure-preserving transformations of a standard probability space, written as {T_\gamma}_{\gamma \in \Gamma} acting on (X,{\cal F}, \mu). We say {T_\gamma}_{\gamma\in\Gamma} has completely positive entropy (or simply cpe for short) if for any finite and nontrivial partition P of X the entropy h(T,P) is not zero. Our goal is to demonstrate what is well known for actions of \Bbb Z and even \Bbb Z^d, that actions of completely positive entropy have very strong mixing properties. Let S_i be a list of finite subsets of \Gamma. We say the S_i spread if any particular \gamma \neq id belongs to at most finitely many of the sets S_i S_i^{-1}. Theorem 0.1. For {T_\gamma}_{\gamma \in \Gamma} an action of \Gamma of completely positive entropy and P any finite partition, for any sequence of finite sets S_i\subseteq \Gamma which spread we have \frac 1{\# S_i} h(\spans{S_i}{P}){\mathop{\to}_i} h(P). The proof uses orbit equivalence theory in an essential way and represents the first significant application of these methods to classical entropy and mixing.

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Article

Let $J$ be a semisimple Lie group with all simple factors of real rank at least two. Let $\Gamma<J$ be a lattice. We prove a very general local rigidity result about actions of $J$ or $\Gamma$. This shows that almost all so-called "standard actions" are locally rigid. As a special case, we see that any action of $\Gamma$ by toral automorphisms is locally rigid. More generally, given a manifold $M$ on which $\Gamma$ acts isometrically and a torus $\Ta^n$ on which it acts by automorphisms, we show that the diagonal action on $\Ta^n{\times}M$ is locally rigid. This paper is the culmination of a series of papers and depends heavily on our work in \cite{FM1,FM2}. The reader willing to accept the main results of those papers as "black boxes" should be able to read the present paper without referring to them.

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Article

We classify smooth locally free actions of the real affine group on closed
orientable three-dimensional manifolds up to smooth conjugacy. As a corollary,
there exists a non-homogeneous action when the manifold is the unit tangent
bundle of a closed surface with a hyperbolic metric.

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Article

In this paper we show that the cohomology of a connected CW complex is periodic if and only if it is the base space of an orientable spherical fibration with total space that is homotopically finite dimensional. As applications we characterize those discrete groups that act freely and properly on a cartesian product of euclidean space and a sphere; we construct non-standard free actions of rank two simple groups on finite complexes Y homotopy equivalent to a product of two spheres and we prove that a finite p-group P acts freely on such a complex if and only if it does not contain a subgroup isomorphic to Z/p X Z/p X Z/p.

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Article

Given a bounded valence, bushy tree T, we prove that any cobounded quasi-action of a group G on T is quasiconjugate to an action of G on another bounded valence, bushy tree T'. This theorem has many applications: quasi-isometric rigidity for fundamental groups of finite, bushy graphs of coarse PD(n) groups for each fixed n; a generalization to actions on Cantor sets of Sullivan's Theorem about uniformly quasiconformal actions on the 2-sphere; and a characterization of locally compact topological groups which contain a virtually free group as a cocompact lattice. Finally, we give the first examples of two finitely generated groups which are quasi-isometric and yet which cannot act on the same proper geodesic metric space, properly discontinuously and cocompactly by isometries.

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Article

This paper introduces a new measure-conjugacy invariant for actions of free groups. Using this invariant, it is shown that two Bernoulli shifts over a finitely generated free group are measurably conjugate if and only if their base measures have the same entropy. This answers a question of Ornstein and Weiss.

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Article

Let G be a connected semisimple Lie group without compact factors whose real rank is at least 2, and let \Gamma \subset G be an irreducible lattice. We provide a C^\infty classification for volume-preserving Cartan actions of \Gamma and G. Also, if G has real rank at least 3, we provide a C^\infty classification for volume-preserving, multiplicity free, trellised, Anosov actions on compact manifolds.

