Annals of mathematics

Annals of Mathematics This paper is the third in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed 3-manifold. In [CM3]–[CM5] we describe the case where the surfaces are topologically disks on any fixed small scale. Although the focus of this paper, general planar domains, is more in line with [CM6], we will prove a result here (namely, Corollary III.

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Annals of Mathematics In our first article [2] we developed a new view of Gauss composition of binary quadratic forms which led to several new laws of composition on various other spaces of forms. Moreover, we showed that the groups arising from these composition laws were closely related to the class groups of orders in quadratic number fields, while the spaces underlying those composition laws were closely related to certain exceptional Lie groups.

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Annals of Mathematics By S. Artstein, V. Milman, and S. J. Szarek For two convex bodies K and T in Rn , the covering number of K by T , denoted N (K, T ), is defined as the minimal number of translates of T needed to cover K. Let us denote by K ◦ the polar body of K and by D the euclidean unit ball in Rn . We prove that the two functions of t, N (K, tD) and N (D, tK ◦ ), are equivalent in the appropriate sense, uniformly over symmetric convex bodies K ⊂...

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Annals of Mathematics By Curtis T. McMullen* .Annals of Mathematics, 165 (2007), 397–456 Dynamics of SL2(R) over moduli space in genus two By Curtis T. McMullen* Abstract This paper classifies orbit closures and invariant measures for the natural action of SL2 (R) on ΩM2 , the bundle of holomorphic 1-forms over the moduli space of Riemann surfaces of genus two. Contents 1. Introduction 2. Dynamics and Lie groups 3. Riemann surfaces and holomorphic 1-forms 4. Abelian varieties with real multiplication 5. Recognizing eigenforms 6. Algebraic sums of 1-forms 7. Connected sums of 1-forms 8.

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We prove that if f (x) = n−1 ak xk is a polynomial with no cyclotomic k=0 factors whose coefficients satisfy ak ≡ 1 mod 2 for 0 ≤ k 1 + log 3 , 2n resolving a conjecture of Schinzel and Zassenhaus [21] for this class of polynomials. More generally, we solve the problems of Lehmer and Schinzel and Zassenhaus for the class of polynomials

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Annals of Mathematics We study the motion of an incompressible perfect liquid body in vacuum. This can be thought of as a model for the motion of the ocean or a star. The free surface moves with the velocity of the liquid and the pressure vanishes on the free surface. This leads to a free boundary problem for Euler’s equations, where the regularity of the boundary enters to highest order. We prove local existence in Sobolev spaces assuming a “physical condition”, related to the fact that the pressure of a fluid has to be positive. ...

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Annals of Mathematics In this paper we will solve one of the central problems in dynamical systems: Theorem 1 (Density of hyperbolicity for real polynomials). Any real polynomial can be approximated by hyperbolic real polynomials of the same degree. Here we say that a real polynomial is hyperbolic or Axiom A, if the real line is the union of a repelling hyperbolic set, the basin of hyperbolic attracting periodic points and the basin of infinity.

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We verify an old conjecture of G. P´lya and G. Szeg˝ saying that the o o regular n-gon minimizes the logarithmic capacity among all n-gons with a fixed area. 1. Introduction The logarithmic capacity cap E of a compact set E in R2 , which we identify with the complex plane C, is defined by (1.1) − log cap E = lim (g(z, ∞) − log |z|), z→∞ where g(z, ∞) denotes the Green function of a connected component Ω(E) ∞ of C \ E having singularity at z = ∞; see [4, Ch. 7], [7, §11.1]. By an n-gon with...

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We completely classify diffeomorphism covariant local nets of von Neumann algebras on the circle with central charge c less than 1. The irreducible ones are in bijective correspondence with the pairs of A-D2n-E6,8 Dynkin diagrams such that the difference of their Coxeter numbers is equal to 1. We first identify the nets generated by irreducible representations of the Virasoro algebra for c

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We show that unital simple C ∗ -algebras with tracial topological rank zero which are locally approximated by subhomogeneous C ∗ -algebras can be classified by their ordered K-theory. We apply this classification result to show that certain simple crossed products are isomorphic if they have the same ordered K-theory. In particular, irrational higher dimensional noncommutative tori of the form C(Tk ) ×θ Z are in fact inductive limits of circle algebras. Introduction In recent years there has been rapid progress in classification of nuclear simple C ...

