Annals of Mathematics

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Print ISSN: 0003-486X


Figure 1. 
Representation theoretic patterns in three dimensional Cryo-Electron Microscopy I: The intrinsic reconstitution algorithm
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September 2011


65 Reads


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.

Noise stability of functions with low influences: Invariance and optimality

November 2005


80 Reads

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.

Constructible exponential functions, motivic Fourier transform and transfer principle, math.AG/0512022

December 2005


51 Reads

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.

The weak type $(1,1)$ bounds for the maximal function associated to cubes grow to infinity with the dimension

May 2008


208 Reads

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.

Chern slopes of simply connected complex surfaces of general type are dense in [2,3]

February 2014


66 Reads

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 $q$.

Figure 1: Construction from the proof of Proposition 4.4. 
Ergodicity of the 2D Navier-Stokes Equations with Degenerate Stochastic Forcing

June 2004


173 Reads

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

The local converse theorem for SO(2n+1) and applications

March 2004


61 Reads

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 $\SO_{2n+1}(k)$.

Closed Characteristics on Compact Convex Hypersurfaces in R 2n

October 2001


124 Reads

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.

Imaginary geometry III: reversibility of SLE_\kappa\ for \kappa \in (4,8)

January 2012


46 Reads

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.

5n Minkowski Symmetrizations Suffice to Arrive at an Approximate Euclidean Ball

May 2002


27 Reads

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.

Derived equivalences for symmetric groups and sl\_2-categorification

August 2004


51 Reads

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.

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

April 2010


67 Reads

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.

Some Results on Green's Higher Abel-Jacobi Map

February 1999


37 Reads

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

Hodge Structures on Abelian Varieties of Type III

November 2004


27 Reads

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.

The Brauer-Manin obstruction for subvarieties of abelian varieties over function fields

January 2007


37 Reads

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.

The Cassels-Tate Pairing on Polarized Abelian Varieties

October 1999


20 Reads

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

Absence of mixing in area-preserving flows on surfaces

March 2009


31 Reads

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

The absolutely continuous spectrum of Jacobi matrices

June 2007


33 Reads

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

On Seminorms and Probabilities, and Abstract Wiener Spaces

July 2010


90 Reads

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.

On Mott's formula for the ac-conductivity in the Anderson model

August 2005


32 Reads

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 $C \nu^2 (\log \frac 1 \nu)^{d+1}$.

Groups acting properly on "bolic" spaces and the Novikov conjecture

March 2004


43 Reads

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.

Isolation, equidistribution, and orbit closures for the SL(2,R) action on Moduli space

May 2013


287 Reads

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.

Complete Moduli in the Presence of Semiabelian Group Action

June 1999


27 Reads

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.

First cohomology and local rigidity of group actions

May 2005


61 Reads

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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