Person: Muñoz Velázquez, Vicente
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First Name
Vicente
Last Name
Muñoz Velázquez
Affiliation
Universidad Complutense de Madrid
Faculty / Institute
Ciencias Matemáticas
Department
Area
Geometría y Topología
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8 results
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Item A lax monoidal Topological Quantum Field Theory for representation varieties(Bulletin des Sciences Mathématiques, 2020) González Prieto, José Ángel; Logares Jiménez, Marina Lucía; Muñoz Velázquez, VicenteWe construct a lax monoidal Topological Quantum Field Theory that computes Deligne-Hodge polynomials of representation varieties of the fundamental group of any closed manifold into any complex algebraic group G. As byproduct, we obtain formulas for these polynomials in terms of homomorphisms between the space of mixed Hodge modules on G. The construction is developed in a categorical-theoretic framework allowing its application to other situations.Item Representation Varieties of Twisted Hopf Links(Mediterranean Journal of Mathematics, 2023) González Prieto, José Ángel; Muñoz Velázquez, VicenteIn this paper, we study the representation theory of the fundamental group of the complement of a Hopf link with n twists. A general framework is described to analyze the -representation varieties of these twisted Hopf links as byproduct of a combinatorial problem and equivariant Hodge theory. As application, close formulas of their E-polynomials are provided for ranks 2 and 3, both for the representation and character varieties.Item Moduli spaces of parabolic U(p,q) -Higgs bundles.(Quarterly Journal of Mathematics, 2009) García Prada, O.; Logares Jiménez, Marina Lucía; Muñoz Velázquez, VicenteUsing the L2-norm of the Higgs field as a Morse function, we study the moduli space of parabolic U(p, q)-Higgs bundles over a Riemann surface with a finite number of marked points, under certain genericity conditions on the parabolic structure. When the parabolic degree is zero this space is homeomorphic to the moduli space of representations of the fundamental group of the punctured surface in U(p, q), with fixed compact holonomy classes around the marked points. By means of this homeomorphism we count the number of connected components of this moduli space of representations. Finally, we apply our results to the study of representations of the fundamental group of elliptic surfaces of general type.Item Motive of the representation varietes of torus knots for low rank affine groups(2022) González Prieto, José Ángel; Logares Jiménez, Marina Lucía; Muñoz Velázquez, VicenteWe compute the motive of the variety of representations of the torus knot of type (m, n) into the affine groups AGL1(C) and AGL2(C). For this, we stratify the varieties and show that the motives lie in the subring generated by the Lefschetz motive q = [C].Item Rationality of the moduli space of stable pairs over a complex curve(Nonlinear Analysis:Stability, Approximation, and Inequalities, 2012) Biswas, Indranil; Logares Jiménez, Marina Lucía; Muñoz Velázquez, Vicente; Pardalos, Panos M.; Georgiev, Pando G.; Srivastava, Hari M.Let X be a smooth complex projective curve of genus g≥2. A pair on X is formed by a vector bundle E→X and a global non-zero section ϕ∈H 0(E). There is a concept of stability for pairs depending on a real parameter τ, giving rise to moduli spaces of τ-stable pairs of rank r and fixed determinant Λ. In this paper, we prove that the moduli spaces are in many cases rational.Item Hodge polynomials of SL (2,C)-character varieties for curves of small genus(Revista matemática complutense, 2013) Logares Jiménez, Marina Lucía; Muñoz Velázquez, Vicente; Newstead, P. E.We compute the E-polynomials of the moduli spaces of representations of the fundamental group of a complex curve into for the case of small genus and allowing the holonomy around a fixed point to be any matrix of that is diagonalisable, or of either of the two Jordan types. For this, we introduce a new geometric technique, based on stratifying the space of representations, and on the analysis of the behaviour of the E-polynomial under fibrationsItem Hodge polynomials of the SL(2, C)-character variety of an elliptic curve with two marked points(International journal of mathematics, 2014) Logares Jiménez, Marina Lucía; Muñoz Velázquez, VicenteWe compute the Hodge polynomials for the moduli space of representations of an elliptic curve with two marked points into SL(2, C). When we fix the conjugacy classes of the representations around the marked points to be diagonal and of modulus one, the character variety is diffeomorphic to the moduli space of strongly parabolic Higgs bundles, whose Betti numbers are known. In that case we can recover some of the Hodge numbers of the character variety. We extend this result to the moduli space of doubly periodic instantons.Item Representation Variety for the Rank One Affine Group(Mathematical Analysis in Interdisciplinary Research, 2021) González Prieto, José Ángel; Logares Jiménez, Marina Lucía; Muñoz Velázquez, VicenteThe aim of this chapter is to study the virtual classes of representation varieties of surface groups onto the rank one affine group. We perform this calculation by three different approaches: the geometric method, based on stratifying the representation variety into simpler pieces; the arithmetic method, focused on counting their number of points over finite fields; and the quantum method, which performs the computation by means of a Topological Quantum Field Theory. We also discuss the corresponding moduli spaces of representations and character varieties, which turn out to be non-equivalent due to the non-reductiveness of the underlying group.