Algebra

URI permanente para esta comunidadhttps://hdl.handle.net/11441/10803

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2º Congreso de Jóvenes Investigadores
(Real Sociedad Matematica Española, 2013) Gancedo García, Francisco; Muro Jiménez, Fernando; Universidad de Sevilla. Departamento de Análisis Matemático; Universidad de Sevilla. Departamento de álgebra; Universidad de Sevilla. FQM104: Análisis Matemático; Universidad de Sevilla. FQM218: Geometría Algebraica, Sistemas Diferenciales y Singularidades
7o Congreso Europeo de Matemáticas
(Real Sociedad Matemática Española, 2016) Rojas León, Antonio; Universidad de Sevilla. Departamento de álgebra; Universidad de Sevilla. FQM218: Singularidades, Geometría Algebraica Aritmética, Grupos y Homotopía
La séptima edición del Congreso Europeo de Matemáticas, evento que tiene lugar cada cuatro años bajo los auspicios de la Sociedad Matemática Europea, se celebró entre los pasados días 18 y 22 de julio en la sede de la Universidad Técnica de Berlín (TU Berlin). Este acontecimiento reunió a algunos de los principales protagonistas de las matemáticas europeas de los últimos años, y sirvió de punto de encuentro de numerosos investigadores que tuvieron ocasión de exponer y discutir los últimos avances en sus respectivas áreas.
A combinatorial overview of the Hopf algebra of MacMahon symmetric functions
(Springer, 2002-11) Rosas Celis, Mercedes Helena; Rota, Gian-Carlo; Stein, Joel; Universidad de Sevilla. Departamento de álgebra; Universidad de Sevilla. FQM333: Algebra Computacional en Anillos no Conmutativos y Aplicaciones
A MacMahon symmetric function is a formal power series in a finite number of alphabets that is invariant under the diagonal action of the symmetric group. In this article, we give a combinatorial overview of the Hopf algebra structure of the MacMahon symmetric functions relying on the construction of a Hopf algebra from any alphabet of neutral letters obtained in [18 G.-C. Rota and J. Stein, Plethystic Hopf algebras, Proc. Natl. Acad. Sci. USA 91 (1994) 13057–13061. 19. G.-C. Rota and J. Stein, Plethystic algebras and vector symmetric functions, Proc. Natl. Acad. Sci. USA 91 (1994) 13062–13066].
A comment of the combinatorics of the vertex operator Γ(t|X)
(Project euclid, 2019-12-08) Rosas Celis, Mercedes Helena; Universidad de Sevilla. Departamento de Álgebra; Universidad de Sevilla. FQM333: Algebra Computacional en Anillos no Conmutativos y Aplicaciones
The Jacobi--Trudi identity associates a symmetric function to any integer sequence. Let Γ(t|X) be the vertex operator defined by Γ(t|X)sα=∑n∈Zs(n,α)[X]tn. We provide a combinatorial proof for the identity Γ(t|X)sα=σ[tX]sα[x−1/t] due to Thibon et al. We include an overview of all the combinatorial ideas behind this beautiful identity, including a combinatorial description for the expansion of s(n,α)[X] in the Schur basis, for any integer value of n.
A complete diophantine characterization of the rational torsion of an elliptic curve
(Springer, 2012-01) García Selfa, Irene; Tornero Sánchez, José María; Universidad de Sevilla. Departamento de álgebra
We give a complete characterization for the rational torsion of an elliptic curve in terms of the (non–)existence of integral solutions of a system of diophantine equations.
A computational approach to the D-module of meromorphic functions
(2001) Castro Jiménez, Francisco Jesús; Ucha Enríquez, José María; Universidad de Sevilla. Departamento de álgebra
Let D be a divisor in Cn. We present methods to compare the D-module of the meromorphic functions O[∗D] to some natural approximations. We show how the analytic case can be treated with computations in the Weyl algebra.
A duality approach to the symmetry of Bernstein-Sato polynomials of free divisors
(Elsevier, 2015-08-20) Narváez Macarro, Luis; Universidad de Sevilla. Departamento de álgebra; Ministerio de Ciencia e Innovación (MICIN). España; European Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER); Universidad de Sevilla. FQM218: Geometria Algebraica, Sistemas Diferenciales y Singularidades
In this paper we prove that the Bernstein-Sato polynomial of any free divisor for which the D[s]-module D[s]h s admits a Spencer logarithmic resolution satisfies the symmetry property b(−s−2) = ±b(s). This applies in particular to locally quasi-homogeneous free divisors (for instance, to free hyperplane arrangements), or more generally, to free divisors of linear Jacobian type. We also prove that the Bernstein-Sato polynomial of an integrable logarithmic connection E and of its dual E ∗ with respect to a free divisor of linear Jacobian type are related by the equality bE(s) = ±bE∗ (−s − 2). Our results are based on the behaviour of the modules D[s]h s and D[s]E[s]h s under duality.
