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Jul 14, 2014
07/14

Jul 14, 2014
by
Denis Auroux

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We construct infinitely many families of monotone Lagrangian tori in $\mathbb{R}^6$, no two of which are related by Hamiltonian isotopies (or symplectomorphisms). These families are distinguished by the (arbitrarily large) numbers of families of Maslov index 2 pseudo-holomorphic discs that they bound.

Topics: Symplectic Geometry, Mathematics

Source: http://arxiv.org/abs/1407.3725

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Jul 18, 2013
07/13

Jul 18, 2013
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

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Mar 8, 2013
03/13

Mar 8, 2013
by
Denis Auroux; J. Elisenda Grigsby; Stephan M. Wehrli

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In 2001, Khovanov and Seidel constructed a faithful action of the (m+1)-strand braid group on the derived category of left modules over a quiver algebra, A_m. We interpret the Hochschild homology of the Khovanov-Seidel braid invariant as a direct summand of the sutured Khovanov homology of the annular braid closure.

Source: http://arxiv.org/abs/1303.1986v1

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Jan 29, 2013
01/13

Jan 29, 2013
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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Apr 30, 2012
04/12

Apr 30, 2012
by
Mohammed Abouzaid; Denis Auroux; Ludmil Katzarkov

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We consider mirror symmetry for (essentially arbitrary) hypersurfaces in (possibly noncompact) toric varieties from the perspective of the Strominger-Yau-Zaslow (SYZ) conjecture. Given a hypersurface $H$ in a toric variety $V$ we construct a Landau-Ginzburg model which is SYZ mirror to the blowup of $V\times\C$ along $H\times 0$, under a positivity assumption. This construction also yields SYZ mirrors to affine conic bundles, as well as a Landau-Ginzburg model which can be naturally viewed as a...

Source: http://arxiv.org/abs/1205.0053v1

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Mar 22, 2011
03/11

Mar 22, 2011
by
Mohammed Abouzaid; Denis Auroux; Alexander I. Efimov; Ludmil Katzarkov; Dmitri Orlov

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We prove that the wrapped Fukaya category of a punctured sphere ($S^2$ with an arbitrary number of points removed) is equivalent to the triangulated category of singularities of a mirror Landau-Ginzburg model, proving one side of the homological mirror symmetry conjecture in this case. By investigating fractional gradings on these categories, we conclude that cyclic covers on the symplectic side are mirror to orbifold quotients of the Landau-Ginzburg model.

Source: http://arxiv.org/abs/1103.4322v1

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Jul 28, 2010
07/10

Jul 28, 2010
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

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Mar 15, 2010
03/10

Mar 15, 2010
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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Mar 2, 2009
03/09

Mar 2, 2009
by
Denis Auroux

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We calculate the monodromies of the canonical Lefschetz pencils on a pair of homeomorphic Horikawa surfaces. We show in particular that the (pluri)canonical pencils on these surfaces have the same monodromy groups, and are related by a "partial twisting" operation.

Source: http://arxiv.org/abs/math/0605692v2

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Feb 10, 2009
02/09

Feb 10, 2009
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

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Mar 18, 2008
03/08

Mar 18, 2008
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

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Jul 10, 2007
07/07

Jul 10, 2007
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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Apr 28, 2006
04/06

Apr 28, 2006
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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Jun 9, 2005
06/05

Jun 9, 2005
by
Denis Auroux; Ludmil Katzarkov; Dmitri Orlov

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We study homological mirror symmetry for Del Pezzo surfaces and their mirror Landau-Ginzburg models. In particular, we show that the derived category of coherent sheaves on a Del Pezzo surface X_k obtained by blowing up CP^2 at k points is equivalent to the derived category of vanishing cycles of a certain elliptic fibration W_k:M_k\to\C with k+3 singular fibers, equipped with a suitable symplectic form. Moreover, we also show that this mirror correspondence between derived categories can be...

Source: http://arxiv.org/abs/math/0506166v1

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Jun 1, 2005
06/05

Jun 1, 2005
by
Denis Auroux; Simon K Donaldson; Ludmil Katzarkov

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We consider structures analogous to symplectic Lefschetz pencils in the context of a closed 4-manifold equipped with a `near-symplectic' structure (ie, a closed 2-form which is symplectic outside a union of circles where it vanishes transversely). Our main result asserts that, up to blowups, every near-symplectic 4-manifold (X,omega) can be decomposed into (a) two symplectic Lefschetz fibrations over discs, and (b) a fibre bundle over S^1 which relates the boundaries of the Lefschetz fibrations...

