Search

We provide an effective version of Katz’ criterion for finiteness of the monodromy group of a lisse, pure of weight zero, -adic sheaf on a normal variety over a finite field, depending on the numerical complexity of the sheaf.

A meromorphic function on a compact complex analytic manifold defines a C∞ locally trivial bundle over the complement to a finite subset of the projective line CP1, the bifurcation set. The monodromy transformations of this bundle correspond to loops around the points of the bifurcation set. In this paper we show that the zeta functions of these monodromy transformations {reviewer's remark: the inverse of the one defined by A'Campo} can be expressed in local terms, namely as integrals of the zeta functions of meromorphic germs with respect to the Euler characteristic. A special case of a meromorphic function on the projective space CPn is a function defined by a polynomial in n variables. We describe some applications of our technique to polynomial functions., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

We provide an effective version of Katz’ criterion for finiteness of the monodromy group of a lisse, pure of weight zero, -adic sheaf on a normal variety over a finite field, depending on the numerical complexity of the sheaf.

The aim of this paper is to prove that a large class of quaternionic slice regular functions result to be (ramified) covering maps. By means of the topological implications of this fact and by providing further topological structures, we are able to give suitable natural conditions for the existence of k-th ⋆ -roots of a slice regular function. Moreover, we are also able to compute all the solutions which, quite surprisingly, in the most general case, are in number of k2. The last part is devoted to compute the monodromy and to present a technique to compute all the k2 roots starting from one of them.

We formulate and prove Cutkosky’s Theorem regarding the discontinuity of Feynman integrals in the massive one-loop case up to the involved intersection index. This is done by applying the techniques to treat singular integrals developed in Fotiadi et al. (Topology 4(2):159–191, 1965) . We write one-loop integrals as an integral of a holomorphic family of holomorphic forms over a compact cycle. Then, we determine at which points simple pinches occur and explicitly compute a representative of the corresponding vanishing sphere. This also yields an algorithm to compute the Landau surface of a one-loop graph without explicitly solving the Landau equations. We also discuss the bubble, triangle and box graph in detail., Peer Reviewed

We prove that the mod 2 Betti numbers of double coverings of a complex hyperplane arrangement complement are combinatorially determined. The proof is based on a relation between the mod 2 Aomoto complex and the transfer long exact sequence. Applying the above result to the icosidodecahedral arrangement (16 planes in the three-dimensional space related to the icosidodecahedron), we conclude that the first homology of the Milnor fiber has non-trivial 2-torsion.

We prove that the mod 2 Betti numbers of double coverings of a complex hyperplane arrangement complement are combinatorially determined. The proof is based on a relation between the mod 2 Aomoto complex and the transfer long exact sequence. Applying the above result to the icosidodecahedral arrangement (16 planes in the three-dimensional space related to the icosidodecahedron), we conclude that the first homology of the Milnor fiber has non-trivial 2-torsion.

Given a prime p and an integer d > 1, we give a numerical criterion to decide whether the ℓ-adic sheaf associated to the one-parameter exponential sums t 7→P x ψ(xd + tx) over Fp has finite monodromy or not, and work out some explicit cases where this is computable.

We exhibit a rigid local system of rank six on the affine line in characteristic p = 5 whose arithmetic and geometric monodromy groups are the finite group 2.J2 (J2 the Hall-Janko sporadic group) in one of its two (Galois-conjugate) irreducible representation of degree six.

In this article we study numerically and theoretically the asymptotics of the algebraic part of the spec- trum for the quasi-exactly solvable sextic potential m,b(x) = x6 + 2bx4 + (b2 − (4m + 3))x2, its level crossing points, and its monodromy in the complex plane of parameter b. Here m is a xed positive integer. We also discuss the connection between the special sequence of quasi-exactly solvable sextics with increasing m and the classical quartic potential.

In this article we study numerically and theoretically the asymptotics of the algebraic part of the spec- trum for the quasi-exactly solvable sextic potential m,b(x) = x6 + 2bx4 + (b2 − (4m + 3))x2, its level crossing points, and its monodromy in the complex plane of parameter b. Here m is a xed positive integer. We also discuss the connection between the special sequence of quasi-exactly solvable sextics with increasing m and the classical quartic potential.

In this article we study numerically and theoretically the asymptotics of the algebraic part of the spec- trum for the quasi-exactly solvable sextic potential m,b(x) = x6 + 2bx4 + (b2 − (4m + 3))x2, its level crossing points, and its monodromy in the complex plane of parameter b. Here m is a xed positive integer. We also discuss the connection between the special sequence of quasi-exactly solvable sextics with increasing m and the classical quartic potential.