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Article

We give necessary and sufficient conditions for an affine deformation of a
Schottky subgroup of O(2,1) to act properly on affine space. There exists a
real-valued biaffine map between the cohomology of the Schottky group and the
space of geodesic currents on the corresponding hyperbolic surface S. For a
fixed cohomology class, this map is uniformly positive or uniformly negative on
the space of geodesic currents if and only if the corresponding affine
deformation is proper. As a corollary, the deformation space of proper affine
deformations is an open convex cone.

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Article

Convergence of non-uniform spherical averages is obtained for measure-preserving actions of free groups. This result generalizes theorems of Grigorichuk, Nevo and Stein [in particular, a simpler proof of the Nevo-Stein theorem about uniform spherical averages is obtained.] The proof uses the Markov operator approach, first proposed by R.I. Grigorchuk. To a measure-preserving action of a free group and a matrix of weights, a Markov operator is assigned in such a way that convergence of spherical averages with corresponding weights is equivalent to convergence of powers of the Markov operator. That last is obtained using Rota's "Alternierende Verfahren"; the 0-2 law for Markov operators in the form of Kaimanovich; and a suitable maximal inequality.

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Article

This is the first in a series of papers showing that Haken manifolds have hyperbolic structures; this first was published, the second two have existed only in preprint form, and later preprints were never completed. This eprint is only an approximation to the published version, which is the definitive form for part I, and is provided for convenience only. All references and quotations should be taken from the published version, since the theorem numbering is different and not all corrections have been incorporated into the present version. Parts II and III will be made available as eprints shortly.

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Article

The purpose of this paper is to give a counterexample of Theorem 10.4 in [Ann. of Math. 102 (1975), 223-290]. In the Harvey-Lawson paper, a global result is claimed, but only a local result is proven. This theorem has had a big impact on CR geometry for almost a quarter of a century because one can use the theory of isolated singularities to study the theory of CR manifolds and vice versa. Comment: 2 pages, published version

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Conference Paper

A graph property is monotone if it is closed under removal of vertices and edges. We consider the following algorithmic problem, called the edge-deletion problem; given a monotone property P and a graph G, compute the smallest number of edge deletions that are needed in order to turn G into a graph satisfying P. We denote this quantity by E′P
(G). Our first result states that the edge-deletion problem can be efficiently approximated for any monotone property.

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Article

On a Riemann surface $S$ of finite type containing a family of $N$ disjoint disks $D_i$ (``islands''), we consider several natural conformal invariants measuring the distance from the islands to $\di S$ and separation between different islands. In a near degenerate situation we establish a relation between them called the Quasi-Additivity Law. We then generalize it to a Quasi-Invariance Law providing us with a transformation rule of the moduli in question under covering maps. This rule (and in particular, its special case called the Covering Lemma) has important applications in holomorphic dynamics which will be addressed in the forthcoming notes.

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Article

For primes p greater than 3, 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.

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Article

In this paper we prove a finiteness result concerning the Chow group of zero-cycles for varieties over $p$-adic local fields. In this final version, there are several corrections concerning mathematical symbols and reference to related known results. Comment: 42 pages (final version, to appear in Ann. of Math., accepted on September 2007) There is no mathematical change from the previous version. Several minor points has been improved, and a new reference [CT5] has been added

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Article

We prove that the canonical dimension of a coadmissible representation of a
semisimple $p$-adic Lie group in a $p$-adic Banach space is either zero or at
least half the dimension of a non-zero coadjoint orbit. To do this we establish
analogues for $p$-adically completed enveloping algebras of Bernstein's
inequality for modules over Weyl algebras, the Beilinson-Bernstein localisation
theorem and Quillen's Lemma about the endomorphism ring of a simple module over
an enveloping algebra.

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Article

We define a new cohomology set for an affine algebraic group G and a
multiplicative finite central subgroup Z, both defined over a local field of
characteristic zero, which is an enlargement of the usual first Galois
cohomology set of G. We show how this set can be used to normalize the
Langlands-Shelstad endoscopic transfer factors and to give a conjectural
description of the internal structure and endoscopic transfer of L-packets for
arbitrary connected reductive groups that extends the well-known conjectural
description for quasi-split groups. In the real case, we show that this
description is correct using Shelstad's work.

