Abstact：These are short summer courses for both graduate and undergraduate students. We will offer lectures on five topics, each of which contain 3 lectures. More details can be found in the following.
Lecture 1: Basic deformation theory and intrinsic normal cone
Speaker: Huai-Liang Chang
Abstract: We introduce basic deformation theory with an emphasis on obstruction theories. From them we discuss intrinsic normal cones. We shall then discuss cotangent complexes, its relation with deformation theory, and introduce the concept of perfect obstruction theory.
The audience should have read Hartshorne AG textbook Chap 1-3 or equivalent context. Knowledge with derived category will help a lot.
Lecture 2: The theory of virtual fundamental classes and Gromov Witten theory
Speaker: Huai-Liang Chang
Abstract: This is a introduction to the virtual cycle theory of Li-Tian and Behrend-Fantechi. Using perfect obstruction theory, we cover the construction of virtual fundamental classes in both Li-Tian and Behrend-Fantechi setups.
As a main example we focus on Gromov Witten theory. We shall explain its origins including Witten's path integral, Kuranishi model, and their comparison with algebraic geometry constructions.
If time permit we shall give more examples and briefly introduce S or T duality.
Lecture 3: Landau Ginzburg theory in algebraic geometry
Speaker: Huai-Liang Chang
Abstract: Enumerative curves "in" Landau Ginzburg space is now a on-going research topic in both symplectic and algebraic geometry. We shall introduce algebraic geometry part including r spin curve, cosection localization, Witten top Chern class (FJRW invariant) and P field. We shall discuss all their relations with symplectic setup as Witten equations. Relations with path integral and Gromov Witten theory will be also discussed, if time permitted.
Lecture 4 : Symmetric obstruction theory, Behrend function and applications to Vafa-Witten invariants
Speaker: Yunfeng Jiang
Abstract: Following Hua-Liang's talk on virtual fundamental classes, I will introduce the symmetric obstruction theory, the Behrend function for any scheme or Deligne-Mumford (DM) stack, and Behrend's theorem equating the virtual count of a symmetric obstruction theory on a scheme or DM stack to its Euler characteristic weighted by the Behrend function. This is the fundamental tool to define Donaldson-Thomas invariants. Then I will talk about several ways to define virtual signed Euler characteristics for an existing perfect obstruction theory. The virtual signed Euler characteristic has applications to define Vafa-Witten invariants recently developed by Tanaka-Thomas. If time permits we will talk about how to define Vafa-Witten invariants and a possible way to relate it to mirror symmetry for the Higgs bundles on algebraic curves.
Lecture 5: Torus Localization
Speaker: Weiping Li
Abstract: I will start with equivariant cohomology, Atiyah-Bott localization theorem, Graber-Pandharipande virtual torus localization, and some applications to Gromov-Witten invariants.
SCHEDULE
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Time\Date
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6.18
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6.19
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6.20
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6.21
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6.22
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10:00-10:45
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——
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Basic deformation theory and intrinsic normal cone (2)
Huai-Liang Chang
Room 1304
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The theory of virtual fundamental classes and Gromov-Witten theory (1)
Huai-Liang Chang
Room 1304
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The theory of virtual fundamental classes and Gromov-Witten theory (2)
Huai-Liang Chang
Room 1303
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The theory of virtual fundamental classes and Gromov-Witten theory (3)
Huai-Liang Chang
Room 1303
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10:55-11:40
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——
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Basic deformation theory and intrinsic normal cone (2)
Huai-Liang Chang
Room 1304
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The theory of virtual fundamental classes and Gromov-Witten theory (1)
Huai-Liang Chang
Room 1304
|
Symmetric obstruction theory, Behrend function and applications to Vafa-Witten invariants (1)
Yunfeng Jiang
Room 1303
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The theory of virtual fundamental classes and Gromov-Witten theory (3)
Huai-Liang Chang
Room 1303
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15:00-15:45
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Basic deformation theory and intrinsic normal cone (1)
Huai-Liang Chang
16:10-16:55
Room 1303
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Basic deformation theory and intrinsic normal cone (3)
Huai-Liang Chang
Room 1303
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The theory of virtual fundamental classes and Gromov-Witten theory (2)
Huai-Liang Chang
Room 1303
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Symmetric obstruction theory, Behrend function and applications to Vafa-Witten invariants (2)
Yunfeng Jiang
Room 1303
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Symmetric obstruction theory, Behrend function and applications to Vafa-Witten invariants (3)
Yunfeng Jiang
Room 1303
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|
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15:55-16:40
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Basic deformation theory and intrinsic normal cone (1)
Huai-Liang Chang
17:05-17:50
Room 1303
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Basic deformation theory and intrinsic normal cone (3)
Huai-Liang Chang
Room 1303
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Symmetric obstruction theory, Behrend function and applications to Vafa-Witten invariants (1)
Yunfeng Jiang
Room 1303
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Symmetric obstruction theory, Behrend function and applications to Vafa-Witten invariants (2)
Yunfeng Jiang
Room 1303
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Symmetric obstruction theory, Behrend function and applications to Vafa-Witten invariants (3)
Yunfeng Jiang
Room 1303
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Time\Date
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6.24
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6.25
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6.26
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10:00-10:45
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Landau-Ginzburg theory in algebraic geometry (1)
Huai-Liang Chang
Room 1303
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Landau-Ginzburg theory in algebraic geometry (3)
Huai-Liang Chang
Room 1303
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Torus Localization (2)
Weiping Li
Room 1304
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——
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|
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10:55-11:40
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Landau-Ginzburg theory in algebraic geometry (1)
Huai-Liang Chang
Room 1303
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Landau-Ginzburg theory in algebraic geometry (3)
Huai-Liang Chang
Room 1303
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Torus Localization (2)
Weiping Li
Room 1304
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——
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|
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15:00-15:45
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Landau-Ginzburg theory in algebraic geometry (2)
Huai-Liang Chang
Room 1303
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Torus Localization (1)
Weiping Li
Room 1303
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Torus Localization (3)
Weiping Li
Room 1303
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|
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15:55-16:40
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Landau-Ginzburg theory in algebraic geometry (2)
Huai-Liang Chang
Room 1303
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Torus Localization (1)
Weiping Li
Room 1303
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Torus Localization (3)
Weiping Li
Room 1303
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