We exhibit a rigid local system of rank six on the affine line in characteristic p = 5 whose arithmetic and geometric monodromy groups are the finite group 2.J2 (J2 the Hall-Janko sporadic group) in one of its two (Galois-conjugate) irreducible representation of degree six.

In this article we study numerically and theoretically the asymptotics of the algebraic part of the spec- trum for the quasi-exactly solvable sextic potential m,b(x) = x6 + 2bx4 + (b2 − (4m + 3))x2, its level crossing points, and its monodromy in the complex plane of parameter b. Here m is a xed positive integer. We also discuss the connection between the special sequence of quasi-exactly solvable sextics with increasing m and the classical quartic potential.

We exhibit a rigid local system of rank six on the affine line in characteristic p = 5 whose arithmetic and geometric monodromy groups are the finite group 2.J2 (J2 the Hall-Janko sporadic group) in one of its two (Galois-conjugate) irreducible representation of degree six.

We exhibit a rigid local system of rank six on the affine line in characteristic p = 5 whose arithmetic and geometric monodromy groups are the finite group 2.J2 (J2 the Hall-Janko sporadic group) in one of its two (Galois-conjugate) irreducible representation of degree six.

We exhibit a rigid local system of rank six on the affine line in characteristic p = 5 whose arithmetic and geometric monodromy groups are the finite group 2.J2 (J2 the Hall-Janko sporadic group) in one of its two (Galois-conjugate) irreducible representation of degree six.

In this article we study numerically and theoretically the asymptotics of the algebraic part of the spec- trum for the quasi-exactly solvable sextic potential m,b(x) = x6 + 2bx4 + (b2 − (4m + 3))x2, its level crossing points, and its monodromy in the complex plane of parameter b. Here m is a xed positive integer. We also discuss the connection between the special sequence of quasi-exactly solvable sextics with increasing m and the classical quartic potential.

In this article we study numerically and theoretically the asymptotics of the algebraic part of the spec- trum for the quasi-exactly solvable sextic potential m,b(x) = x6 + 2bx4 + (b2 − (4m + 3))x2, its level crossing points, and its monodromy in the complex plane of parameter b. Here m is a xed positive integer. We also discuss the connection between the special sequence of quasi-exactly solvable sextics with increasing m and the classical quartic potential.

We define what it means for a Laurent polynomial in two variables to be mutable. For a mutable Laurent polynomial we prove several results about f and its period pi(f) in terms of the Newton polygon of f. In particular, we give an in principle complete description of the monodromy of pi(f) around the origin. Special attention is given to the class of maximally mutable Laurent polynomials, which has applications to the conjectured classification of Fano manifolds via mirror symmetry.

Producción Científica, The purpose of this note is to give a brief overview on zeta functions of curve singularities and to provide some evidences on how these and global zeta functions associated to singular algebraic curves over perfect fields relate to each other, Ministerio de Economía, Industria y Competitividad ( grants MTM2012-36917-C03-03 / MTM2015-65764- C3-2-P)

Producción Científica, The purpose of this note is to give a brief overview on zeta functions of curve singularities and to provide some evidences on how these and global zeta functions associated to singular algebraic curves over perfect fields relate to each other, Ministerio de Economía, Industria y Competitividad ( grants MTM2012-36917-C03-03 / MTM2015-65764- C3-2-P)

Producción Científica, The purpose of this note is to give a brief overview on zeta functions of curve singularities and to provide some evidences on how these and global zeta functions associated to singular algebraic curves over perfect fields relate to each other, Ministerio de Economía, Industria y Competitividad ( grants MTM2012-36917-C03-03 / MTM2015-65764- C3-2-P)

The possibility that the asymptotic quasi-normal mode (QNM) frequencies can be used to obtain the Bekenstein-Hawking entropy for the Schwarzschild black hole — commonly referred to as Hod’s conjecture — has received considerable attention. To test this conjecture, using monodromy technique, attempts have been made to analytically compute the asymptotic frequencies for a large class of black hole spacetimes. In an earlier work, two of the current authors computed the high frequency QNMs for scalar perturbations of (D+2)-dimensional spherically symmetric, asymptotically flat, single horizon spacetimes with generic power-law singularities. In this work, we extend these results to asymptotically non-flat spacetimes. Unlike the earlier analyses, we treat asymptotically flat and de Sitter spacetimes in a unified manner, while the asymptotic anti-de Sitter spacetimes is considered separately. We obtain master equations for the asymptotic QNM frequency for all the three cases. We show that for all the three cases, the real part of the asymptotic QNM frequency – in general – is not proportional to ln(3) thus indicating that the Hod’s conjecture may be restrictive.