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Article

Let p be an odd prime and g an integer greater or equal to 2. We prove that a
finite slope Siegel cuspidal eigenform of genus g can be p-adically deformed
over the g-dimensional weight space. The proof of this result relies on the
construction of a family of sheaves of locally analytic overconvergent modular
forms.

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Article

The u-invariant of a field is the maximum dimension of ansiotropic quadratic
forms over the field. It is an open question whether the u-invariant of
function fields of p-aidc curves is 8. In this paper, we answer this question
in the affirmative for function fields of non-dyadic p-adic curves.

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Article

We produce a canonical filtration for locally free sheaves on an open p-adic annulus equipped with a Frobenius structure. Using this filtration, we deduce a conjecture of Crew on p-adic differential equations, analogous to Grothendieck's local monodromy theorem (also a consequence of results of Andre and of Mebkhout). Namely, given a finite locally free sheaf on an open p-adic annulus with a connection and a compatible Frobenius structure, the corresponding module admits a basis over a finite cover of the annulus on which the connection acts via a nilpotent matrix. Note: this preprint improves on results from our previous preprints math.AG/0102173, math.AG/0105244, math.AG/0106192, math.AG/0106193 but does not explicitly invoke any results from these preprints.

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Article

Freire, Lopes and Mane proved that for any rational map f there exists a natural invariant measure \mu_f [5]. Mane showed there exists an n>0 such that (f^n, \mu_f) is measurably conjugate to the one-sided $d^n$-shift, with Bernoulli measure $(\frac 1{d^n},... ,\frac 1{d^n})$ \[15]. In this paper we show that (f,\mu_f)is conjugate to the one-sided Bernoulli $d$-shift. This verifies a conjecture of Freire, Lopes and Mane [5] and Lyubich [11].

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Article

The study of non-abelian fundamental groups renders it plausible that the principle of Birch and Swinnerton-Dyer, whereby non-vanishing of L-values, in some appropriate sense, accounts for the finiteness of integral points, can eventually be extended to hyperbolic curves. Here we will discuss the very simple case of a genus 1 hyperbolic curve X/Q obtained by removing the origin from an elliptic curve E defined over Q with complex multiplication by an imaginary quadratic field K. Denote by E a Weierstrass minimal model of E and by X the integral model of X obtained as the complement in E of the closure of the origin. Let S be a set of primes including the infinite place and those of bad-reduction for E. We wish to examine the theorem of Siegel, asserting the finiteness of X(ZS), the S-integral points of X, from the point of view of fundamental groups and Selmer varieties. In particular, we show how the finiteness of points can be proved using ‘the method of Coates and Wiles’ which, in essence, makes use of the non-vanishing of p-adic L-functions arising from the situation. That is to say, in studying the set E(ZS)( = E(Z) = E(Q)), Coates and Wiles showed the special case of the conjecture of Birch and Swinnerton-Dyer by deriving the finiteness of E(ZS) from the non-vanishing of L(E/Q, s) at s = 1. Of course the L-function can vanish at 1 in general, in which case E(ZS) is supposed to be infinite. But we know that X(ZS) is always finite. From the perspective

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Article

We prove that the zero locus of an admissible normal function over an algebraic parameter space S is algebraic in the case where S is a curve.

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Article

Let us consider a specialization of an untwisted quantum affine algebra of type $ADE$ at a nonzero complex number, which may or may not be a root of unity. The Grothendieck ring of its finite dimensional representations has two bases, simple modules and standard modules. We identify entries of the transition matrix with special values of ``computable'' polynomials, similar to Kazhdan-Lusztig polynomials. At the same time we ``compute'' $q$-characters for all simple modules. The result is based on ``computations'' of Betti numbers of graded/cyclic quiver varieties. (The reason why we put `` '' will be explained in the end of the introduction.)

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Article

In this paper, we prove global second derivative estimates for solutions of the Dirichlet problem for the Monge-Ampere equation when the inhomogeneous term is only assumed to be Holder continuous. As a consequence of our approach, we also establish the existence and uniqueness of globally smooth solutions to the second boundary value problem for the affine maximal surface equation and affine mean curvature equation.

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