68
68

Nov 14, 2013
11/13

by
Denis Auroux

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eye 68

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1. Pseudoholomorphic Curves ...

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

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

80
80

Nov 14, 2013
11/13

by
Denis Auroux

texts

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eye 80

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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

86
86

Nov 14, 2013
11/13

by
J. Bernstein

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eye 86

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

Source: http://www.flooved.com/reader/1954

88
88

Nov 14, 2013
11/13

by
Denis Auroux

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eye 88

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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

58
58

Nov 14, 2013
11/13

by
Denis Auroux

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eye 58

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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

87
87

Nov 14, 2013
11/13

by
Denis Auroux

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eye 87

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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

107
107

Nov 14, 2013
11/13

by
Emma Carberry

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eye 107

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In order to study regular surfaces globally, we need some global hypothesis to ensure that the surface cannot be extended further as a regular surface. Compactness serves this purpose, but it would be useful to have a weaker hypothesis than compctness which could still have the same e_ect.

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

Source: http://www.flooved.com/reader/1905

84
84

Nov 14, 2013
11/13

by
N. Rosenblyum

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eye 84

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

Source: http://www.flooved.com/reader/1958

99
99

Nov 14, 2013
11/13

by
K. Venkatram

texts

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eye 99

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

Source: http://www.flooved.com/reader/1949

74
74

Nov 14, 2013
11/13

by
K. Venkatram

texts

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eye 74

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

Source: http://www.flooved.com/reader/1953

71
71

Nov 14, 2013
11/13

by
Denis Auroux

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eye 71

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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

89
89

Nov 14, 2013
11/13

by
Denis Auroux

texts

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eye 89

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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

99
99

Nov 14, 2013
11/13

by
Denis Auroux

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eye 99

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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

82
82

Nov 14, 2013
11/13

by
Denis Auroux

texts

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eye 82

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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

320
320

Nov 14, 2013
11/13

by
James S. Milne

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eye 320

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This is an introduction to the arithmetic theory of modular functions and modular forms, with a greater emphasis on the geometry than most accounts.

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

Source: http://www.flooved.com/reader/3434

88
88

Nov 14, 2013
11/13

by
Denis Auroux

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eye 88

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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

78
78

Nov 14, 2013
11/13

by
Denis Auroux

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eye 78

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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

79
79

Nov 14, 2013
11/13

by
Denis Auroux

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eye 79

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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
Y. Lekili

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eye 89

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

Source: http://www.flooved.com/reader/1957

84
84

Nov 14, 2013
11/13

by
Emma Carberry

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eye 84

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Two De�nitions of Completeness We�ve already seen do Carmo�s de�nition of a complete surface � one where every partial geodesic is extendable to a geodesic de�ned on all of R. Osserman uses a di_erent de�nition of complete, which we will show to be equivalent (this is also exercise 7 on page 336 of do Carmo).

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

Source: http://www.flooved.com/reader/1907

84
84

Nov 14, 2013
11/13

by
K. Venkatram

texts

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eye 84

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

Source: http://www.flooved.com/reader/1945

76
76

Nov 14, 2013
11/13

by
J. Bernstein

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eye 76

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

Source: http://www.flooved.com/reader/1959

95
95

Nov 14, 2013
11/13

by
Denis Auroux

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eye 95

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1. Mirror Symmetry Conjecture - Last time, we said that if we have a large complex structure limit (LCSL) degeneration, then we have a special basis...

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

Source: http://www.flooved.com/reader/1984

88
88

Nov 14, 2013
11/13

by
Denis Auroux

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eye 88

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1. Homological Mirror Symmetry (cntd.) - Last time, we studied homological mirror symmetry on T^2 (with area form _) on the one hand ...

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

Source: http://www.flooved.com/reader/1973

77
77

Nov 14, 2013
11/13

by
Denis Auroux

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eye 77

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1. Coherent Sheaves on a Complex Manifold (contd.) - We now recall the following de�nitions from category theory. De�nition 1. An additive category is one in which Hom(A, B) are abelian groups, composition is distributive, and...

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

Source: http://www.flooved.com/reader/1969

101
101

Nov 14, 2013
11/13

by
K. Venkatram

texts

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eye 101

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

Source: http://www.flooved.com/reader/1950

77
77

Nov 14, 2013
11/13

by
Denis Auroux

texts

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eye 77

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1. Gromov-Witten Invariants Recall that if (X, _) is a symplectic manifold

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

Source: http://www.flooved.com/reader/1981

83
83

Nov 14, 2013
11/13

by
Denis Auroux

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eye 83

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1. Degenerations and Monodromy (contd.) - Last time, we considered families..

