1,720,996 research outputs found
Large N Duality, Lagrangian Cycles, and Algebraic Knots
Full text linkWe consider knot invariants in the context of large N transitions of topological strings. In particular we consider aspects of Lagrangian cycles associated to knots in the conifold geometry. We show how these can be explicitly constructed in the case of algebraic knots. We use this explicit construction to explain a recent conjecture relating study of stable pairs on algebraic curves with HOMFLY polynomials. Furthermore, for torus knots, using the explicit construction of the Lagrangian cycle, we also give a direct A-model computation and recover the HOMFLY polynomial for this case.Engineering and Physical Sciences Research CouncilSimons Foundatio
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Developments in the mathematics of the A-model: constructing Calabi-Yau structures and stability conditions on target categories
Full text linkThis dissertation is an exposition of the work conducted by the author in the later years of graduate school, when two main projects were completed. Both projects concern the application of sheaf-theoretic techniques to construct geometric structures on categories appearing in the mathematical description of the A-model, which are of interest to symplectic geometers and mathematicians working in mirror symmetry. This dissertation starts with an introduction to the aspects of the physics of mirror symmetry that will be needed for the exposition of the techniques and results of these two projects. The first project concerns the construction of Calabi-Yau structures on topological Fukaya categories, using the microlocal model of Nadler and others for these categories. The second project introduces and studies a similar local-to-global technique, this time used to construct Bridgeland stability conditions on Fukaya categories of marked surfaces, extending some results of Haiden, Katzarkov and Kontsevich on the relation between stability of Fukaya categories and geometry of holomorphic differentials
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Symplectic geometric methods in microlocal sheaf theory
Full text linkThe main goal of this dissertation is to import symplectic geometric methods into microlocal sheaf theory based on the foundational sheaf quantization construction by Guillermou, Kashiwara, and Schapira. This construction provides the notion of isotopies of sheaves and a sheaf-theoretic analogue of the notion of continuation maps in Lagrangian Floer theory.Based on previous work by Ganatra, Pardon, and Shende, we make further ex- amination on the category of unbounded sheaves microsupported in some singular isotropic Λ in the cosphere bundle. We show that various categorical constructions concerning this category can be described in symplectic geometric terms by using isotopies of sheaves.The main construction is a sheaf-theoretic analogue of the wrapped Fukaya category, by localizing a category of sheaves microsupported away from some given Λ along continuation maps. When Λ is a subanalytic singular isotropic, we also construct a comparison map to the category of compact objects in the category mentioned above, and show that it is an equivalence. The last statement can be seen as a sheaf-theoretical incarnation of the sheaf-Fukaya comparison theorem of Ganatra-Pardon-Shende
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Applications of the Intersection Theory of Singular Varieties
Full text linkWe develop tools for computing invariants of singular varieties and apply them to the classical theory of nodal curves and the complexity analysis of non-convex optimization problems.The first result provides a method for computing the Segre class of a closed embedding X → Y in terms of the Segre classes of X and Y in an ambient space Z. This method is used to extend the classical Riemann-Kempf formula to the case of nodal curves.Next we focus on techniques for computing the ED degree of a complex projective variety associated to an optimization problem. As a first application we consider the problem of scene reconstruction and find a degree 3 polynomial that computes the ED degree of the multiview variety as a function of the number of cameras. Our second application concerns the problem of weighted low rank approximation. We provide a characterization of the weight matrices for which the weighted 1-rank approximation problem has maximal ED degree
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Combinatorial Topology and Applications to Quantum Field Theory
Full text linkTopology has become increasingly important in the study of many-body quantum mechanics, in both high energy and condensed matter applications. While the importance of smooth topology has long been appreciated in this context, especially with the rise of index theory, torsion phenomena and discrete group symmetries are relatively new directions. In this thesis, I collect some mathematical results and conjectures that I have encountered in the exploration of these new topics. I also give an introduction to some quantum field theory topics I hope will be accessible to topologists
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A Hodge-theoretic study of augmentation varieties associated to Legendrian knots/tangles
Full text linkIn this article, we give a tangle approach in the study of Legendrian knots in the standardcontact three-space. On the one hand, we define and construct Legenrian isotopy invariantsincluding ruling polynomials and Legendrian contact homology differential graded algebras(LCH DGAs) for Legendrian tangles, generalizing those of Legendrian knots. Ruling polynomialsare the Legendrian analogues of Jones polynomials in topological knot theory, in thesense that they satisfy the composition axiom.On the other hand, we study certain aspects of the Hodge theory of the “representationvarieties (of rank 1)” of the LCH DGAs, called augmentation varieties, associated to Legendriantangles. The augmentation variety (with fixed boundary conditions), hence its mixedHodge structure on the compactly supported cohomology, is a Legendrian isotopy invariantup to a normalization. This gives a generalization of ruling polynomials in the followingsense: the point-counting/weight (or E-) polynomial of the variety, up to a normalized factor,is the ruling polynomial. This tangle approach in particular provides a generalizationand a more natural proof to the previous known results of M.Henry and D.Rutherford. Italso leads naturally to a ruling decomposition of this variety, which then induces a spectralsequence converging to the MHS. As some applications, we show that the variety is of Hodge-Tatetype, show a vanishing result on its cohomology, and provide an example-computationof the MHSs
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe 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
Full text link“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
Full text linkWe 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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