The possibility that the asymptotic quasi-normal mode (QNM) frequencies can be used to obtain the Bekenstein-Hawking entropy for the Schwarzschild black hole — commonly referred to as Hod’s conjecture — has received considerable attention. To test this conjecture, using monodromy technique, attempts have been made to analytically compute the asymptotic frequencies for a large class of black hole spacetimes. In an earlier work, two of the current authors computed the high frequency QNMs for scalar perturbations of (D+2)-dimensional spherically symmetric, asymptotically flat, single horizon spacetimes with generic power-law singularities. In this work, we extend these results to asymptotically non-flat spacetimes. Unlike the earlier analyses, we treat asymptotically flat and de Sitter spacetimes in a unified manner, while the asymptotic anti-de Sitter spacetimes is considered separately. We obtain master equations for the asymptotic QNM frequency for all the three cases. We show that for all the three cases, the real part of the asymptotic QNM frequency – in general – is not proportional to ln(3) thus indicating that the Hod’s conjecture may be restrictive.

The possibility that the asymptotic quasi-normal mode (QNM) frequencies can be used to obtain the Bekenstein-Hawking entropy for the Schwarzschild black hole — commonly referred to as Hod’s conjecture — has received considerable attention. To test this conjecture, using monodromy technique, attempts have been made to analytically compute the asymptotic frequencies for a large class of black hole spacetimes. In an earlier work, two of the current authors computed the high frequency QNMs for scalar perturbations of (D+2)-dimensional spherically symmetric, asymptotically flat, single horizon spacetimes with generic power-law singularities. In this work, we extend these results to asymptotically non-flat spacetimes. Unlike the earlier analyses, we treat asymptotically flat and de Sitter spacetimes in a unified manner, while the asymptotic anti-de Sitter spacetimes is considered separately. We obtain master equations for the asymptotic QNM frequency for all the three cases. We show that for all the three cases, the real part of the asymptotic QNM frequency – in general – is not proportional to ln(3) thus indicating that the Hod’s conjecture may be restrictive.

We prove John Hubbard's conjecture on the topological complexity of the hyperbolic horseshoe locus of the complex Henon map. In fact, we show that there exist several non-trivial loops in the locus which generate infinitely many mutually different monodromies. Furthermore, we prove that the dynamics of the real Henon map is completely determined by the monodromy of the complex Henon map, providing the parameter of the map is contained in the hyperbolic horseshoe locus.

We prove John Hubbard's conjecture on the topological complexity of the hyperbolic horseshoe locus of the complex Henon map. In fact, we show that there exist several non-trivial loops in the locus which generate infinitely many mutually different monodromies. Furthermore, we prove that the dynamics of the real Henon map is completely determined by the monodromy of the complex Henon map, providing the parameter of the map is contained in the hyperbolic horseshoe locus.

The possibility that the asymptotic quasi-normal mode (QNM) frequencies can be used to obtain the Bekenstein-Hawking entropy for the Schwarzschild black hole — commonly referred to as Hod’s conjecture — has received considerable attention. To test this conjecture, using monodromy technique, attempts have been made to analytically compute the asymptotic frequencies for a large class of black hole spacetimes. In an earlier work, two of the current authors computed the high frequency QNMs for scalar perturbations of (D+2)-dimensional spherically symmetric, asymptotically flat, single horizon spacetimes with generic power-law singularities. In this work, we extend these results to asymptotically non-flat spacetimes. Unlike the earlier analyses, we treat asymptotically flat and de Sitter spacetimes in a unified manner, while the asymptotic anti-de Sitter spacetimes is considered separately. We obtain master equations for the asymptotic QNM frequency for all the three cases. We show that for all the three cases, the real part of the asymptotic QNM frequency – in general – is not proportional to ln(3) thus indicating that the Hod’s conjecture may be restrictive.

The possibility that the asymptotic quasi-normal mode (QNM) frequencies can be used to obtain the Bekenstein-Hawking entropy for the Schwarzschild black hole — commonly referred to as Hod’s conjecture — has received considerable attention. To test this conjecture, using monodromy technique, attempts have been made to analytically compute the asymptotic frequencies for a large class of black hole spacetimes. In an earlier work, two of the current authors computed the high frequency QNMs for scalar perturbations of (D+2)-dimensional spherically symmetric, asymptotically flat, single horizon spacetimes with generic power-law singularities. In this work, we extend these results to asymptotically non-flat spacetimes. Unlike the earlier analyses, we treat asymptotically flat and de Sitter spacetimes in a unified manner, while the asymptotic anti-de Sitter spacetimes is considered separately. We obtain master equations for the asymptotic QNM frequency for all the three cases. We show that for all the three cases, the real part of the asymptotic QNM frequency – in general – is not proportional to ln(3) thus indicating that the Hod’s conjecture may be restrictive.