A fast solution to the conjugacy problem in the four-strand braid group
(De Gruyter, 2014-09) Calvez, Matthieu; Wiest, Bert; Universidad de Sevilla. Departamento de álgebra; Universidad de Sevilla. FQM218: Geometria Algebraica, Sistemas Diferenciales y Singularidades
We present an algorithm for solving the conjugacy search problem in the fourstrand braid group. The computational complexity is cubic with respect to the braid length.
A flatness property for filtered D-modules
(European Mathematical Society, 2007) Castro Jiménez, Francisco Jesús; Granger, Michel; Universidad de Sevilla. Departamento de Álgebra; Universidad de Sevilla. FQM333: Algebra Computacional en Anillos no Conmutativos y Aplicaciones
Let M be a coherent module over the ring DX of linear differential operators on an analytic manifold X and let Z1, · · · , Zk be k germs of transverse hypersurfaces at a point x ∈ X. The Malgrange-Kashiwara V-filtrations along these hypersurfaces, associated with a given presentation of the germ of M at x, give rise to a multifiltration U•(M) of Mx as in Sabbah’s paper [9] C. Sabbah, Proximité evanescente I. La structure polaire d’un D–module Compositio Math. 62 (1987) 283-319 and to an analytic standard fan in a way similar to [3] A. Assi., F. Castro-Jiménez and M. Granger, The analytic standard fan of a D-module, J. Pure Appl. Algebra 164 (2001) 3-21. We prove here that this standard fan is adapted to the multifiltration, in the sense of C. Sabbah. This result completes the proof of the existence of an adapted fan in [9] C. Sabbah, Proximité evanescente I. La structure polaire d’un D–module Compositio Math. 62 (1987) 283-319, for which the use of [8] C. Sabbah and F. Castro, Appendice à “proximité evanescente” I. La structure polaire d’un D–module, Compositio Math. 62 (1987) 320-328. is not possible.
A generalisation of the Phase Kick-Back
(Springer, 2023-03-13) Ossorio Castillo, J.; Pastor Díaz, Ulises; Tornero Sánchez, José María; Universidad de Sevilla. FQM218: Singularidades, Geometría Algebraica Aritmética, Grupos y Homotopía
In this paper, we present a generalisation of the Phase Kick-Back technique, which is central to some of the classical algorithms in quantum computing. We will begin by recalling the Phase Kick-Back technique to then introduce the new generalised version for f:{0,1}n→{0,1}m functions using the eigenvalues of the oracle function Uf. After that, we will present a new generalised version of the Deutsch–Jozsa problem and how it can be solved using the previously defined technique. We will also deal with a generalised version of the Bernstein–Vazirani problem and solve it using the generalised Phase Kick-Back. Finally, we show how we can use this technique to obtain an algorithm for Simon’s problem that improves the classical one.
A geometric characterization of the upper bound for the span of the Jones polynomial
(World Scientific, 2011-07) González-Meneses López, Juan; González Manchón, Pedro María; Universidad de Sevilla. Departamento de álgebra
Let D be a link diagram with n crossings, sA and sB its extreme states and |sAD| (resp. |sBD|) the number of simple closed curves that appear when smoothing D according to sA (resp. sB). We give a general formula for the sum |sAD| + |sBD| for a k-almost alternating diagram D, for any k, characterizing this sum as the number of faces in an appropriate triangulation of an appropriate surface with boundary. When D is dealternator connected, the triangulation is especially simple, yielding |sAD| + |sBD| = n + 2 − 2k. This gives a simple geometric proof of the upper bound of the span of the Jones polynomial for dealternator connected diagrams, a result first obtained by Zhu. Another upper bound of the span of the Jones polynomial for dealternator connected and dealternator reduced diagrams, discovered historically first by Adams et al, is obtained as a corollary. As a new application, we prove that the Turaev genus is equal to the number k of dealternator crossings for any dealternator connected diagram.
A geometric description of the extreme Khovanov cohomology
(Cambridge University Press, 2018-06) González-Meneses López, Juan; González Manchón, Pedro María; Silvero Casanova, Marithania; Universidad de Sevilla. Departamento de álgebra; Ministerio de Economía y Competitividad (MINECO). España; European Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER); Universidad de Sevilla. FQM218: Singularidades, Geometría Algebraica Aritmética, Grupos y Homotopía
We prove that the hypothetical extreme Khovanov cohomology of a link is the cohomology of the independence simplicial complex of its Lando graph. We also provide a family of knots having as many non-trivial extreme Khovanov cohomology modules as desired, that is, examples of H-thick knots which are as far of being H-thin as desired.
A note on K-theory and triangulated derivators
(Elsevier, 2011-08-01) Muro Jiménez, Fernando; Raptis, George; Universidad de Sevilla. Departamento de álgebra
In this paper we show an example of two differential graded algebras that have the same derivator K-theory but non-isomorphic Waldhausen K-theory. We also prove that Maltsiniotis’s comparison and localization conjectures for derivator K-theory cannot be simultaneously true.
