106
106

Nov 14, 2013
11/13

by
Denis Auroux

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In this section and the next we give a di_erent way of looking at Green�s theorem which both shows its signi�cance for �ow �elds and allows us to give an intuitive physical meaning for this rather mysterious equality between integrals.

Topics: Maths, Analysis and Calculus, Calculus, Mathematics

Source: http://www.flooved.com/reader/1823

82
82

Nov 14, 2013
11/13

by
Denis Auroux

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Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics

Source: http://www.flooved.com/reader/2122

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45

Sep 18, 2013
09/13

by
Denis Auroux; Ivan Smith

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This set of lectures aims to give an overview of Donaldson's theory of linear systems on symplectic manifolds and the algebraic and geometric invariants to which they give rise. After collecting some of the relevant background, we discuss topological, algebraic and symplectic viewpoints on Lefschetz pencils and branched covers of the projective plane. The later lectures discuss invariants obtained by combining this theory with pseudo-holomorphic curve methods.

Source: http://arxiv.org/abs/math/0401021v1

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132

Jul 20, 2013
07/13

by
Denis Auroux; J. Elisenda Grigsby; Stephan M. Wehrli

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We discuss a relationship between Khovanov- and Heegaard Floer-type homology theories for braids. Explicitly, we define a filtration on the bordered Heegaard-Floer homology bimodule associated to the double-branched cover of a braid and show that its associated graded bimodule is equivalent to a similar bimodule defined by Khovanov and Seidel.

Source: http://arxiv.org/abs/1107.2841v2

85
85

Nov 14, 2013
11/13

by
Denis Auroux

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1. Symplectic Manifolds...

Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics

Source: http://www.flooved.com/reader/2121

109
109

Nov 14, 2013
11/13

by
Denis Auroux

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1. Homeomorphism Classification of Simply Connected Compact 4-Manifolds...

Topics: Maths, Linear Algebra and Geometry, Linear Algebra, Geometry, Geometry of Manifolds, Complex...

Source: http://www.flooved.com/reader/2119

59
59

Sep 23, 2013
09/13

by
Denis Auroux

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We give an overview of various recent results concerning the topology of symplectic 4-manifolds and singular plane curves, using branched covers and isotopy problems as a unifying theme. While this paper does not contain any new results, we hope that it can serve as an introduction to the subject, and will stimulate interest in some of the open questions mentioned in the final section.

Source: http://arxiv.org/abs/math/0411233v1

78
78

Nov 14, 2013
11/13

by
Denis Auroux

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1. Spin Structures...

Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics

Source: http://www.flooved.com/reader/2120

251
251

Nov 14, 2013
11/13

by
Denis Auroux

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1. K�hler Geometry

Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics

Source: http://www.flooved.com/reader/2108

78
78

Nov 14, 2013
11/13

by
Denis Auroux

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Let (M, _, J) be a compact K�hler manifold, .... Then we can �nd a line bundle L_M with �rst Chern class ...

Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics

Source: http://www.flooved.com/reader/2112

129
129

Nov 14, 2013
11/13

by
Denis Auroux

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Theorems about homogeneous and inhomogeneous systems: On the basis of our work so far, we can formulate a few general results about square systems of linear equations. They are the theorems most frequently referred to in the applications.

Topics: Maths, Analysis and Calculus, Calculus, Mathematics

Source: http://www.flooved.com/reader/1809

72
72

Nov 14, 2013
11/13

by
Denis Auroux

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1. Pseudoholomorphic Curves ...

Topics: Maths, Linear Algebra and Geometry, Geometry, Mathematics

Source: http://www.flooved.com/reader/1980

78
78

Sep 20, 2013
09/13

by
Denis Auroux

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Using the recent results of Siebert and Tian about the holomorphicity of genus 2 Lefschetz fibrations with irreducible singular fibers, we show that any genus 2 Lefschetz fibration becomes holomorphic after fiber sum with a holomorphic fibration.

Source: http://arxiv.org/abs/math/0204285v1

62
62

Sep 24, 2013
09/13

by
Denis Auroux

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We study the classification of Lefschetz fibrations up to stabilization by fiber sum operations. We show that for each genus there is a `universal' fibration f^0_g with the property that, if two Lefschetz fibrations over S^2 have the same Euler-Poincare characteristic and signature, the same numbers of reducible singular fibers of each type, and admit sections with the same self-intersection, then after repeatedly fiber summing with f^0_g they become isomorphic. As a consequence, any two...

