Igusa's $p$-Adic Local Zeta Function and the Monodromy Conjecture for Non-Degenerate Surface Singularities PDF Download
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Author: Bart Bories
Publisher: American Mathematical Soc.
ISBN: 147041841X
Category : Mathematics
Languages : en
Pages : 146
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Book Description
In 2011 Lemahieu and Van Proeyen proved the Monodromy Conjecture for the local topological zeta function of a non-degenerate surface singularity. The authors start from their work and obtain the same result for Igusa's p-adic and the motivic zeta function. In the p-adic case, this is, for a polynomial f∈Z[x,y,z] satisfying f(0,0,0)=0 and non-degenerate with respect to its Newton polyhedron, we show that every pole of the local p-adic zeta function of f induces an eigenvalue of the local monodromy of f at some point of f−1(0)⊂C3 close to the origin. Essentially the entire paper is dedicated to proving that, for f as above, certain candidate poles of Igusa's p-adic zeta function of f, arising from so-called B1-facets of the Newton polyhedron of f, are actually not poles. This turns out to be much harder than in the topological setting. The combinatorial proof is preceded by a study of the integral points in three-dimensional fundamental parallelepipeds. Together with the work of Lemahieu and Van Proeyen, this main result leads to the Monodromy Conjecture for the p-adic and motivic zeta function of a non-degenerate surface singularity.
Author: Bart Bories
Publisher: American Mathematical Soc.
ISBN: 147041841X
Category : Mathematics
Languages : en
Pages : 146
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Book Description
In 2011 Lemahieu and Van Proeyen proved the Monodromy Conjecture for the local topological zeta function of a non-degenerate surface singularity. The authors start from their work and obtain the same result for Igusa's p-adic and the motivic zeta function. In the p-adic case, this is, for a polynomial f∈Z[x,y,z] satisfying f(0,0,0)=0 and non-degenerate with respect to its Newton polyhedron, we show that every pole of the local p-adic zeta function of f induces an eigenvalue of the local monodromy of f at some point of f−1(0)⊂C3 close to the origin. Essentially the entire paper is dedicated to proving that, for f as above, certain candidate poles of Igusa's p-adic zeta function of f, arising from so-called B1-facets of the Newton polyhedron of f, are actually not poles. This turns out to be much harder than in the topological setting. The combinatorial proof is preceded by a study of the integral points in three-dimensional fundamental parallelepipeds. Together with the work of Lemahieu and Van Proeyen, this main result leads to the Monodromy Conjecture for the p-adic and motivic zeta function of a non-degenerate surface singularity.
Author: Yakir Aharonov
Publisher: American Mathematical Soc.
ISBN: 1470423243
Category : Mathematics
Languages : en
Pages : 120
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Book Description
In the past 50 years, quantum physicists have discovered, and experimentally demonstrated, a phenomenon which they termed superoscillations. Aharonov and his collaborators showed that superoscillations naturally arise when dealing with weak values, a notion that provides a fundamentally different way to regard measurements in quantum physics. From a mathematical point of view, superoscillating functions are a superposition of small Fourier components with a bounded Fourier spectrum, which result, when appropriately summed, in a shift that can be arbitrarily large, and well outside the spectrum. The purpose of this work is twofold: on one hand the authors provide a self-contained survey of the existing literature, in order to offer a systematic mathematical approach to superoscillations; on the other hand, they obtain some new and unexpected results, by showing that superoscillating sequences can be seen of as solutions to a large class of convolution equations and can therefore be treated within the theory of analytically uniform spaces. In particular, the authors will also discuss the persistence of the superoscillatory behavior when superoscillating sequences are taken as initial values of the Schrödinger equation and other equations.
Author: Th. De Pauw
Publisher: American Mathematical Soc.
ISBN: 1470423359
Category : Mathematics
Languages : en
Pages : 128
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Book Description
The authors consider a category of pairs of compact metric spaces and Lipschitz maps where the pairs satisfy a linearly isoperimetric condition related to the solvability of the Plateau problem with partially free boundary. It includes properly all pairs of compact Lipschitz neighborhood retracts of a large class of Banach spaces. On this category the authors define homology and cohomology functors with real coefficients which satisfy the Eilenberg-Steenrod axioms, but reflect the metric properties of the underlying spaces. As an example they show that the zero-dimensional homology of a space in our category is trivial if and only if the space is path connected by arcs of finite length. The homology and cohomology of a pair are, respectively, locally convex and Banach spaces that are in duality. Ignoring the topological structures, the homology and cohomology extend to all pairs of compact metric spaces. For locally acyclic spaces, the authors establish a natural isomorphism between their cohomology and the Čech cohomology with real coefficients.
Author: Ivan Cheltsov
Publisher: American Mathematical Soc.
