Automorphism group of the moduli space of parabolic vector bundles over a curve

Tesis Doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 26-10-2018The main objective of this thesis is the computation of the automorphism group of the moduli space of parabolic vector bundles over a smooth complex projective curve. We will start by de ning the notion of a parabolic -module { a module over a sheaf of rings of di erential operators with a parabolic structure at certain marked points { and building their moduli space. This will provide us a common theoretical framework that allows us to work with several kinds of moduli spaces of bundles with parabolic structure such parabolic vector bundles, parabolic (L-twisted) Higgs bundles, parabolic connections or parabolic -connections. As an application, we build the parabolic Hodge moduli space and the parabolic Deligne{Hitchin moduli space. Then, we will address the computation of the automorphism group of the moduli space of parabolic bundles. Let X and X0 be irreducible smooth complex projective curves with sets of marked points D X and D0 X0 and genus g 6 and g0 6 respectively. LetM(X; r; ; ) be the moduli space of rank r stable parabolic vector bundles on (X;D) with parabolic weights and determinant . We classify the possible isomorphisms : M(X; r; ; ) ���! M(X0; r0; 0; 0). First, a Torelli type theorem is proved, implying that for to exist it is necessary that (X;D) = (X0;D0) and r = r0. Then we prove that the possible isomorphisms are generated by automorphisms of the pointed curve (X;D), tensorization with suitable line bundles, dualization of parabolic vector bundles and Hecke transformations at the parabolic points. These results are extended to birational equivalences : M(X; r; ; ) 99K M(X0; r0; 0; 0) which are de ned over \big" open subsets. The particular case of \concentrated" weights (corresponding to \small" stability parameters) is studied further. In this case Hecke transformations give rise to birational morphisms that do not extend to automorphisms of the moduli space. Moreover, an analysis of the stability chambers for the weights allows us to determine an explicit computable presentation of the group of automorphisms of the moduli space for arbitrary generic weights. Finally, the automorphism group of the moduli space of framed bundles over a smooth complex projective curve X of genus g > 2 with a framing over a point x 2 X is also described. It is shown that this group is generated by pullbacks using automorphisms of the curve X that x the marked point x, tensorization with certain line bundles over X and the action of PGLr(C) by composition with the framing

Automatic aspect extraction in information retrieval diversity

In this master thesis we describe a new automatic aspect extraction algorithm by incorporating relevance information to the dynamics of the Probabilistic Latent Semantic Analysis. An utility-biased likelihood statistical framework is described to formalize the incorporation of prior relevance information to the dynamics of the algorithm intrinsically. Moreover, a general abstract algorithm is presented to incorporate any arbitrary new feature variables to the analysis. A tempering procedure is inferred for this general algorithm as an entropic regularization of the utility-biased likelihood functional and a geometric interpretation of the algorithm is described, showing the intrinsic changes in the information space of the problem produced when di erent sources of prior utility estimations are provided over the same data. The general algorithm is applied to several information retrieval, recommendation and personalization tasks. Moreover, a set of post-processing aspect lters is presented. Some characteristics of the aspect distributions such as sparsity or low entropy are identi ed to enhance the overall diversity attained by the diversi cation algorithm. Proposed lters assure that the nal aspect space has those properties, thus leading to better diversity levels. An experimental setup over TREC web track 09-12 data shows that the algorithm surpasses classic pLSA as an aspect extraction tool for the search diversi cation. Additional theoretical applications of the general procedure to information retrieval, recommendation and personalization tasks are given, leading to new relevanceaware models incorporating several variables to the latent semantic analysis. Finally the problem of optimizing the aspect space size for diversi cation is addressed. Analytical formulas for the dependency of diversity metrics on the choice of an automatically extracted aspect space are given under a simpli ed generative model for the relation between system aspects and evaluation true aspects. An experimental analysis of this dependence is performed over TREC web track data using pLSA as aspect extraction algorithm