Source: http://arxiv.org/abs/math/0412120v2

42
42

Sep 21, 2013
09/13

by
Denis Auroux

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In this survey paper, we briefly review various aspects of the SYZ approach to mirror symmetry for non-Calabi-Yau varieties, focusing in particular on Lagrangian fibrations and wall-crossing phenomena in Floer homology. Various examples are presented, some of them new.

Source: http://arxiv.org/abs/0902.1595v1

90
90

Nov 14, 2013
11/13

by
Denis Auroux

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1. The Quintic (contd.) Recall that we had a quintic mirror family...

Topics: Maths, Linear Algebra and Geometry, Geometry, Mathematics

Source: http://www.flooved.com/reader/1962

56
56

Sep 18, 2013
09/13

by
Denis Auroux

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The first part of this paper is a review of the Strominger-Yau-Zaslow conjecture in various settings. In particular, we summarize how, given a pair (X,D) consisting of a Kahler manifold and an anticanonical divisor, families of special Lagrangian tori in X-D and weighted counts of holomorphic discs in X can be used to build a Landau-Ginzburg model mirror to X. In the second part we turn to more speculative considerations about Calabi-Yau manifolds with holomorphic involutions and their...

Source: http://arxiv.org/abs/0803.2734v1

218
218

Nov 14, 2013
11/13

by
Denis Auroux

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However in real-world applications this is frequently not so. Computing partial derivatives then becomes confusing, but it is better to face these complications now while you are still in a calculus course, than wait to be hit with them at the same time that you are struggling to cope with the thermodynamics or economics or whatever else is involved.

Topics: Maths, Analysis and Calculus, Analysis, Calculus, Differentiation from ? m to ??, Differentials,...

Source: http://www.flooved.com/reader/1810

298
298

Nov 14, 2013
11/13

by
Denis Auroux

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As we pointed out in the introduction, vectors will be used throughout the course. The basic concepts are straight forward, but you will have to master some new terminology. Another important point we made earlier is that we can view vectors in two di_erent ways: geometrically and algebraically. We will start with the geometric view and introduce terminology along the way.

Topics: Maths, Analysis and Calculus, Calculus, Mathematics

Source: http://www.flooved.com/reader/1826

83
83

Nov 14, 2013
11/13

by
Denis Auroux

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Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics

Source: http://www.flooved.com/reader/2127

286
286

Nov 14, 2013
11/13

by
Denis Auroux

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An important application of the higher partial derivatives is that they are used in partial di_erential equations to express some laws of physics which are basic to most science and engineering subjects. In this section, we will give examples of a few such equations. The reason is partly cultural, so you meet these equations early and learn to recognize them, and partly technical: to give you a little more practice with the chain rule and computing higher derivatives.

Topics: Maths, Differential Equations (ODEs & PDEs), Methods, Partial Differential Equations (PDEs),...

Source: http://www.flooved.com/reader/1811

81
81

Nov 14, 2013
11/13

by
Denis Auroux

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0.1. Lagrangian Floer Homology (contd). Let (M, _) be a symplectic man_ifold, L0, L1 compact Lagrangian submanifolds.

Topics: Maths, Linear Algebra and Geometry, Geometry, Mathematics

Source: http://www.flooved.com/reader/1963

88
88

Nov 14, 2013
11/13

by
Denis Auroux

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Goal of the class: this is not always going to be the most rigorous class, but the goal is to tell the story of mirror symmetry.

Topics: Maths, Linear Algebra and Geometry, Geometry, Mathematics

Source: http://www.flooved.com/reader/1961

74
74

Nov 14, 2013
11/13

by
Denis Auroux

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1. Lagrangian Floer Homology (contd) Recall �rst our approaches to CF _(L, L) with the A� algebraic structure:

Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics

Source: http://www.flooved.com/reader/1967

59
59

Nov 14, 2013
11/13

by
Denis Auroux

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Last time, we say that a deformation of (X, J) is given by...

Topics: Maths, Linear Algebra and Geometry, Geometry, Mathematics

Source: http://www.flooved.com/reader/1979

126
126

Nov 14, 2013
11/13

by
Denis Auroux

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1. Hodge Theory - Theorem 1 (Hodge)....

Topics: Maths, Linear Algebra and Geometry, Analysis and Calculus, Differential Equations (ODEs &...

Source: http://www.flooved.com/reader/2109

115
115

Nov 14, 2013
11/13

by
Denis Auroux

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1. Branched Covers...

Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics

Source: http://www.flooved.com/reader/2118

94
94

Nov 14, 2013
11/13

by
Denis Auroux

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We now return to the complex K�hler case. Let (M, _, J) be a complex K�hler manifold...

Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics

Source: http://www.flooved.com/reader/2113

55
55

Sep 20, 2013
09/13

by
Denis Auroux; Ludmil Katzarkov; Dmitri Orlov

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We study the derived categories of coherent sheaves of weighted projective spaces and their noncommutative deformations, and the derived categories of Lagrangian vanishing cycles of their mirror Landau-Ginzburg models. In particular, we show that the derived category of coherent sheaves (B-branes) on the weighted projective plane $\CP^2(a,b,c)$ is equivalent to the derived category of vanishing cycles (A-branes) on the affine hypersurface $X=\{x^ay^bz^c=1\}\subset (\C^*)^3$ equipped with an...

Source: http://arxiv.org/abs/math/0404281v1

61
61

Sep 18, 2013
09/13

by
Denis Auroux; Ludmil Katzarkov

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Every compact symplectic 4-manifold can be realized as a branched cover of the complex projective plane branched along a symplectic curve with cusp and node singularities; the covering map is induced by a triple of sections of a "very ample" line bundle. In this paper, we give an explicit formula describing the behavior of the braid monodromy invariants of the branch curve upon degree doubling of the linear system (from a very ample bundle $L^k$ to $L^{2k}$). As a consequence, we...

Source: http://arxiv.org/abs/math/0605001v1

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74

Sep 21, 2013
09/13

by
Denis Auroux

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The goal of these notes is to give a short introduction to Fukaya categories and some of their applications. The first half of the text is devoted to a brief review of Lagrangian Floer (co)homology and product structures. Then we introduce the Fukaya category (informally and without a lot of the necessary technical detail), and briefly discuss algebraic concepts such as exact triangles and generators. Finally, we mention wrapped Fukaya categories and outline a few applications to symplectic...

Source: http://arxiv.org/abs/1301.7056v1

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127

Sep 18, 2013
09/13

by
Denis Auroux; Viktor S. Kulikov; Vsevolod V. Shevchishin

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We prove that any two irreducible cuspidal Hurwitz curves $C_0$ and $C_1$ (or more generally, curves with A-type singularities) in the Hirzebruch surface $F_N$ with coinciding homology classes and sets of singularities are regular homotopic; and symplectically regular homotopic if $C_0$ and $C_1$ are symplectic with respect to a compatible symplectic form.

Source: http://arxiv.org/abs/math/0401172v1

61
61

Sep 21, 2013
09/13

by
Denis Auroux

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We study the geometry of complexified moduli spaces of special Lagrangian submanifolds in the complement of an anticanonical divisor in a compact Kahler manifold. In particular, we explore the connections between T-duality and mirror symmetry in concrete examples, and show how quantum corrections arise in this context.

Source: http://arxiv.org/abs/0706.3207v2

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48

Sep 22, 2013
09/13

by
Denis Auroux

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The main goal of this paper is to discuss a symplectic interpretation of Lipshitz, Ozsvath and Thurston's bordered Heegaard-Floer homology in terms of Fukaya categories of symmetric products and Lagrangian correspondences. More specifically, we give a description of the algebra A(F) which appears in the work of Lipshitz, Ozsvath and Thurston in terms of (partially wrapped) Floer homology for product Lagrangians in the symmetric product, and outline how bordered Heegaard-Floer homology itself...

Source: http://arxiv.org/abs/1001.4323v3

161
161

Nov 14, 2013
11/13

by
Denis Auroux

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Topics: Maths, Analysis and Calculus, Calculus, Mathematics

Source: http://www.flooved.com/reader/1816

168
168

Nov 14, 2013
11/13

by
Denis Auroux

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The three theorems we have studied: the divergence theorem and Stokes� theorem in space, and Green�s theorem in the plane (which is really just a special case of Stokes� theo_rem) are widely used in physics and continuum mechanics, in the study of �elds, potentials, heat �ow, wave motion in liquids, gases, and solids, and thermodynamics, to name some of the uses. Often partial di_erential equations which model some physical situation are de_rived using the vector integral calculus...

Topics: Maths, Analysis and Calculus, Calculus, Mathematics

Source: http://www.flooved.com/reader/1821

131
131

Nov 14, 2013
11/13

by
Denis Auroux

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Gradient �elds and path-independence: The two-dimensional theory developed for line integrals in the plane generalizes easily to three-space. For the part where no new ideas are involved, we will be brief, just stating the results, and in places sketching the proofs.