The possibility that the asymptotic quasi-normal mode (QNM) frequencies can be used to obtain the Bekenstein-Hawking entropy for the Schwarzschild black hole — commonly referred to as Hod’s conjecture — has received considerable attention. To test this conjecture, using monodromy technique, attempts have been made to analytically compute the asymptotic frequencies for a large class of black hole spacetimes. In an earlier work, two of the current authors computed the high frequency QNMs for scalar perturbations of (D+2)-dimensional spherically symmetric, asymptotically flat, single horizon spacetimes with generic power-law singularities. In this work, we extend these results to asymptotically non-flat spacetimes. Unlike the earlier analyses, we treat asymptotically flat and de Sitter spacetimes in a unified manner, while the asymptotic anti-de Sitter spacetimes is considered separately. We obtain master equations for the asymptotic QNM frequency for all the three cases. We show that for all the three cases, the real part of the asymptotic QNM frequency – in general – is not proportional to ln(3) thus indicating that the Hod’s conjecture may be restrictive.

An invertible polynomial in n variables is a quasi-homogeneous polynomial consisting of n monomials so that the weights of the variables and the quasi-degree are well defined. In the framework of the construction of mirror symmetric orbifold Landau–Ginzburg models, Berglund, Hübsch and Henningson considered a pair (f,G) consisting of an invertible polynomial f and an abelian group G of its symmetries together with a dual pair (Figure presented.). Here we study the reduced orbifold zeta functions of dual pairs (f,G) and (Figure presented.) and show that they either coincide or are inverse to each other depending on the number n of variables. Copyright © Edinburgh Mathematical Society 2016

The theme of this dissertation is the study of fundamental groups and classifying spaces in the context of the étale topology of schemes. The main result is the existence of K(pi,1) neighborhoods in the case of semistable (or more generally log smooth) reduction, generalizing a result of Gerd Faltings. As an application to p-adic Hodge theory, we use the existence of these neighborhoods to compare the cohomology of the geometric generic fiber of a semistable scheme over a discrete valuation ring with the cohomology of the associated Faltings topos. The other results include comparison theorems for the cohomology and homotopy types of several types of Milnor fibers. We also prove an l-adic version of a formula of Ogus, describing the monodromy action on the complex of nearby cycles of a log smooth family in terms of the log structure.

The theme of this dissertation is the study of fundamental groups and classifying spaces in the context of the étale topology of schemes. The main result is the existence of K(pi,1) neighborhoods in the case of semistable (or more generally log smooth) reduction, generalizing a result of Gerd Faltings. As an application to p-adic Hodge theory, we use the existence of these neighborhoods to compare the cohomology of the geometric generic fiber of a semistable scheme over a discrete valuation ring with the cohomology of the associated Faltings topos. The other results include comparison theorems for the cohomology and homotopy types of several types of Milnor fibers. We also prove an l-adic version of a formula of Ogus, describing the monodromy action on the complex of nearby cycles of a log smooth family in terms of the log structure.

This paper looks at the L-function of the kth symmetric power of the -sheaf Aif over the affine line associated to the generalized Airy family of exponential sums. Using ℓ-adic techniques, we compute the degree of this rational function as well as the local factors at infinity. Using p-adic techniques, we study the q-adic Newton polygon of the L-function.

This paper looks at the L-function of the kth symmetric power of the -sheaf Aif over the affine line associated to the generalized Airy family of exponential sums. Using ℓ-adic techniques, we compute the degree of this rational function as well as the local factors at infinity. Using p-adic techniques, we study the q-adic Newton polygon of the L-function.

This paper looks at the L-function of the kth symmetric power of the -sheaf Aif over the affine line associated to the generalized Airy family of exponential sums. Using ℓ-adic techniques, we compute the degree of this rational function as well as the local factors at infinity. Using p-adic techniques, we study the q-adic Newton polygon of the L-function.

