1,721,042 research outputs found
Lie algebroids in derived differential topology
No full textA classical principle in deformation theory asserts that any formal deformation problem is controlled by a differential graded Lie algebra. This thesis studies a generalization of this principle to Lie algebroids, and uses this to examine the interactions between the theory of Lie algebroids and the derived geometry of moduli spaces. The first half of the thesis develops the homotopy theory of differential graded Lie algebroids over a fixed affine derived manifold. We prove that any deformation problem over such a derived manifold is controlled by a Lie algebroid, by constructing an equivalence between the homotopy theory of Lie algebroids and the homotopy theory of formal moduli problems. This equivalence furthermore extends to an equivalence between representations of Lie algebroids and quasi-coherent sheaves over formal moduli problems. In the second half of the thesis, we develop the theory of derived differential topology and apply it to study Lie algebroids arising from derived differentiable stacks. Using the results of the first half, we show that the relative tangent bundle of a derived manifold over a derived stack has a Lie algebroid structure. We then provide a criterion for maps between Lie algebroids to integrate to maps between stacks, generalizing classical theorems of Lie and Van Est. This result is applied to show that finite-dimensional L-infinity algebras can be integrated to higher Lie groups
Stable operations and topological modular forms
No full textThis document expands our structural knowledge of topological modular forms TMF in two directions: the first, by extending the functoriality inherent to the definition of TMF, and the second, being tools to calculate the effect that endomorphisms of TMF have on homotopy groups. These structural statements allow us to lift classical operations on modular forms, such as Adams operations, Hecke operators, and Atkin–Lehner involutions, to stable operations on TMF. Some novel applications of these operations are then found, including a derivation of some congruences of Ramanujan in a purely homotopy theoretic manner, improvements upon known bounds of Maeda’s conjecture, as well as some applications in homotopy theory. These techniques serve as teasers for the potential of these operations
Lie algebroids in derived differential topology
No full textA classical principle in deformation theory asserts that any formal deformation problem is controlled by a differential graded Lie algebra. This thesis studies a generalization of this principle to Lie algebroids, and uses this to examine the interactions between the theory of Lie algebroids and the derived geometry of moduli spaces. The first half of the thesis develops the homotopy theory of differential graded Lie algebroids over a fixed affine derived manifold. We prove that any deformation problem over such a derived manifold is controlled by a Lie algebroid, by constructing an equivalence between the homotopy theory of Lie algebroids and the homotopy theory of formal moduli problems. This equivalence furthermore extends to an equivalence between representations of Lie algebroids and quasi-coherent sheaves over formal moduli problems. In the second half of the thesis, we develop the theory of derived differential topology and apply it to study Lie algebroids arising from derived differentiable stacks. Using the results of the first half, we show that the relative tangent bundle of a derived manifold over a derived stack has a Lie algebroid structure. We then provide a criterion for maps between Lie algebroids to integrate to maps between stacks, generalizing classical theorems of Lie and Van Est. This result is applied to show that finite-dimensional L-infinity algebras can be integrated to higher Lie groups
Stable homotopy theory of dendroidal sets
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139941.pdf (Publisher’s version ) (Open Access)Radboud Universiteit Nijmegen, 23 april 2015Promotor : Moerdijk, I.III, 170 p
A Mysterious Tensor Product in Topology
Full text linkWe describe the construction of the tensor product of topological operads introduced by Boardman and Vogt, and explain some questions about it that have been puzzling Vogt and others for many years
On the relation between connection and sprays
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128972.pdf (Publisher’s version ) (Closed access
Stable operations and topological modular forms
Full text linkThis document expands our structural knowledge of topological modular forms TMF in two directions: the first, by extending the functoriality inherent to the definition of TMF, and the second, being tools to calculate the effect that endomorphisms of TMF have on homotopy groups. These structural statements allow us to lift classical operations on modular forms, such as Adams operations, Hecke operators, and Atkin–Lehner involutions, to stable operations on TMF. Some novel applications of these operations are then found, including a derivation of some congruences of Ramanujan in a purely homotopy theoretic manner, improvements upon known bounds of Maeda’s conjecture, as well as some applications in homotopy theory. These techniques serve as teasers for the potential of these operations
Cohomology Theories In Synthetic Differential Geometry
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128969.pdf (Publisher’s version ) (Open Access
Continuous fibrations and inverse limits of toposes
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128971.pdf (Publisher’s version ) (Open Access
Unstable Periodic Homotopy Theory
Full text linkIn this thesis we study unstable vₕ-periodic homotopy theory, where h is a natural number; here ``unstable'' refers to the homotopy theory of topological spaces. The work consists of two parts. In Part I we give a detailed exposition of the foundations of unstable vₕ-periodic homotopy theory, sharpen an existing result about vₕ-periodic equivalences of H-spaces, and pose concrete questions and conjectures for future studies. The expository part follows paper by Bousfield, Dror Farjoun and Heuts and aims to assemble in one place the central notions and theorems of unstable localisations with a focus on unstable periodic homotopy theory. The goal of Part II is to understand unstable vₕ-periodic phenomena from the point of view of Lie algebras in the stable vₕ-periodic homotopy category. We analyse the costabilisation of vₕ-periodic homotopy types and obtain a universal property of the Bousfield--Kuhn functor
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