Spelling suggestions: "subject:"gott"" "subject:"mott""
1 |
Bott PeriodicityMurrugarra Tomairo, David Manuel 05 June 2007 (has links)
Bott periodicity plays a fundamental role in the definition and understanding of K-theory, the generalized cohomology theory defined by vector bundles. This paper examines the proof, given by Atiyah and Bott[3], of the periodicity theorem for the complex case.
We also describe the long exact sequence for K-cohomology in the category of connected finite CW-complexes. / Master of Science
|
2 |
The moment graph for Bott-Samelson varieties and applications to quantum cohomologyWithrow, Camron Michael 29 June 2018 (has links)
We give a description of the moment graph for Bott-Samelson varieties in arbitrary Lie type. We use this, along with curve neighborhoods and explicit moduli space computations, to compute a presentation for the small quantum cohomology ring of a particular Bott-Samelson variety in Type A. / Ph. D.
|
3 |
Um invariante para sistemas com integral primeira Morse-Bott / A invariant for systems with a Morse-Bott first integralSarmiento, Ingrid Sofia Meza 16 August 2011 (has links)
Nesta dissertação são investigados os sistemas diferenciais com integral primeira do tipo Morse-Bott definidos em superfícies compactas e orientáveis. A cada sistema, nas condições acima descritas, associa-se um grafo de modo que a correspondência entre os grafos e as classes de equivalência topologica orbital dos campos investigados seja bijetiva. Portanto, apresenta-se um invariante completo, chamado aqui de grafo de Bott, para essa classe de sistemas. Essa abordagem surgiu como uma iniciativa de generalizar o estudo realizado para sistemas Hamiltonianos com um grau de liberdade com integral primeira do tipo Morse definidos em superfícies 2-dimensionais compactas, onde os conceitos de átomos e fluxos gradiente foram aplicados por A.V. Bolsinov em [4] / In this dissertation are studied differential systems with a Morse-Bott first integral defined on compact orientable surfaces. For each system, under the conditions described above, is associated a graph so that the correspondence between graphs and the orbital topological equivalence classes of the systems are bijective. Therefore, we present a complete invariant, called here Bott graph for this class of systems. This approach has emerged as an initiative to generalize the study to systems Hamiltonian with one degree of freedom having a Morse first integral in 2-dimensional compact surfaces, where the concepts of atoms and gradient flows were applied by A.V. Bolsinov in [4]
|
4 |
Um invariante para sistemas com integral primeira Morse-Bott / A invariant for systems with a Morse-Bott first integralIngrid Sofia Meza Sarmiento 16 August 2011 (has links)
Nesta dissertação são investigados os sistemas diferenciais com integral primeira do tipo Morse-Bott definidos em superfícies compactas e orientáveis. A cada sistema, nas condições acima descritas, associa-se um grafo de modo que a correspondência entre os grafos e as classes de equivalência topologica orbital dos campos investigados seja bijetiva. Portanto, apresenta-se um invariante completo, chamado aqui de grafo de Bott, para essa classe de sistemas. Essa abordagem surgiu como uma iniciativa de generalizar o estudo realizado para sistemas Hamiltonianos com um grau de liberdade com integral primeira do tipo Morse definidos em superfícies 2-dimensionais compactas, onde os conceitos de átomos e fluxos gradiente foram aplicados por A.V. Bolsinov em [4] / In this dissertation are studied differential systems with a Morse-Bott first integral defined on compact orientable surfaces. For each system, under the conditions described above, is associated a graph so that the correspondence between graphs and the orbital topological equivalence classes of the systems are bijective. Therefore, we present a complete invariant, called here Bott graph for this class of systems. This approach has emerged as an initiative to generalize the study to systems Hamiltonian with one degree of freedom having a Morse first integral in 2-dimensional compact surfaces, where the concepts of atoms and gradient flows were applied by A.V. Bolsinov in [4]
|
5 |
Abelianization and Floer homology of Lagrangians in clean intersectionSchmäschke, Felix 10 April 2017 (has links) (PDF)
This thesis is split up into two parts each revolving around Floer
homology and quantum cohomology of closed monotone symplectic
manifolds. In the first part we consider symplectic manifolds obtained
by symplectic reduction. Our main result is that a quantum version of
an abelianization formula of Martin holds, which relates
the quantum cohomologies of symplectic quotients by a group and by its
maximal torus. Also we show a quantum version of the Leray-Hirsch
theorem for Floer homology of Lagrangian intersections in the
quotient.