On the poster new findings on two different topics are presented --- related separately to geometric (Berry) phases in non-Hermitian quantum systems in higher-dimensional Hilbert spaces and to the specific realization properties of PT-symmetric quantum brachistochrones in two-qubit systems fulfilling the Aharonov-Anandan lower bound on quantum evolution times. Specifically, in the first part of the poster we present new results on the holonomy properties of eigenvector bundles of non-Hermitian operators which can be mapped into simplest versally deformed n-th order Jordan blocks. (This generalizes earlier considerations on similar setups with 2-dimensional Jordan blocks [1].) In the second part of the poster, we start from the embedding of the PT-symmetric brachistochrone into a Hermitian two-qubit system (as recently proposed in [2]) and show that the resulting evolution of the two-qubit system is itself a conventional quantum brachistochrone lying exactly on the non-local Lie-group orbit induced by one of the non-trivial entanglement generators. [1] U. Günther, I. Rotter and B. Samsonov, "Projective Hilbert space structures at exceptional points", J. Phys. A 40, 8815 (2007), arXiv:0704.1291[math-ph]. [2] U. Günther and B. Samsonov, "Naimark-dilated PT-symmetric brachistochrone", Phys. Rev. Lett. 101, 230404 (2008), arXiv:0807.3643[quant-ph].

On the poster new findings on two different topics are presented --- related separately to geometric (Berry) phases in non-Hermitian quantum systems in higher-dimensional Hilbert spaces and to the specific realization properties of PT-symmetric quantum brachistochrones in two-qubit systems fulfilling the Aharonov-Anandan lower bound on quantum evolution times. Specifically, in the first part of the poster we present new results on the holonomy properties of eigenvector bundles of non-Hermitian operators which can be mapped into simplest versally deformed n-th order Jordan blocks. (This generalizes earlier considerations on similar setups with 2-dimensional Jordan blocks [1].) In the second part of the poster, we start from the embedding of the PT-symmetric brachistochrone into a Hermitian two-qubit system (as recently proposed in [2]) and show that the resulting evolution of the two-qubit system is itself a conventional quantum brachistochrone lying exactly on the non-local Lie-group orbit induced by one of the non-trivial entanglement generators. [1] U. Günther, I. Rotter and B. Samsonov, "Projective Hilbert space structures at exceptional points", J. Phys. A 40, 8815 (2007), arXiv:0704.1291[math-ph]. [2] U. Günther and B. Samsonov, "Naimark-dilated PT-symmetric brachistochrone", Phys. Rev. Lett. 101, 230404 (2008), arXiv:0807.3643[quant-ph].

We construct several geometric representatives for the C-n/Z(m) fractional branes on either a partially or the completely resolved orbifold. In the process we use large radius and conifold-type monodromies and provide a strong consistency check. In particular, for C-3/Z(5) we give three different sets of geometric representatives. We also find the explicit Seiberg duality which connects our fractional branes to the ones given by the McKay correspondence.

In this paper we are concerned with the monodromy of Picard-Fuchs differential equations associated with one-parameter families of Calabi-Yau threefolds. Our results show that in the hypergeometric cases the matrix representations of monodromy relative to the Frobenius bases can be expressed in terms of the geometric invariants of the underlying Calabi-Yau threefolds. This phenomenon is also verified numerically for other families of Calabi-Yau threefolds in the paper. Furthermore, we discover that under a suitable change of bases the monodromy groups are contained in certain congruence subgroups of Sp(4, ℤ) of finite index and whose levels are related to the geometric invariants of the Calabi-Yau threefolds. © Walter de Gruyter.

we study the Maslov index of the monodromy matrix of periodic Hamiltonian orbit, extending substantially results of other authors, Vi undesöker maslov indexet av monodromimatrisen av periodiska Hamiltonska banor

Monodromy groups, i.e. the groups of isometries of the intersection lattice Lx:= H2/torsion generated by the monodromy action of all deformation families of a given surface, have been computed by the author for any minimal elliptic surface with pg > q = 0. New and refined methods are now employed to address the cases of minimal elliptic surfaces with pg ≥ q > 0. Thereby we get explicit families such that any isometry is in the group generated by their monodromies or does not respect the invariance of the canonical class or the spinor norm. The monodromy is also shown to act by the full symplectic group on the first homology modulo torsion.

We show that some hypergeometric monodromy groups in Sp(4,Z) split as free or amalgamated products and hence by cohomological considerations give examples of Zariski dense, non-arithmetic monodromy groups of real rank 2. In particular, we show that the monodromy of the natural quotient of the Dwork family of quintic threefolds in P^{4} splits as Z*Z/5. As a consequence, for a smooth quintic threefold X we show that a certain group of autoequivalences of the bounded derived category of coherent sheaves is an Artin group of dihedral type.

This thesis deals with the implementation of an algorithm to compute the monodromy of an isolated hypersurface singularity in the computer algebra system Singular.

This thesis deals with the implementation of an algorithm to compute the monodromy of an isolated hypersurface singularity in the computer algebra system Singular.

This thesis deals with the implementation of an algorithm to compute the monodromy of an isolated hypersurface singularity in the computer algebra system Singular.