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Abstract: We construct examples of p-adic L-functions over universal deformation spaces for \({{\,\mathrm{GL}\,}}_2\) . We formulate a conjecture predicting that the natural parameter spaces for p-adic L-functions and Euler systems are not the usual eigenvarieties (parametrising nearly-ordinary families of automorphic representations), but other, larger spaces depending on a choice of a parabolic subgroup, which we call ‘big parabolic eigenvarieties’. PubDate: 2021-12-13
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Abstract: Let \(G=A *B\) be a free product of freely indecomposable groups. We explicitly construct quasimorphisms on G which are invariant with respect to all automorphisms of G. We also prove that the space of such quasimorphisms is infinite-dimensional whenever G is not the infinite dihedral group. As an application we prove that an invariant analogue of stable commutator length recently introduced by Kawasaki and Kimura is non-trivial for these groups. PubDate: 2021-12-03
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Abstract: We consider the case of scattering by several obstacles in \({\mathbb {R}}^d\) , \(d \ge 2\) for the Laplace operator \(\Delta \) with Dirichlet boundary conditions imposed on the obstacles. In the case of two obstacles, we have the Laplace operators \(\Delta _1\) and \(\Delta _2\) obtained by imposing Dirichlet boundary conditions only on one of the objects. The relative operator \(g(\Delta ) - g(\Delta _1) - g(\Delta _2) + g(\Delta _0)\) was introduced in Hanisch, Waters and one of the authors in (A relative trace formula for obstacle scattering. arXiv:2002.07291, 2020) and shown to be trace-class for a large class of functions g, including certain functions of polynomial growth. When g is sufficiently regular at zero and fast decaying at infinity then, by the Birman–Krein formula, this trace can be computed from the relative spectral shift function \(\xi _\mathrm {rel}(\lambda ) = -\frac{1}{\pi } {\text {Im}}(\Xi (\lambda ))\) , where \(\Xi (\lambda )\) is holomorphic in the upper half-plane and fast decaying. In this paper we study the wave-trace contributions to the singularities of the Fourier transform of \(\xi _\mathrm {rel}\) . In particular we prove that \({\hat{\xi }}_\mathrm {rel}\) is real-analytic near zero and we relate the decay of \(\Xi (\lambda )\) along the imaginary axis to the first wave-trace invariant of the shortest bouncing ball orbit between the obstacles. The function \(\Xi (\lambda )\) is important in the physics of quantum fields as it determines the Casimir interactions between the objects. PubDate: 2021-11-25
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Abstract: We consider an automorphism of an arbitrary CAT(0) cube complex. We study its combinatorial displacement and we show that either the automorphism has a fixed point or it preserves some combinatorial axis. It follows that when a f.g. group contains a distorted cyclic subgroup, it admits no proper action on a discrete space with walls. As an application Baumslag-Solitar groups and Heisenberg groups provide examples of groups having a proper action on measured spaces with walls, but no proper action on a discrete space with wall. PubDate: 2021-11-24
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Abstract: This paper is devoted to the analysis of Steklov eigenvalues and Steklov eigenfunctions on a class of warped product Riemannian manifolds (M, g) whose boundary \(\partial M\) consists in two distinct connected components \(\Gamma _0\) and \(\Gamma _1\) . First, we show that the Steklov eigenvalues can be divided into two families \((\lambda _m^\pm )_{m \ge 0}\) which satisfy accurate asymptotics as \(m \rightarrow \infty \) . Second, we consider the associated Steklov eigenfunctions which are the harmonic extensions of the boundary Dirichlet to Neumann eigenfunctions. In the case of symmetric warped product, we prove that the Steklov eigenfunctions are exponentially localized on the whole boundary \(\partial M\) as \(m \rightarrow \infty \) . When we add an asymmetric perturbation of the metric to a symmetric warped product, we observe in almost all cases a flea on the elephant effect. Roughly speaking, we prove that “half” the Steklov eigenfunctions are exponentially localized on one connected component of the boundary, say \(\Gamma _0\) , and the other half on the other connected component \(\Gamma _1\) as \(m \rightarrow \infty \) . PubDate: 2021-11-20
