1,720,958 research outputs found
The cyclic theory of Hopf algebroids
No full textWe give a systematic description of the cyclic cohomology theory of Hopf algebroids
in terms of its associated category of modules. Then we introduce a dual cyclic homology
theory by applying cyclic duality to the underlying cocyclic object. We derive general structure
theorems for these theories in the special cases of commutative and cocommutative Hopf
algebroids. Finally, we compute the cyclic theory in examples associated to Lie–Rinehart
algebras and étale groupoids
Equivariant theory of Lie groupoids from the perspective of non-commutative geometry
No full textIn this dissertation we describe connections between the equivariant theory of Lie groupoids and the non-commutative geometry of the convolution algebra. Lie groupoids are objects that encode symmetries of a space. They are a generalization of Lie groups in the following sense: Lie groups describe symmetries that are globally defined on the space, while Lie groupoids describe symmetries whose application is place-dependent. These objects are interesting because of their applications in physics. The existence of solutions to a physical system can be proven or disproven by exhibiting geometric properties of the space on which the system is applied. When there is a symmetry under which the system is invariant, you can shrink the space on which one needs to solve the system by factoring out the symmetry. In this philosophy we are interested in the ‘geometry of the space that is invariant under the symmetry’. In case where there is an action of a Lie group on the space, this is something we understand reasonably well, and in this dissertation we try the generalize these ideas to Lie groupoids. We want to understand the non-commutative geometry of convolution algebras, and that is the mathematical content of this dissertation. We describe connections between various mathematical properties of a Lie groupoid and the non-commutative geometry of the convolution algebra, with the goal to sketch a complete picture of this non-commutative geometry
Equivariant theory of Lie groupoids from the perspective of non-commutative geometry
No full textIn this dissertation we describe connections between the equivariant theory of Lie groupoids and the non-commutative geometry of the convolution algebra. Lie groupoids are objects that encode symmetries of a space. They are a generalization of Lie groups in the following sense: Lie groups describe symmetries that are globally defined on the space, while Lie groupoids describe symmetries whose application is place-dependent. These objects are interesting because of their applications in physics. The existence of solutions to a physical system can be proven or disproven by exhibiting geometric properties of the space on which the system is applied. When there is a symmetry under which the system is invariant, you can shrink the space on which one needs to solve the system by factoring out the symmetry. In this philosophy we are interested in the ‘geometry of the space that is invariant under the symmetry’. In case where there is an action of a Lie group on the space, this is something we understand reasonably well, and in this dissertation we try the generalize these ideas to Lie groupoids. We want to understand the non-commutative geometry of convolution algebras, and that is the mathematical content of this dissertation. We describe connections between various mathematical properties of a Lie groupoid and the non-commutative geometry of the convolution algebra, with the goal to sketch a complete picture of this non-commutative geometry
Proper Lie groupoids and their orbit spaces
Get PDFThis thesis studies proper Lie groupoids on three levels: the groupoids themselves, their induced foliations, and their orbit spaces. Proper Lie groupoids are shown to admit desingularisations via a successive blow-up procedure, whereby orbits are systematically added to achieve regularity. Regarding their foliations, a thorough treatment of the known integration results for singular foliations is included. Moreover, the underlying orbit spaces of proper Lie groupoids are studied. This is done by first providing an intrinsic definition of a so-called orbispace using atlases, both in the language of Morita bibundles, and in the language of Morita equivalences through fractions. An orbispace is said to be proper if it admits a proper defining atlas. It is then shown that proper Lie groupoids, up to a precise notion of Morita equivalence, correspond exactly to such proper orbispaces. This can be interpreted as the statement that proper orbispaces form a subcategory of all differentiable stacks. All of these developments mirror the well-known correspondences between regular proper Lie groupoids, regular foliations, and orbifolds. The above results are further shown to hold in the setting of proper Riemannian groupoids. In particular the desingularisation procedure can be performed in such a way that the regularised groupoid has arbitrarily small Gromov—Hausdorff distance from the original groupoid. Moreover, proper Riemannian orbispaces are defined and shown to correspond precisely to appropriate equivalence classes of proper Riemannian groupoids. This thesis contains various other results, including those on holonomy groupoids of orbit-like foliations, and a de Rham theorem for orbispaces
Cyclic theory of Lie algebroids
Get PDFIn this thesis we study the cyclic theory of universal enveloping algebras of Lie algebroids. Lie algebroids are geometrical objects that encode infinitesimal symmetries, and the concept encompasses many classical objects from geometry, such as Poisson manifolds, foliations and actions of Lie algebras on manifolds. The study of geometrical objects is in many cases equivalent to the study of the algebras of functions on these objects, and this observation led to the field of noncommutative geometry, where one studies noncommutative algebras, that are not necessarily related to geometrical objects, with techniques from geometry. For each Lie algebroid one can define a noncommutative algebra, called the universal enveloping algebra, which generalizes the algebra of differential operators on a manifold and the universal enveloping algebra of a Lie algebra. In this thesis we show that the cyclic theory of this algebra is equal to the Poisson (co)homology of the dual of the Lie algebroid, which in turn is equal to the Lie algebroid cohomology with values in the symmetric algebra of the adjoint representation up to homotopy (twisted by a line bundle). Moreover, we define a trace-density map from the cyclic theory of the universal enveloping algebra to the de Rham complex of the Lie algebroid, which generalizes known results for the tangent bundle of a manifold. We use a Čech resolution of the de Rham complex, which makes the construction suitable for holomorphic Lie algebroids as well. Both the calculation of the cyclic theory of the universal enveloping algebra as well as the construction of the trace-density map is based on the Poincaré–Birkhoff–Witt theorem for Lie algebroids, which we therefore prove first
Cohomological field theories and global spectral curves
Get PDFIn my thesis I consider interplay between several different structures in mathematical physics. These structures are used to solve a large class of problems in enumerative algebraic geometry and combinatorics in a universal way. The problems can range from counting certain one-dimensional drawings on two-dimensional surfaces to counting maps of certain type from a two-dimensional surface to some higher-dimensional space. The structures that we study in this thesis allow to encode the solutions to this type of enumerative and combinatorial problems in some general compact form. In one approach the solutions to the enumerative problems are encoded in a complex algebraic curve with certain functions on it. From this initial small set of data one can reconstruct the full solution with the help of a recursive procedure that is absolutely universal and does not depend on a particular problem. In another approach the solutions to the enumerative problems are encoded as certain integrals over some complicated spaces that parametrize different complex structures on two-dimensional surfaces. This reveals that the solutions to the enumerative problems reflect the geometric properties of the space of complex structures, also in a universal way. These two approaches turn out to be related in many different ways. In this thesis their relation is studied in the framework of an advanced differential geometric structure called Frobenius manifold
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Get PDF“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Appropriate Similarity Measures for Author Cocitation Analysis
Get PDFWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
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