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

Source: http://www.flooved.com/reader/1983

98
98

Nov 14, 2013
11/13

by
Denis Auroux

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eye 98

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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

96
96

Nov 14, 2013
11/13

by
J. Pascale?

texts

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eye 96

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

Source: http://www.flooved.com/reader/1955

102
102

Nov 14, 2013
11/13

by
Denis Auroux

texts

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eye 102

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1. Homological Mirror Symmetry - Conjecture 1. X, X_ are mirror Calabi-Yau varieties

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

Source: http://www.flooved.com/reader/1971

120
120

Nov 14, 2013
11/13

by
Emma Carberry

texts

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eye 120

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Weierstrass-Enneper Representations of Minimal Surfaces - Let M be a minimal surface de�ned by an isothermal parameterization...

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

Source: http://www.flooved.com/reader/1906

97
97

Nov 14, 2013
11/13

by
Denis Auroux

texts

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eye 97

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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/1964

73
73

Nov 14, 2013
11/13

by
Denis Auroux

texts

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eye 73

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Recall that given an (almost) Calabi-Yau manifold (X, J, _, �), we de�ned M to be the set of pairs (L, _), L _ X a special Lagrangian torus, _ a �at U(1) connection on C _ L modulo gauge equivalence....

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

Source: http://www.flooved.com/reader/1976

79
79

Nov 14, 2013
11/13

by
Denis Auroux

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eye 79

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1. SYZ Conjecture (cntd.) - Recall: Proposition 1. First order deformations of special Lagrangian L in a strict (resp. almost) Calabi-Yau manifold are given by...where

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

Source: http://www.flooved.com/reader/1975

153
153

Nov 14, 2013
11/13

by
Nizameddin H. Ordulu

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eye 153

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Introduction - The �Plateau�s Problem� is the problem of �nding a surface with minimal area among all surfaces which have the same prescribed boundary. Let x be a solution to Plateau�s problem t for a closed curve...

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

Source: http://www.flooved.com/reader/1910

83
83

Nov 14, 2013
11/13

by
J. Bernstein;A. Rita

texts

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eye 83

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

Source: http://www.flooved.com/reader/1951

81
81

Nov 14, 2013
11/13

by
K. Venkatram

texts

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eye 81

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

Source: http://www.flooved.com/reader/1946

80
80

Nov 14, 2013
11/13

by
K. Venkatram

texts

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eye 80

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Last time, we talked about the geometry of a connected lie group G. Speci�cally, for any a in the corresponding Lie algebra g, one can de�ne...

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

Source: http://www.flooved.com/reader/1960

64
64

Nov 14, 2013
11/13

by
Denis Auroux

texts

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eye 64

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1. Derived Fukaya Category - Last time: derived categories for abelian categories (e.g. DbCoh(X)). This time: the derived Fukaya category.

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

Source: http://www.flooved.com/reader/1970

75
75

Nov 14, 2013
11/13

by
K. Venkatram

texts

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eye 75

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

Source: http://www.flooved.com/reader/1947

102
102

Nov 14, 2013
11/13

by
C. Kottke

texts

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eye 102

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

Source: http://www.flooved.com/reader/1956

99
99

Nov 14, 2013
11/13

by
Juha Valkama

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eye 99

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Let V be a Euclidean space, i.e. a real �nite dimensional linear space with a symmetric positive de�nite inner product _,_.

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

Source: http://www.flooved.com/reader/1666

108
108

Nov 14, 2013
11/13

by
Kiran S. Kedlaya

texts

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eye 108

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

Source: http://www.flooved.com/reader/2066

88
88

Nov 14, 2013
11/13

by
Abhinav Kumar

texts

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eye 88

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

Source: http://www.flooved.com/reader/2206

97
97

Nov 14, 2013
11/13

by
Abhinav Kumar

texts

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eye 97

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

Source: http://www.flooved.com/reader/2204

121
121

Nov 14, 2013
11/13

by
Kiran S. Kedlaya

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eye 121

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

Source: http://www.flooved.com/reader/2056

130
130

Nov 14, 2013
11/13

by
Abhinav Kumar

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eye 130

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

Source: http://www.flooved.com/reader/2196

131
131

Nov 14, 2013
11/13

by
Kiran S. Kedlaya

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eye 131

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Hartshorne only treats �atness after cohomology (so see III.9) and doesn�t talk about descent at all. The EGA reference for �atness is EGA IV, part 2, �2. I�m not sure if descent is discussed at all in EGA, so I gave references to SGA 1 instead.

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

Source: http://www.flooved.com/reader/2060

102
102

Nov 14, 2013
11/13

by
Abhinav Kumar

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eye 102

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

Source: http://www.flooved.com/reader/2207