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Abstract: Let \(\ell \) be a rational prime. Previously, abelian \(\ell \) -towers of multigraphs were introduced which are analogous to \({\mathbb {Z}}_{\ell }\) -extensions of number fields. It was shown that for a certain class of towers of bouquets, the growth of the \(\ell \) -part of the number of spanning trees behaves in a predictable manner (analogous to a well-known theorem of Iwasawa for \({\mathbb {Z}}_{\ell }\) -extensions of number fields). In this paper, we give a generalization to a broader class of regular abelian \(\ell \) -towers of bouquets than was originally considered. To carry this out, we observe that certain shifted Chebyshev polynomials are members of a continuously parametrized family of power series with coefficients in \({\mathbb {Z}}_{\ell }\) and then study the special value at \(u=1\) of the Artin-Ihara L-function \(\ell \) -adically. PubDate: 2021-11-20
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Abstract: Our goal is the discussion of the problem of mathematical interpretation of basic postulates (or “principles”) of Quantum Mechanics: transitions to quantum stationary orbits, the wave-particle duality, and the probabilistic interpretation, in the context of semiclassical self-consistent Maxwell–Schrödinger equations. We discuss possible dynamical interpretation of these postulates relying on a new general mathematical conjecture on global attractors of G-invariant nonlinear Hamiltonian partial differential equations with a Lie symmetry group G. This conjecture is inspired by the results on global attractors of nonlinear Hamiltonian PDEs obtained by the author together with his collaborators since 1990 for a list of model equations with three basic symmetry groups: the trivial group, the group of translations, and the unitary group \(\mathbf {U}(1)\) . We sketch these results. PubDate: 2021-11-01
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Abstract: In this note we give a detailed construction of a \(\Lambda \) -adic \(\mathfrak d\) th Shintani lifting. We obtain a \(\Lambda \) -adic version of Kohnen’s formula relating Fourier coefficients of half-integral weight modular forms and special values of twisted L-series. As a by-product, we derive a mild generalization of such classical formulae, and also point out a relation between Fourier coefficients of \(\Lambda \) -adic \(\mathfrak d\) th Shintani liftings and Stark–Heegner points. PubDate: 2021-10-30
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Abstract: We review the concept of the limiting absorption principle and its connection to virtual levels of operators in Banach spaces. PubDate: 2021-10-28
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Abstract: This paper deals with a version of the two-timing method which describes various ‘slow’ effects caused by externally imposed ‘fast’ oscillations. Such small oscillations are often called vibrations and the research area can be referred as vibrodynamics. The governing equations represent a generic system of first-order ODEs containing a prescribed oscillating velocity \({\varvec{u}}\) , given in a general form. Two basic small parameters stand in for the inverse frequency and the ratio of two time-scales; they appear in equations as regular perturbations. The proper connections between these parameters yield the distinguished limits, leading to the existence of closed systems of asymptotic equations. The aim of this paper is twofold: (i) to clarify (or to demystify) the choices of a slow variable, and (ii) to give a coherent exposition which is accessible for practical users in applied mathematics, sciences and engineering. We focus our study on the usually hidden aspects of the two-timing method such as the uniqueness or multiplicity of distinguished limits and universal structures of averaged equations. The main result is the demonstration that there are two (and only two) different distinguished limits. The explicit instruction for practically solving ODEs for different classes of \({\varvec{u}}\) is presented. The key roles of drift velocity and the qualitatively new appearance of the linearized equations are discussed. To illustrate the broadness of our approach, two examples from mathematical biology are shown. PubDate: 2021-10-21
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Abstract: Two fluid configurations along a flow are conjugate if there is a one parameter family of geodesics (fluid flows) joining them to infinitesimal order. Geometrically, they can be seen as a consequence of the (infinite dimensional) group of volume preserving diffeomorphisms having sufficiently strong positive curvatures which ‘pull’ nearby flows together. Physically, they indicate a form of (transient) stability in the configuration space of particle positions: a family of flows starting with the same configuration deviate initially and subsequently re-converge (resonate) with each other at some later moment in time. Here, we first establish existence of conjugate points in an infinite family of Kolmogorov flows—a class of stationary solutions of the Euler equations—on the rectangular flat torus of any aspect ratio. The analysis is facilitated by a general criterion for identifying conjugate points in the group of volume preserving diffeomorphisms. Next, we show non-existence of conjugate points along Arnold stable steady states on the annulus, disk and channel. Finally, we discuss cut points, their relation to non-injectivity of the exponential map (impossibility of determining a flow from a particle configuration at a given instant) and show that the closest cut point to the identity is either a conjugate point or the midpoint of a time periodic Lagrangian fluid flow. PubDate: 2021-10-20
