379 research outputs found
Bernstein-Sato polynomial of plane curves and Yano's conjecture
The main aim of this thesis is the study of the Bernstein-Sato polynomial of plane curve singularities. In this context, we prove a conjecture posed by Yano about the generic b-exponents of a plane irreducible curve.
In a part of the thesis, we study the Bernstein-Sato polynomial through the analytic continuation of the complex zeta function of a singularity. We obtain several results on the vanishing and non-vanishing of the residues of the complex zeta function. Using these results we obtain a proof of Yano's conjecture under the hypothesis that the eigenvalues of the monodromy are pair-wise different. In another part of the thesis, we study the periods of integrals in the Milnor fiber and their asymptotic expansion. These periods of integrals can be related to the b-exponents and can be constructed in terms of resolution of singularities. Using these techniques, we can present a proof for the general case of Yano's conjecture.
In addition to the Bernstein-Sato polynomial, we also study the minimal Tjurina number of a plane irreducible curve and we answer in the positive a question raised by Dimca and Greuel on the quotient between the Milnor and Tjurina numbers. More precisely, we prove a formula for the minimal Tjurina number of a plane irreducible curve in terms of the multiplicities of the strict transform along its minimal resolution. From this formula, we obtain the positive answer to Dimca and Greuel question.
This thesis also contains computational results for the theory of singularities on smooth complex surfaces. First, we describe an algorithm to compute log-resolutions of ideals on a smooth complex surface. Secondly, we provide an algorithm to compute generators for complete ideals on a smooth complex surface. These algorithms have several applications, for instance, in the computation of the multiplier ideals associated to an ideal on a smooth complex surface.El principal objectiu d'aquesta tesi és l'estudi del polinomi de Bernstein-Sato de singularitats de corbes planes. En aquest context, es demostra una conjectura proposada per Yano el 1982 sobre els -exponents genèrics d'una corba plana irreductible. En una part d'aquesta tesi, s'estudia el polinomi de Bernstein-Sato utilitzant la continuació analítica de la funció zeta complexa d'una singularitat. S'obtenen diversos resultat sobre l'anul·lació i no anul·lació del residu de la funció zeta complexa d'una corba plana. Utilitzant aquests resultats, s'obté una demostració de la conjectura de Yano sota la hipòtesi de que els valors propis de la monodromia siguin diferents dos a dos. En un altre part de la tesi, s'estudien els períodes d'integrals en la fibra de Milnor i la seva expansió asimptòtica. Aquesta expansió asimptòtica dels períodes pot ser relacionada amb els b-exponents i pot ser construïda en termes de la resolució de singularitats. Utilitzant aquestes tècniques, es presenta una prova del cas general de la conjectura de Yano. A més a més del polinomi de Bernstein-Sato, també s'estudia el nombre de Tjurina mínim d'una corba plana irreductible i responem positivament a una pregunta formulada per Dimca i Greuel sobre el quocient entre els nombres de Milnor i Tjurina. Concretament, es demostra una fórmula pel nombre de Tjurina mínim en un classe d'equisingularitat de corbes planes irreductibles en termes de la seqüència de multiplicitats de la transformada estricta al llarg de la resolució minimal. A partir d'aquesta fórmula, s'obté la resposta positiva a la pregunta de Dimca i Greuel. Aquesta tesi també conté resultats computacionals per la teoria de singularitats en superfícies complexes llises. Primer, es descriu un algorisme que calcula la log-resolució d'ideals en un superfície complexa llisa. En segon lloc, es dona un algorisme per calcular generadors per ideals complets en una superfície complexa llisa. Aquests algorismes tenen diverses aplicacions, com per exemple, en el càlcul d'ideals multiplicadors associats a un ideal en una superfície complexa llisa.Postprint (published version
Bernstein-Sato polynomial of plane curves and Yano's conjecture
The main aim of this thesis is the study of the Bernstein-Sato polynomial of plane curve singularities. In this context, we prove a conjecture posed by Yano about the generic b-exponents of a plane irreducible curve.
In a part of the thesis, we study the Bernstein-Sato polynomial through the analytic continuation of the complex zeta function of a singularity. We obtain several results on the vanishing and non-vanishing of the residues of the complex zeta function. Using these results we obtain a proof of Yano's conjecture under the hypothesis that the eigenvalues of the monodromy are pair-wise different. In another part of the thesis, we study the periods of integrals in the Milnor fiber and their asymptotic expansion. These periods of integrals can be related to the b-exponents and can be constructed in terms of resolution of singularities. Using these techniques, we can present a proof for the general case of Yano's conjecture.
In addition to the Bernstein-Sato polynomial, we also study the minimal Tjurina number of a plane irreducible curve and we answer in the positive a question raised by Dimca and Greuel on the quotient between the Milnor and Tjurina numbers. More precisely, we prove a formula for the minimal Tjurina number of a plane irreducible curve in terms of the multiplicities of the strict transform along its minimal resolution. From this formula, we obtain the positive answer to Dimca and Greuel question.