ISBN: 1470423162
Category : Mathematics
Languages : en
Pages : 130
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Book Description
The authors prove that every quasi-smooth weighted Fano threefold hypersurface in the 95 families of Fletcher and Reid is birationally rigid.
Author: Thierry Daudé
Publisher: American Mathematical Soc.
ISBN: 1470423766
Category : Mathematics
Languages : en
Pages : 126
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Book Description
In this paper, the authors study the direct and inverse scattering theory at fixed energy for massless charged Dirac fields evolving in the exterior region of a Kerr-Newman-de Sitter black hole. In the first part, they establish the existence and asymptotic completeness of time-dependent wave operators associated to our Dirac fields. This leads to the definition of the time-dependent scattering operator that encodes the far-field behavior (with respect to a stationary observer) in the asymptotic regions of the black hole: the event and cosmological horizons. The authors also use the miraculous property (quoting Chandrasekhar)—that the Dirac equation can be separated into radial and angular ordinary differential equations—to make the link between the time-dependent scattering operator and its stationary counterpart. This leads to a nice expression of the scattering matrix at fixed energy in terms of stationary solutions of the system of separated equations. In a second part, the authors use this expression of the scattering matrix to study the uniqueness property in the associated inverse scattering problem at fixed energy. Using essentially the particular form of the angular equation (that can be solved explicitly by Frobenius method) and the Complex Angular Momentum technique on the radial equation, the authors are finally able to determine uniquely the metric of the black hole from the knowledge of the scattering matrix at a fixed energy.
Author: Xin-Rong Dai
Publisher: American Mathematical Soc.
ISBN: 1470420155
Category : Mathematics
Languages : en
Pages : 116
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Book Description
A longstanding problem in Gabor theory is to identify time-frequency shifting lattices aZ×bZ and ideal window functions χI on intervals I of length c such that {e−2πinbtχI(t−ma): (m,n)∈Z×Z} are Gabor frames for the space of all square-integrable functions on the real line. In this paper, the authors create a time-domain approach for Gabor frames, introduce novel techniques involving invariant sets of non-contractive and non-measure-preserving transformations on the line, and provide a complete answer to the above abc-problem for Gabor systems.
Author: Toshihiko Masuda
Publisher: American Mathematical Soc.
ISBN: 1470420163
Category : Mathematics
Languages : en
Pages : 128
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Book Description
The authors will classify Rohlin flows on von Neumann algebras up to strong cocycle conjugacy. This result provides alternative approaches to some preceding results such as Kawahigashi's classification of flows on the injective type II1 factor, the classification of injective type III factors due to Connes, Krieger and Haagerup and the non-fullness of type III0 factors. Several concrete examples are also studied.
Author: Hans Lundmark
Publisher: American Mathematical Soc.
ISBN: 1470420260
Category : Mathematics
Languages : en
Pages : 102
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Book Description
The authors solve a spectral and an inverse spectral problem arising in the computation of peakon solutions to the two-component PDE derived by Geng and Xue as a generalization of the Novikov and Degasperis-Procesi equations. Like the spectral problems for those equations, this one is of a "discrete cubic string" type, but presents some interesting novel features.
Author: Reiner Hermann:
Publisher: American Mathematical Soc.
ISBN: 1470419955
Category : Mathematics
Languages : en
Pages : 158
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Book Description
In this monograph, the author extends S. Schwede's exact sequence interpretation of the Gerstenhaber bracket in Hochschild cohomology to certain exact and monoidal categories. Therefore the author establishes an explicit description of an isomorphism by A. Neeman and V. Retakh, which links Ext-groups with fundamental groups of categories of extensions and relies on expressing the fundamental group of a (small) category by means of the associated Quillen groupoid. As a main result, the author shows that his construction behaves well with respect to structure preserving functors between exact monoidal categories. The author uses his main result to conclude, that the graded Lie bracket in Hochschild cohomology is an invariant under Morita equivalence. For quasi-triangular bialgebras, he further determines a significant part of the Lie bracket's kernel, and thereby proves a conjecture by L. Menichi. Along the way, the author introduces n-extension closed and entirely extension closed subcategories of abelian categories, and studies some of their properties.
Author: Joseph Hundley
Publisher: American Mathematical Soc.
ISBN: 1470416670
Category : Mathematics
Languages : en
Pages : 138
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Book Description
In this paper the authors provide an extension of the theory of descent of Ginzburg-Rallis-Soudry to the context of essentially self-dual representations, that is, representations which are isomorphic to the twist of their own contragredient by some Hecke character. The authors' theory supplements the recent work of Asgari-Shahidi on the functorial lift from (split and quasisplit forms of) GSpin2n to GL2n.