The second part is devoted to Floer homology of a pair of monotone
Lagrangian submanifolds in clean intersection. Under these assumptions
the symplectic action functional is degenerated. Nevertheless
Frauenfelder defines a version of Floer
homology, which is in a certain sense an infinite dimensional analogon
of Morse-Bott homology. Via natural filtrations on the chain level we
were able to define two spectral sequences which serve as a tool to
compute Floer homology. We show how these are used to obtain new
intersection results for simply connected Lagrangians in the product
of two complex projective spaces.
The link between both parts is that in the background the same
technical methods are applied; namely the theory of holomorphic strips
with boundary on Lagrangians in clean intersection. Since all our
constructions rely heavily on these methods we also give a detailed
account of this theory although in principle many results are not new
or require only straight forward generalizations.
|
6 |
Abelianization and Floer homology of Lagrangians in clean intersectionSchmäschke, Felix 14 December 2016 (has links)
This thesis is split up into two parts each revolving around Floer
homology and quantum cohomology of closed monotone symplectic
manifolds. In the first part we consider symplectic manifolds obtained
by symplectic reduction. Our main result is that a quantum version of
an abelianization formula of Martin holds, which relates
the quantum cohomologies of symplectic quotients by a group and by its
maximal torus. Also we show a quantum version of the Leray-Hirsch
theorem for Floer homology of Lagrangian intersections in the
quotient.
The second part is devoted to Floer homology of a pair of monotone
Lagrangian submanifolds in clean intersection. Under these assumptions
the symplectic action functional is degenerated. Nevertheless
Frauenfelder defines a version of Floer
homology, which is in a certain sense an infinite dimensional analogon
of Morse-Bott homology. Via natural filtrations on the chain level we
were able to define two spectral sequences which serve as a tool to
compute Floer homology. We show how these are used to obtain new
intersection results for simply connected Lagrangians in the product
of two complex projective spaces.