This thesis also contains computational results for the theory of singularities on smooth complex surfaces. First, we describe an algorithm to compute log-resolutions of ideals on a smooth complex surface. Secondly, we provide an algorithm to compute generators for complete ideals on a smooth complex surface. These algorithms have several applications, for instance, in the computation of the multiplier ideals associated to an ideal on a smooth complex surface.El principal objectiu d'aquesta tesi és l'estudi del polinomi de Bernstein-Sato de singularitats de corbes planes. En aquest context, es demostra una conjectura proposada per Yano el 1982 sobre els -exponents genèrics d'una corba plana irreductible. En una part d'aquesta tesi, s'estudia el polinomi de Bernstein-Sato utilitzant la continuació analítica de la funció zeta complexa d'una singularitat. S'obtenen diversos resultat sobre l'anul·lació i no anul·lació del residu de la funció zeta complexa d'una corba plana. Utilitzant aquests resultats, s'obté una demostració de la conjectura de Yano sota la hipòtesi de que els valors propis de la monodromia siguin diferents dos a dos. En un altre part de la tesi, s'estudien els períodes d'integrals en la fibra de Milnor i la seva expansió asimptòtica. Aquesta expansió asimptòtica dels períodes pot ser relacionada amb els b-exponents i pot ser construïda en termes de la resolució de singularitats. Utilitzant aquestes tècniques, es presenta una prova del cas general de la conjectura de Yano. A més a més del polinomi de Bernstein-Sato, també s'estudia el nombre de Tjurina mínim d'una corba plana irreductible i responem positivament a una pregunta formulada per Dimca i Greuel sobre el quocient entre els nombres de Milnor i Tjurina. Concretament, es demostra una fórmula pel nombre de Tjurina mínim en un classe d'equisingularitat de corbes planes irreductibles en termes de la seqüència de multiplicitats de la transformada estricta al llarg de la resolució minimal. A partir d'aquesta fórmula, s'obté la resposta positiva a la pregunta de Dimca i Greuel. Aquesta tesi també conté resultats computacionals per la teoria de singularitats en superfícies complexes llises. Primer, es descriu un algorisme que calcula la log-resolució d'ideals en un superfície complexa llisa. En segon lloc, es dona un algorisme per calcular generadors per ideals complets en una superfície complexa llisa. Aquests algorismes tenen diverses aplicacions, com per exemple, en el càlcul d'ideals multiplicadors associats a un ideal en una superfície complexa llisa
Bernstein-Sato polynomial of plane curves and Yano's conjecture
The main aim of this thesis is the study of the Bernstein-Sato polynomial of plane curve singularities. In this context, we prove a conjecture posed by Yano about the generic b-exponents of a plane irreducible curve.
In a part of the thesis, we study the Bernstein-Sato polynomial through the analytic continuation of the complex zeta function of a singularity. We obtain several results on the vanishing and non-vanishing of the residues of the complex zeta function. Using these results we obtain a proof of Yano's conjecture under the hypothesis that the eigenvalues of the monodromy are pair-wise different. In another part of the thesis, we study the periods of integrals in the Milnor fiber and their asymptotic expansion. These periods of integrals can be related to the b-exponents and can be constructed in terms of resolution of singularities. Using these techniques, we can present a proof for the general case of Yano's conjecture.
In addition to the Bernstein-Sato polynomial, we also study the minimal Tjurina number of a plane irreducible curve and we answer in the positive a question raised by Dimca and Greuel on the quotient between the Milnor and Tjurina numbers. More precisely, we prove a formula for the minimal Tjurina number of a plane irreducible curve in terms of the multiplicities of the strict transform along its minimal resolution. From this formula, we obtain the positive answer to Dimca and Greuel question.
