Annals of mathematics

The dissertation is written on the basis of the paper published in Annales Polonici Mathematici, the paper published in Acta Mathematica Vietnamica and the paper published in Results in Mathematics.

65p caphesuadathemtieu 02-03-2022 23 6   Download

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 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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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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Annals of Mathematics This is the first in a series of papers whereby we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat’s Last Theorem. In this paper we give new improved bounds for linear forms in three logarithms. We also apply a combination of classical techniques with the modular approach to show that the only perfect powers in the Fibonacci sequence are 0, 1, 8 and 144 and the only perfect powers in the Lucas sequence are 1...

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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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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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Let X be a smooth quasiprojective subscheme of Pn of dimension m ≥ 0 over Fq . Then there exist homogeneous polynomials f over Fq for which the intersection of X and the hypersurface f = 0 is smooth. In fact, the set of such f has a positive density, equal to ζX (m + 1)−1 , where ζX (s) = ZX (q −s ) is the zeta function of X. An analogue for regular quasiprojective schemes over Z is proved, assuming the abc conjecture and another conjecture. 1. Introduction The classical Bertini theorems say that if a subscheme...

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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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R´sum´ anglais e e For a hyperbolic metric on a 3-dimensional manifold, the boundary of its convex core is a surface which is almost everywhere totally geodesic, but which is bent along a family of disjoint geodesics. The locus and intensity of this bending is described by a measured geodesic lamination, which is a topological object.

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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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In this paper we treat the two-variable positive extension problem for trigonometric polynomials where the extension is required to be the reciprocal of the absolute value squared of a stable polynomial. This problem may also be interpreted as an autoregressive filter design problem for bivariate stochastic processes. We show that the existence of a solution is equivalent to solving a finite positive definite matrix completion problem where the completion is required to satisfy an additional low rank condition. ...

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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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We prove an identity of Kloosterman integrals which is the fundamental lemma of a relative trace formula for the general linear group in n variables. 1. Introduction One of the simplest examples of Langlands’ principle of functoriality is the quadratic base change. Namely, let E/F be a quadratic extension of global fields and z
→ z the corresponding Galois conjugation. The base change associates to every automorphic representation π of GL(n, F) an automorphic representation Π of GL(n,E). If n = 1 then π is an id`ele class character and Π(z) = π(zz)....

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We prove that the existence of an automorphism of finite order on a Q-variety X implies the existence of algebraic linear relations between the logarithm of certain periods of X and the logarithm of special values of the Γ-function. This implies that a slight variation of results by Anderson, Colmez and Gross on the periods of CM abelian varieties is valid for a larger class of CM motives. In particular, we prove a weak form of the period conjecture of Gross-Deligne [11, p. 205]1 .

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We study the integral points on surfaces by means of a new method, relying on the Schmidt Subspace Theorem. This method was recently introduced in [CZ] for the case of curves, leading to a new proof of Siegel’s celebrated theorem that any affine algebraic curve defined over a number field has only finitely many S-integral points, unless it has genus zero and not more than two points at infinity. Here, under certain conditions involving the intersection matrix of the divisors at infinity, we shall conclude that the integral points on a surface all lie on a curve. We shall...

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We show that a tensor product of irreducible, finite dimensional representations of a simple Lie algebra over a field of characteristic zero determines the individual constituents uniquely. This is analogous to the uniqueness of prime factorisation of natural numbers. 1. Introduction 1.1. Let g be a simple Lie algebra over C. The main aim of this paper is to prove the following unique factorisation of tensor products of irreducible, finite dimensional representations of g:

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This paper should be regarded as a sequel to [7]. There it was shown that the geometric Langlands conjecture for GLn follows from a certain vanishing conjecture. The goal of the present paper is to prove this vanishing conjecture. Let X be a smooth projective curve over a ground field k. Let E be an m-dimensional local system on X, and let Bunm be the moduli stack of rank m vector bundles on X.

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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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