The link between both parts is that in the background the same
technical methods are applied; namely the theory of holomorphic strips
with boundary on Lagrangians in clean intersection. Since all our
constructions rely heavily on these methods we also give a detailed
account of this theory although in principle many results are not new
or require only straight forward generalizations.:1. Introduction
2. Overview of the main results
2.1. Abelianization .
2.2. Quantum Leray-Hirsch theorem
2.3. Floer homology of Lagrangians in clean intersection
3. Background
3.1. Symplectic geometry .
3.2. Hamiltonian action functional
3.3. Morse homology .
3.4. Floer homology
4. Asymptotic analysis
4.1. Main statement .
4.2. Mean-value inequality .
4.3. Isoperimetric inequality
4.4. Linear theory
4.5. Proofs
5. Compactness
5.1. Cauchy-Riemann-Floer equation .
5.2. Local convergence .
5.3. Convergence on the ends
5.4. Minimal energy .
5.5. Action, energy and index estimates
6. Fredholm Theory
6.1. Banach manifold .
6.2. Linear theory
7. Transversality
7.1. Setup
7.2. R-dependent structures
7.3. R-invariant structures .
7.4. Regular points .
7.5. Floer’s ε-norm .
8. Gluing
8.1. Setup and main statement
8.2. Pregluing .
8.3. A uniform bounded right inverse
8.4. Quadratic estimate
8.5. Continuity of the gluing map
8.6. Surjectivity of the gluing map
8.7. Degree of the gluing map
8.8. Morse gluing .
9. Orientations
9.1. Preliminaries and notation
9.2. Spin structures and relative spin structures
9.3. Orientation of caps
9.4. Linear theory .
10.Pearl homology
10.1. Overview .
10.2. Pearl trajectories .
10.3. Invariance .
10.4. Spectral sequences
11.Proofs of the main results
11.1. Abelianization Theorem
11.2. Quantum Leray-Hirsch Theorem .
12.Applications
12.1. Quantum cohomology of the complex Grassmannian
12.2. Lagrangian spheres in symplectic quotients
A. Estimates
A.1. Derivative of the exponential map
A.2. Parallel Transport
A.3. Estimates for strips
B. Operators on Hilbert spaces
B.1. Spectral gap
B.2. Flow operator
C. Viterbo index
D. Quotients of principal bundles by maximal tori
D.1. Compact Lie groups
D.2. The cohomology of the quotient of principle bundles by maximal tori
|
7 |
Teorias de Morse e Morse-Bott em sistemas dinâmicosBeltrán, Elmer Rusbert Calderón January 2014 (has links)
Orientadora: Profa. Dra. Mariana Rodrigues da Silveira / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Matemática , 2014. / Neste trabalho apresentamos um estudo das Teorias de Morse e Morse-Bott no contexto
de sistemas dinâmicos. Consideramos uma variedade Riemanniana suave e fechada M
de dimensão finita. Dada f : M ! R uma função de Morse-Smale, associamos a f o
complexo de cadeia de Morse-Smale-Witten, que recupera a homologia da variedade
M (Teorema de Homologia de Morse). Mais geralmente, qualquer função de Morse-
Bott-Smale f :M !R pode ser associada ao complexo de cadeia de Morse-Bott-Smale,
que é um multicomplexo que se reduz ao complexo de cadeia de Morse-Smale-Witten
quando f é uma função de Morse. O Teorema de Homologia de Morse-Bott mostra que a
homologia deste multicomplexo também coincide com a homologia de M sua prova tem
como caso particular uma prova para o Teorema da Homologia de Morse. / In this work we present a study of Morse and Morse-Bott theories in the context of
dynamical systems. We consider a Riemannian smooth, closed n-dimensional manifold
M. Given a Morse-Smale function f :M !R, we associate f to the Morse-Smale-Witten
chain complex, which recovers the homology of the manifold M (Morse Homology
Theorem). More generally, any Morse-Bott-Smale function f :M !R can be associated
to the Morse-Bott-Smale chain complex, which is a multicomplex that coincides with the
Morse-Smale-Witten complex when f is a Morse function. The Morse-Bott Homology
Theorem shows that the homology of thismulticomplex also coincides with the homology
of M and its proof has as a particular case a proof for the Morse Homology Theorem.
|
8 |
Bott\'s periodicity theorem from the algebraic topology viewpoint / O teorema da periodicidade de Bott sob o olhar da topologia algébricaBonatto, Luciana Basualdo 23 August 2017 (has links)
In 1970, Raoul Bott published The Periodicity Theorem for the Classical Groups and Some of Its Applications, in which he uses this famous result as a guideline to present some important areas and tools of Algebraic Topology. This dissertation aims to use the path Bott presented in his article as a guideline to address certain topics on Algebraic Topology. We start this incursion developing important tools used in Homotopy Theory such as spectral sequences and Eilenberg-MacLane spaces, exploring how they can be combined to aid in computation of homotopy groups. We then study important results of Morse Theory, a tool which was in the centre of Botts proof of the Periodicity Theorem. We also develop two extensions: Morse-Bott Theory, and the applications of such results to the loopspace of a manifold. We end by giving an introduction to generalised cohomology theories and K-Theory. / Em 1970, Raoul Bott publicou o artigo The Periodicity Theorem for the Classical Groups and Some of Its Applications no qual usava esse famoso resultado como um guia para apresentar importantes áreas e ferramentas da Topologia Algébrica. O presente trabalho usa o mesmo caminho traçado por Bott em seu artigo como roteiro para explorar tópicos importantes da Topologia Algébrica. Começamos esta incursão desenvolvendo ferramentas importantes da Teoria de Homotopia como sequências espectrais e espaços de Eilenberg-MacLane, explorando como estes podem ser combinados para auxiliar em cálculos de grupos de homotopia. Passamos então a estudar resultados importantes de Teoria de Morse, uma ferramenta que estava no centro da demonstração de Bott do Teorema da Periodicidade. Desenvolvemos ainda, duas extensões: Teoria de Morse-Bott e aplicações destes resultados ao espaço de laços de uma variedade. Terminamos com uma introdução a teorias de cohomologia generalizadas e K-Teoria.