Source: http://arxiv.org/abs/math/0410332v2

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May 5, 2005
05/05

May 5, 2005
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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Jan 21, 2005
01/05

Jan 21, 2005
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

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Nov 10, 2004
11/04

Nov 10, 2004
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

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Jul 8, 2004
07/04

Jul 8, 2004
by
Denis Auroux; Vicente Muñoz; Francisco Presas

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Given a Lagrangian submanifold in a symplectic manifold and a Morse function on the submanifold, we show that there is an isotopic Morse function and a symplectic Lefschetz pencil on the manifold extending the Morse function to the whole manifold. From this construction we define a sequence of symplectic invariants classifying the isotopy classes of Lagrangian spheres in a symplectic 4-manifold.

Source: http://arxiv.org/abs/math/0407126v2

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Apr 15, 2004
04/04

Apr 15, 2004
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

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Jan 14, 2004
01/04

Jan 14, 2004
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

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Jan 4, 2004
01/04

Jan 4, 2004
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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Apr 8, 2003
04/03

Apr 8, 2003
by
Denis Auroux

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This text is a set of lecture notes for a series of four talks given at I.P.A.M., Los Angeles, on March 18-20, 2003. The first lecture provides a quick overview of symplectic topology and its main tools: symplectic manifolds, almost-complex structures, pseudo-holomorphic curves, Gromov-Witten invariants and Floer homology. The second and third lectures focus on symplectic Lefschetz pencils: existence (following Donaldson), monodromy, and applications to symplectic topology, in particular the...

Source: http://arxiv.org/abs/math/0304113v1

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54

Apr 23, 2002
04/02

Apr 23, 2002
by
Denis Auroux

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We describe a simplification of Donaldson's arguments for the construction of symplectic hypersurfaces or Lefschetz pencils that makes it possible to avoid any reference to Yomdin's work on the complexity of real algebraic sets.

Source: http://arxiv.org/abs/math/0204286v1

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Apr 23, 2002
04/02

Apr 23, 2002
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

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44

Oct 5, 2000
10/00

Oct 5, 2000
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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Jul 21, 2000
07/00

Jul 21, 2000
by
Denis Auroux

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After reviewing recent results on symplectic Lefschetz pencils and symplectic branched covers of CP^2, we describe a new construction of maps from symplectic manifolds of any dimension to CP^2 and the associated monodromy invariants. We also show that a dimensional induction process makes it possible to describe any compact symplectic manifold by a series of words in braid groups and a word in a symmetric group.

Source: http://arxiv.org/abs/math/0007130v1

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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

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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

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298

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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

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

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

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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

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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

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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

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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

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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

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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

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1. Branched Covers...

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

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

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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

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0.1. Coherent Sheaves on a Complex Manifold (contd.) - Let X be a com_plex manifold, OX the sheaf of holomorphic functions on X.

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

Source: http://www.flooved.com/reader/1968

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

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

Source: http://www.flooved.com/reader/1965

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

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

Source: http://www.flooved.com/reader/1966

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1. Deformations of Complex Structures An (almost) complex structure (X, J) splits the complexi�ed tangent and (wedge powers of) cotangent bundles as...

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

Source: http://www.flooved.com/reader/1972

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1. Integrability of Almost-Complex Structures...

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

Source: http://www.flooved.com/reader/2107

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

Source: http://www.flooved.com/reader/2126

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The Hodge decomposition stated last time places strong constraints on H* of K�hler manifolds, e.g. dim H^k is even for k odd because C conjugation gives isomorphisms ... (note that this is false for symplectic manifolds in general).

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

Source: http://www.flooved.com/reader/2111

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

Source: http://www.flooved.com/reader/1822

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Polar Coordinates Polar coordinates are a di_erent way of describing points in the plane. The polar coordinates (r, _) are related to the usual rectangular coordinates (x, y) by by x = r cos _, y = r sin _

Topics: Maths, Analysis and Calculus, Calculus, Mathematics

Source: http://www.flooved.com/reader/1805

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In this section, we consider the most common case � �nding a line which goes approximately through a set of data points.

Topics: Maths, Analysis and Calculus, Calculus, Mathematics

Source: http://www.flooved.com/reader/1806

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Topics: Maths, Analysis and Calculus, Calculus, Mathematics

Source: http://www.flooved.com/reader/1827