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This paper is the first in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed Riemannian 3-manifold. The key for understanding such surfaces is to understand the local structure in a ball and in particular the structure of an embedded minimal disk in a ball in R3 (with the flat metric). This study is undertaken here and completed in [CM6]. These local results are then applied in [CM7] where we describe the general structure of fixed genus surfaces in 3-manifolds. There are two local models for...

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The uniform spanning forest (USF) in Zd is the weak limit of random, uniformly chosen, spanning trees in [−n, n]d . Pemantle [11] proved that the USF consists a.s. of a single tree if and only if d ≤ 4. We prove that any two components of the USF in Zd are adjacent a.s. if 5 ≤ d ≤ 8, but not if d ≥ 9. More generally, let N (x, y) be the minimum number of edges outside the USF in a path joining x and y in Zd . Then max N (x, y) : x, y ∈ Zd = (d − 1)/4 a.s.

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This paper is the fourth in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed 3manifold. The key is to understand the structure of an embedded minimal disk in a ball in R3 . This was undertaken in [CM3], [CM4] and the global version of it will be completed here; see the discussion around Figure 12 for the local case and [CM15] for some more details. Our main results are Theorem 0.1 (the lamination theorem) and Theorem 0.2 (the one-sided curvature estimate). ...

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Statement of the main result. Denote by Δ(r) the disk of radius r in C, Δ := Δ(1), and for 0

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Reduction of the singularities of codimension one singular foliations in dimension three By Felipe Cano Contents 0. Introduction 1. Blowing-up singular foliations 1.1. Adapted singular foliations 1.2. Permissible centers 1.3. Vertical invariants 1.4. First properties of presimple singularities 2. Global strategy 2.1. Reduction to presimple singularities. Statement 2.2. Good points. Bad points. Equi-reduction 2.3. Finiteness of bad points 2.4. The influency locus 2.5. The local control theorem 2.6. Destroying cycles 2.7. Global criteria of blowing-up 3. Local control 3.1.

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We 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.

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We introduce and study “isomonodromy” transformations of the matrix linear difference equation Y (z + 1) = A(z)Y (z) with polynomial A(z). Our main result is construction of an isomonodromy action of Zm(n+1)−1 on the space of coefficients A(z) (here m is the size of matrices and n is the degree of A(z)). The (birational) action of certain rank n subgroups can be described by difference analogs of the classical Schlesinger equations, and we prove that for generic initial conditions these difference Schlesinger equations have a unique solution. ...

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In this paper we introduce a method for partial description of the Poisson boundary for a certain class of groups acting on a segment. As an application we find among the groups of subexponential growth those that admit nonconstant bounded harmonic functions with respect to some symmetric (infinitely supported) measure µ of finite entropy H(µ). This implies that the entropy h(µ) of the corresponding random walk is (finite and) positive. As another application we exhibit certain discontinuity for the recurrence property of random walks. ...

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In this paper we prove C k structural stability conjecture for unimodal maps. In other words, we shall prove that Axiom A maps are dense in the space of C k unimodal maps in the C k topology. Here k can be 1, 2, . . . , ∞, ω. 1. Introduction 1.1. The structural stability conjecture. The structural stability conjecture was and remains one of the most interesting and important open problems in the theory of dynamical systems. This conjecture states that a dynamical system is structurally stable if and only if it satisfies Axiom A and the...

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Surfaces of sections are a classical tool in the study of 3-dimensional dynamical systems. Their use goes back to the work of Poincar´ and Birkhoff. e In the present paper we give a natural generalization of this concept by constructing a system of transversal sections in the complement of finitely many distinguished periodic solutions. Such a system is established for nondegenerate Reeb flows on the tight 3-sphere by means of pseudoholomorphic curves.

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