This thesis also contains computational results for the theory of singularities on smooth complex surfaces. First, we describe an algorithm to compute log-resolutions of ideals on a smooth complex surface. Secondly, we provide an algorithm to compute generators for complete ideals on a smooth complex surface. These algorithms have several applications, for instance, in the computation of the multiplier ideals associated to an ideal on a smooth complex surface.El principal objectiu d'aquesta tesi és l'estudi del polinomi de Bernstein-Sato de singularitats de corbes planes. En aquest context, es demostra una conjectura proposada per Yano el 1982 sobre els -exponents genèrics d'una corba plana irreductible. En una part d'aquesta tesi, s'estudia el polinomi de Bernstein-Sato utilitzant la continuació analítica de la funció zeta complexa d'una singularitat. S'obtenen diversos resultat sobre l'anul·lació i no anul·lació del residu de la funció zeta complexa d'una corba plana. Utilitzant aquests resultats, s'obté una demostració de la conjectura de Yano sota la hipòtesi de que els valors propis de la monodromia siguin diferents dos a dos. En un altre part de la tesi, s'estudien els períodes d'integrals en la fibra de Milnor i la seva expansió asimptòtica. Aquesta expansió asimptòtica dels períodes pot ser relacionada amb els b-exponents i pot ser construïda en termes de la resolució de singularitats. Utilitzant aquestes tècniques, es presenta una prova del cas general de la conjectura de Yano. A més a més del polinomi de Bernstein-Sato, també s'estudia el nombre de Tjurina mínim d'una corba plana irreductible i responem positivament a una pregunta formulada per Dimca i Greuel sobre el quocient entre els nombres de Milnor i Tjurina. Concretament, es demostra una fórmula pel nombre de Tjurina mínim en un classe d'equisingularitat de corbes planes irreductibles en termes de la seqüència de multiplicitats de la transformada estricta al llarg de la resolució minimal. A partir d'aquesta fórmula, s'obté la resposta positiva a la pregunta de Dimca i Greuel. Aquesta tesi també conté resultats computacionals per la teoria de singularitats en superfícies complexes llises. Primer, es descriu un algorisme que calcula la log-resolució d'ideals en un superfície complexa llisa. En segon lloc, es dona un algorisme per calcular generadors per ideals complets en una superfície complexa llisa. Aquests algorismes tenen diverses aplicacions, com per exemple, en el càlcul d'ideals multiplicadors associats a un ideal en una superfície complexa llisa
A Panorama on Multiscale Geometric Representations, Intertwining Spatial, Directional and Frequency Selectivity
The richness of natural images makes the quest for optimal representations in
image processing and computer vision challenging. The latter observation has
not prevented the design of image representations, which trade off between
efficiency and complexity, while achieving accurate rendering of smooth regions
as well as reproducing faithful contours and textures. The most recent ones,
proposed in the past decade, share an hybrid heritage highlighting the
multiscale and oriented nature of edges and patterns in images. This paper
presents a panorama of the aforementioned literature on decompositions in
multiscale, multi-orientation bases or dictionaries. They typically exhibit
redundancy to improve sparsity in the transformed domain and sometimes its
invariance with respect to simple geometric deformations (translation,
rotation). Oriented multiscale dictionaries extend traditional wavelet
processing and may offer rotation invariance. Highly redundant dictionaries
require specific algorithms to simplify the search for an efficient (sparse)
representation. We also discuss the extension of multiscale geometric
decompositions to non-Euclidean domains such as the sphere or arbitrary meshed
surfaces. The etymology of panorama suggests an overview, based on a choice of
partially overlapping "pictures". We hope that this paper will contribute to
the appreciation and apprehension of a stream of current research directions in
image understanding.Comment: 65 pages, 33 figures, 303 reference
Quanta of Maths
The work of Alain Connes has cut a wide swath across several areas of math- ematics and physics. Reflecting its broad spectrum and profound impact on the contemporary mathematical landscape, this collection of articles covers a wealth of topics at the forefront of research in operator algebras, analysis, noncommutative geometry, topology, number theory and physics
Legendrian Weaves: N-graph Calculus, Flag Moduli and Applications
We study a class of Legendrian surfaces in contact five-folds by encoding
their wavefronts via planar combinatorial structures. We refer to these
surfaces as Legendrian weaves, and to the combinatorial objects as N-graphs.
First, we develop a diagrammatic calculus which encodes contact geometric
operations on Legendrian surfaces as multi-colored planar combinatorics.
Second, we present an algebraic-geometric characterization for the moduli space
of microlocal constructible sheaves associated to these Legendrian surfaces.
Then we use these N-graphs and the flag moduli description of these Legendrian
invariants for several new applications to contact and symplectic topology.
Applications include showing that any finite group can be realized as a
subfactor of a 3-dimensional Lagrangian concordance monoid for a Legendrian
surface in the 1-jet space of the two-sphere, a new construction of infinitely
many exact Lagrangian fillings for Legendrian links in the standard contact
three-sphere, and performing rational point counts over finite fields that
distinguish Legendrian surfaces in the standard five-dimensional Darboux chart.
In addition, the manuscript develops the notion of Legendrian mutation,
studying microlocal monodromies and their transformations. The appendix
illustrates the connection between our N-graph calculus for Lagrangian
cobordisms and Elias-Khovanov-Williamson's Soergel Calculus.Comment: 114 Pages, 105 Figure
Introduction to the Gopakumar-Vafa Large N Duality
Gopakumar-Vafa large N duality is a correspondence between Chern-Simons
invariants of a link in a 3-manifold and relative Gromov-Witten invariants of a
6-dimensional symplectic manifold relative to a Lagrangian submanifold. We
address the correspondence between the Chern-Simons free energy of S^3 with no
link and the Gromov-Witten invariant of the resolved conifold in great detail.
This case avoids mathematical difficulties in formulating a definition of
relative Gromov-Witten invariants, but includes all of the important ideas.
There is a vast amount of background material related to this duality. We make
a point of collecting all of the background material required to check this
duality in the case of the 3-sphere, and we have tried to present the material
in a way complementary to the existing literature. This paper contains a large
section on Gromov-Witten theory and a large section on quantum invariants of
3-manifolds. It also includes some physical motivation, but for the most part
it avoids physical terminology.Comment: This is the version published by Geometry & Topology Monographs on 21
September 200
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