|
9 |
Bott\'s periodicity theorem from the algebraic topology viewpoint / O teorema da periodicidade de Bott sob o olhar da topologia algébricaLuciana Basualdo Bonatto 23 August 2017 (has links)
In 1970, Raoul Bott published The Periodicity Theorem for the Classical Groups and Some of Its Applications, in which he uses this famous result as a guideline to present some important areas and tools of Algebraic Topology. This dissertation aims to use the path Bott presented in his article as a guideline to address certain topics on Algebraic Topology. We start this incursion developing important tools used in Homotopy Theory such as spectral sequences and Eilenberg-MacLane spaces, exploring how they can be combined to aid in computation of homotopy groups. We then study important results of Morse Theory, a tool which was in the centre of Botts proof of the Periodicity Theorem. We also develop two extensions: Morse-Bott Theory, and the applications of such results to the loopspace of a manifold. We end by giving an introduction to generalised cohomology theories and K-Theory. / Em 1970, Raoul Bott publicou o artigo The Periodicity Theorem for the Classical Groups and Some of Its Applications no qual usava esse famoso resultado como um guia para apresentar importantes áreas e ferramentas da Topologia Algébrica. O presente trabalho usa o mesmo caminho traçado por Bott em seu artigo como roteiro para explorar tópicos importantes da Topologia Algébrica. Começamos esta incursão desenvolvendo ferramentas importantes da Teoria de Homotopia como sequências espectrais e espaços de Eilenberg-MacLane, explorando como estes podem ser combinados para auxiliar em cálculos de grupos de homotopia. Passamos então a estudar resultados importantes de Teoria de Morse, uma ferramenta que estava no centro da demonstração de Bott do Teorema da Periodicidade. Desenvolvemos ainda, duas extensões: Teoria de Morse-Bott e aplicações destes resultados ao espaço de laços de uma variedade. Terminamos com uma introdução a teorias de cohomologia generalizadas e K-Teoria.
|
10 |
A topologia de folheações e sistemas integráveis Morse-Bott em superfícies / The topology of foliations and integrable Morse-Bott systems on surfacesSarmiento, Ingrid Sofia Meza 23 July 2015 (has links)
Nesta tese estudamos os sistemas integráveis definidos em superfícies compactas possuindo uma integral primeira que é uma função Morse-Bott a valores em R. Estes sistemas são aqui chamados de sistemas integráveis Morse-Bott. Classificamos as curvas fechadas e oitos associados a pontos de selas imersos em superfícies compactas. Essa classificação é aplicada ao estudo das folheações Morse-Bott em superfícies e nos permite definir um invariante topológico completo para a classificação topológica global destas folheações. Como uma aplicação desse estudo obtemos a classificação dos sistemas Morse-Bott assim como a classificação topológica das funções Morse-Bott em superfícies compactas e orientáveis. Demonstramos ainda um teorema da realização baseado em duas transformações e numa folheação geradora. Para o caso das funções Morse-Bott também obtivemos um teorema de realização. Finalmente, investigamos a generalização de alguns dos resultados anteriores para sistemas definidos em superfícies não orientáveis. / In this thesis we study integrable systems on compact surfaces with a first integral as a Morse-Bott function with target R. These systems are called here integrable Morse-Bott systems. Initially we present the classification of closed curves and eights associated to saddle points on compact surfaces. This classification is applied to the study of Morse- Bott foliations on surfaces allowing us to define a complete topological invariant for the global topological classification of these foliations. Then as an application of this study we obtain the classification of integrable Morse-Bott systems as well as the topological classification of Morse-Bott functions on compact and orientable surfaces. We also prove a realization theorem based on two transformation and a generating foliation (the foliation on the sphere with two centers). In the case of Morse-Bott functions we also obtain a realization theorem. Finally we investigate generalizations of previous results for systems defined on non-orientable surfaces.
|
Page generated in 0.0286 seconds