Topics: Maths, Analysis and Calculus, Calculus, Integration Theorems, Divergence Theorem, Mathematics

Source: http://www.flooved.com/reader/1815

203
203

Nov 14, 2013
11/13

by
Denis Auroux

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Conversely, given a vector �eld in a region of the xy-plane, it determines a vector function of the type (1), by expressing each vector of the �eld in terms of its i and j components.

Topics: Maths, Analysis and Calculus, Calculus, Mathematics

Source: http://www.flooved.com/reader/1813

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44

Sep 19, 2013
09/13

by
Denis Auroux

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The main theorem of this paper is a result of estimated transversality with respect to stratifications of jet spaces in the approximately holomorphic category over an almost-complex manifold. The notion of asymptotic ampleness of complex vector bundles over an almost-complex manifold is also discussed, as well as applications to the construction of maps with generic singularities from compact symplectic manifolds to projective spaces.

Source: http://arxiv.org/abs/math/0010052v1

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80

Nov 14, 2013
11/13

by
Denis Auroux

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0.1. General Approach to Special Lagrangian Fibrations. The idea is to degenerate X to a union of toric varieties, build a degenerate �bration there, and try to smooth it: the approach is due to Haase-Zharkov, WD Ruan, Gross-Siebert, etc. This is a special type of LCSL. We �rst sketch this in the K3 case: as last time,...

Topics: Maths, Linear Algebra and Geometry, Geometry, Mathematics

Source: http://www.flooved.com/reader/1977

89
89

Nov 14, 2013
11/13

by
Denis Auroux

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1. The Quintic (contd.) - To recall where we were, we had...

Topics: Maths, Linear Algebra and Geometry, Geometry, Mathematics

Source: http://www.flooved.com/reader/1985

79
79

Nov 14, 2013
11/13

by
Denis Auroux

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1. SYZ Conjecture (cntd.) - Question: how does one build a mirror X�of a given Calabi-Yau X?

Topics: Maths, Linear Algebra and Geometry, Geometry, Mathematics

Source: http://www.flooved.com/reader/1974

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99

Nov 14, 2013
11/13

by
Denis Auroux

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We continue our discussion of the Thurston manifold introduced last time. ..

Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics

Source: http://www.flooved.com/reader/2116

75
75

Nov 14, 2013
11/13

by
Denis Auroux

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1. Symplectic Sum...

Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics

Source: http://www.flooved.com/reader/2117

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97

Nov 14, 2013
11/13

by
Denis Auroux

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Recall that we were in the midst of elliptic operator analysis of the Laplace-deRham operator _ = (d + d_)^2 . We claimed that _ was an elliptic operator, i.e. it has an invertible symbol _(_) = _ |_|^2 id. We stated that a di_erential operator

Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics

Source: http://www.flooved.com/reader/2110

54
54

Sep 20, 2013
09/13

by
Denis Auroux

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The topology of symplectic 4-manifolds is related to that of singular plane curves via the concept of branched covers. Thus, various classification problems concerning symplectic 4-manifolds can be reformulated as questions about singular plane curves. Moreover, using braid monodromy, these can in turn be reformulated in the language of braid group factorizations. While the results mentioned in this paper are not new, we hope that they will stimulate interest in these questions, which remain...

Source: http://arxiv.org/abs/math/0410119v2

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55

Sep 17, 2013
09/13

by
Denis Auroux

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We outline an interpretation of Heegaard-Floer homology of 3-manifolds (closed or with boundary) in terms of the symplectic topology of symmetric products of Riemann surfaces, as suggested by recent work of Tim Perutz and Yanki Lekili. In particular we discuss the connection between the Fukaya category of the symmetric product and the bordered algebra introduced by Robert Lipshitz, Peter Ozsvath and Dylan Thurston, and recast bordered Heegaard-Floer homology in this language.

Source: http://arxiv.org/abs/1003.2962v1

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102

Nov 14, 2013
11/13

by
Denis Auroux

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Last time, we were considering CP1 mirror to ... for ... the latter object is a Landau-Ginzburg model, i.e. a K�hler manifold with a holomorphic function called the �superpotential�. Homological mirror symmetry gave...

Topics: Maths, Linear Algebra and Geometry, Geometry, Mathematics

Source: http://www.flooved.com/reader/1978

74
74

Nov 14, 2013
11/13

by
Denis Auroux

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Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics

Source: http://www.flooved.com/reader/2124

75
75

Nov 14, 2013
11/13

by
Denis Auroux

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1. Homology and Cohomology - Recall from last time that, for M a smooth manifold, we produced a graded di_erential algebra...

Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics

Source: http://www.flooved.com/reader/2114