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Abstract: We consider the equation \(q_t+qq_x=q_{xx}\) for \(q:{{\mathbf {R}}}\times (0,\infty )\rightarrow {\mathbf {H}}\) (the quaternions), and show that while singularities can develop from smooth compactly supported data, such situations are non-generic. The singularities will disappear under an arbitrary small “generic” smooth perturbation of the initial data. Similar results are also established for the same equation in \(\mathbf{S}^1\times (0,\infty )\) , where \(\mathbf{S}^1\) is the standard one-dimensional circle. PubDate: 2021-10-20
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Abstract: Our objective in this series of two articles, of which the present article is the first, is to give a Perrin-Riou-style construction of p-adic L-functions (of Bellaïche and Stevens) over the eigencurve. As the first ingredient, we interpolate the Beilinson–Kato elements over the eigencurve (including the neighborhoods of \(\theta \) -critical points). Along the way, we prove étale variants of Bellaïche’s results describing the local properties of the eigencurve. We also develop the local framework to construct and establish the interpolative properties of these p-adic L-functions away from \(\theta \) -critical points. PubDate: 2021-10-08 DOI: 10.1007/s40316-021-00172-8
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Abstract: This paper is a synopsis of the recent book [9]. The latter is dedicated to the stochastic Burgers equation as a model for 1d turbulence, and the paper discusses its content in relation to the Kolmogorov theory of turbulence. PubDate: 2021-10-05 DOI: 10.1007/s40316-021-00174-6
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Abstract: In this computational paper we verify a truncated version of the Buzzard–Calegari conjecture on the Newton polygon of the Hecke operator \(T_2\) for all large enough weights. We first develop a formula for computing p-adic valuations of exponential sums, which we then implement to compute 2-adic valuations of traces of Hecke operators acting on spaces of cusp forms. Finally, we verify that if Newton polygon of the Buzzard–Calegari polynomial has a vertex at \(n\le 15\) , then it agrees with the Newton polygon of \(T_2\) up to n. PubDate: 2021-10-01 DOI: 10.1007/s40316-020-00149-z
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Abstract: We give a criterion for the transversality of a curve embedded in a product of elliptic curves. We then apply our criterion to some explicit classes of curves. The transversality allows us to apply theorems that produce explicit and implementable bounds for the height of the rational points on the curves. PubDate: 2021-10-01 DOI: 10.1007/s40316-021-00161-x
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Abstract: The hyperbolic center of mass of a finite measure on the unit ball with respect to a radially increasing weight is shown to exist, be unique, and depend continuously on the measure. Prior results of this type are extended by characterizing the center of mass as the minimum point of an energy functional that is strictly convex along hyperbolic geodesics. A special case is Hersch’s center of mass lemma on the sphere, which follows from convexity of a logarithmic kernel introduced by Douady and Earle. PubDate: 2021-10-01 DOI: 10.1007/s40316-020-00151-5
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Abstract: In this article, we study the growth of (fine) Selmer groups of elliptic curves in certain infinite Galois extensions where the Galois group G is a uniform, pro-p, p-adic Lie group. By comparing the growth of (fine) Selmer groups with that of class groups, we show that it is possible for the \(\mu \) -invariant of the (fine) Selmer group to become arbitrarily large in a certain class of nilpotent, uniform, pro-p Lie extension. We also study the growth of fine Selmer groups in false Tate curve extensions. PubDate: 2021-10-01 DOI: 10.1007/s40316-020-00147-1
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Abstract: Building on previous results [17, 35], we complete the classification of compact oriented Einstein 4-manifolds with \(\det (W^+) > 0\) . There are, up to diffeomorphism, exactly 15 manifolds that carry such metrics, and, on each of these manifolds, such metrics sweep out exactly one connected component of the corresponding Einstein moduli space. PubDate: 2021-10-01 DOI: 10.1007/s40316-020-00154-2
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