A positive proportion of elements of mapping class groups is pseudo-Anosov
(Wiley, 2018-03-28) Cumplido Cabello, María; Wiest, Bert; Universidad de Sevilla. Departamento de álgebra; Universidad de Sevilla. FQM218: Singularidades, Geometría Algebraica Aritmética, Grupos y Homotopía
In the Cayley graph of the mapping class group of a closed surface, with respect to any generating set, we look at a ball of large radius centered on the identity vertex, and at the proportion among the vertices in this ball representing pseudo-Anosov elements. A well-known conjecture states that this proportion should tend to one as the radius tends to infinity. We prove that it stays bounded away from zero. We also prove similar results for a large class of subgroups of the mapping class group.
A rigid local system with monodromy group 2.J2
(Elsevier, 2019-05) Katz, Nicholas M.; Rojas León, Antonio; Universidad de Sevilla. Departamento de álgebra; Universidad de Sevilla. FQM218: Singularidades, Geometría Algebraica Aritmética, Grupos y Homotopía
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.
Abelian varieties over number fields, tame ramification and big Galois image
(International Press, 2013) Arias de Reyna Domínguez, Sara; Kappen, Christian; Universidad de Sevilla. Departamento de álgebra; Ministerio de Educación y Ciencia (MEC). España; Universidad de Sevilla. FQM218: Geometria Algebraica, Sistemas Diferenciales y Singularidades
Given a natural number n ≥ 1 and a number field K, we show the existence of an integer ℓ0 such that for any prime number ℓ ≥ ℓ0, there exists a finite extension F/K, unramified in all places above ℓ, together with a principally polarized abelian variety A of dimension n over F such that the resulting ℓ-torsion representation ρA,ℓ : GF → GSp(A[ℓ](F)) is surjective and everywhere tamely ramified. In particular, we realize GSp2n(Fℓ) as the Galois group of a finite tame extension of number fields F0/F such that F is unramified above ℓ.
Acylindrical hyperbolicity and Artin-Tits groups of spherical type
(Springer, 2017) Calvez, Matthieu; Wiest, Bert; Universidad de Sevilla. Departamento de álgebra; Universidad de Sevilla. FQM218: Geometría Algebraica, Sistemas Diferenciales y Singularidades
We prove that, for any irreducible Artin-Tits group of spherical type G, the quotient of G by its center is acylindrically hyperbolic. This is achieved by studying the additional length graph associated to the classical Garside structure on G, and constructing a specific element xG of G/Z(G) whose action on the graph is loxodromic and WPD in the sense of Bestvina-Fujiwara; following Osin, this implies acylindrical hyperbolicity. Finally, we prove that “generic” elements of G act loxodromically, where the word “generic” can be understood in either of the two common usages: as a result of a long random walk or as a random element in a large ball in the Cayley graph.
Alexander-Conway polynomial state model and link homology
(World Scientific Publishing, 2016-03) Kauffman, Louis H.; Silvero Casanova, Marithania; Universidad de Sevilla. Departamento de álgebra; Universidad de Sevilla. FQM218: Geometria Algebraica, Sistemas Diferenciales y Singularidades
This paper shows how the Formal Knot Theory state model for the Alexander-Conway polynomial is related to Knot Floer Homology. In particular we prove a parity result about the states in this model that clarifies certain relationships of the model with Knot Floer Homology.
Álgebra computacional y programación entera no lineal
(Real Sociedad Matemática Española, 2016) Gago Vargas, Manuel Jesús; Hartillo Hermoso, Isabel; Universidad de Sevilla. Departamento de álgebra; Universidad de Sevilla. FQM333: Álgebra Computacional en Anillos no Conmutativos y Aplicaciones
Álgebra y combinatoria de las singularidades
(2001) Soto Prieto, Manuel Jesús; Vicente Córdoba, José Luis; Universidad de Sevilla. Departamento de Álgebra
La memoria titulada Algebra y Combinatoria de las Singularidades, dirigida por el Profesor José Luis Vicente Córdoba, contiene los resultados de una profunda reflexión sobre el papel de ciertos objetos combinatorios, así como de los correspondientes aspectos algorítmicos y computacionales, en las relaciones entre el álgebra y la geometría, y más particularmente entre el álgebra y la teoría de singularidades. De hecho, la filosofía que ha impulsado esta meritoria reflexión ha sido ""la construcción de un diccionario práctico entre combinatoria y álgebra"", el deseo de ""poner en un primer plano, tanto conceptual como operativa, la estructura combinatoria subyacente a muchas partes del álgebra"". Los resultados de esta filosofía expuestos en la memoria son substanciales, algunos de una dificultad muy considerable, y en particular muestran que la ""aplicación de los teoremas generales se puede sustituir por técnicas combinatorias o, lo que es lo mismo, por algoritmos, implementables o no en computador.|

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