In the rigid case partial differential equations can be used to define symplectic invariants. As an example, holomorphic curves solutions of the Cauchy-Riemann equations are used to define the so-called Gromov-Witten invariants. Apart from classical mechanics, symplectic structures appear in a few other fields, for example in: Algebraic geometry: Every smooth algebraic subvariety of the complex projective space carries a canonical symplectic form.
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Gauge theory: The space of Yang-Mills instantons over a product of two real surfaces is closely related to the space of gauge equivalence classes of flat connections over a surface, which carries a canonical symplectic form. Differential topology: Certain invariants of smooth real 4-manifolds the Seiberg-Witten invariants are closely related to certain symplectic invariants the Gromov-Witten invariants. Symplectic geometry also has connections to the theory of dynamical systems and string theory.
Some highlights of this course will be the following: A normal form theorem for a submanifold of a symplectic manifold. A special case of this is Darboux's theorem, which states that locally, all symplectic manifolds look the same. Symplectic reduction for a Hamiltonian Lie group action. This corresponds to the reduction of the degrees of freedom of a mechanical system.
It gives rise to many examples of symplectic manifolds.
A construction of symplectic forms on open manifolds, which is based on Gromov's h-principle. The focus will be on a situation where technical difficulties are avoided by excluding holomorphic disc bubbling. Complementary lecture by Jingyu Zhao : Convergence of the differential in family Floer homology Following Mohammed Abouzaid's lectures, I will first recall the differential for the family Floer complex between a tautologically unobstructed Lagrangian and fibres of a Lagrangian torus fibration of the symplectic manifold.
Using the works of Fukaya and Groman-Solomon, we explain that the family Floer differential constructed above can be viewed as a convergent function on an affinoid domain of the rigid analytic mirror. Notes by Kevin Sackel: lecture 1 , lecture 2 , lecture 3 , complements 1 , complements 2.
The goal of this lecture series will be to illustrate homological mirror symmetry by focusing on some simple examples. Along the way we will encounter wrapped and partially wrapped Fukaya categories of these spaces and their mirrors. While the basic examples may seem elementary, they illustrate general features of homological mirror symmetry and provide test cases for work in progress on hypersurfaces in toric varieties. Derived symplectic linear algebra.
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Examples of symplectic and Lagrangian derived stacks. Topological field theories from shifted symplectic structures.
Homological algebra and sheaves 2. Micro-support of sheaves, involutivity, Morse theory 3. Complementary lectures by Pierre Schapira : 1. Complements and examples 2. Examples of microsupports, links with D-modules. Notes by the speaker. Quantization of exact Lagrangians in cotangent bundle I 3. Quantization of exact Lagrangians in cotangent bundle II.
We will associate a sheaf to a given compact exact Lagrangian submanifold of a cotangent bundle and see how to deduce that this Lagrangian has the homotopy type of the base. Complementary lecture by Nicolas Vichery : Examples of quantization of Lagrangian submanifolds, quantization of Hamiltonian isotopies.
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Notes by Sheridan , solutions to exercises by Maydanskiy Notes taken by Dingyu Yang: lecture 1 , lecture 2 , lecture 3 , complements 1 , complements 2. Notes by Amiel Peiffer-Smadja: lectures 1,2,3. Lagrangian Floer cohomology: We will introduce the basics of Lagrangian Floer cohomology, using the Arnold conjecture as motivation.
This will involve discussing the Novikov field, transversality, compactness and gluing. Product structures: We will introduce the Fukaya category, and give example computations. We will discuss obstructions, and how to deal with them.
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Triangulated structure: This talk will focus on the triangulated structure of the derived Fukaya category. Examples will include the relationship between Lagrangian surgery and cones, and Seidel's long exact sequence for a Dehn twist. Complementary lectures by Maksim Maydanskiy : In these supplementary sessions we will take up, in the form of exercise solutions, some additional material on Lagrangian Floer cohomology and Fukaya category as described the first two talks of N.
With audience participation, we will select for discussion a few topics from the following: interpretation of holomorphic strips as gradient flowlines of the action functional, explicit examples of Gromov convergence, gradings and computations of Floer cohomology in real dimension 2, relations of holomorphic discs to displacement energy, identification of Morse-Witten and Floer complexes in the cotangent bundle case , Stasheff associahedra and moduli of holomorphic discs.
Sheng-Fu Chiu : Sheaf-theoretic invariant and non-squeezability of contact balls. We apply microlocal category methods to a contact non-squeezing conjecture proposed by Eliashberg, Kim and Polterovich. In this talk I want to explain such phenomenon as well as the construction of those strutures. The Gromov-Witten invariants play a central role in the mirror symmetry conjecture which in turn gives predictions for these invariants.
Many such predictions for closed Gromov-Witten invariants have been established mathematically. Similar predictions exist for open and real Gromov-Witten invariants and I will discuss some of the difficulties related to understanding the open invariants and recent advances in the real case.
I will then if time permits also discus some consequences of this. Since its discovery in the 60s, Yang-Baxter equation YBE has been studied extensively as the master equation in integrable models in statistical mechanics and quantum field theory. In , Polishchuk discovered a connection between the solutions to Yang-Baxter equations classical, associative and quantum and the Massey products in a Calabi-Yau 1-category, and using this he was able to construct geometrically some of the trigonometric solutions of the YBE coming from simple vector bundles on cycles of projective lines.
We first prove a homological mirror symmetry statement, hence see these trigonometric solutions to YBE via the Fukaya category of punctured tori. Next, we consider Fukaya categories of higher genus square-tiled surfaces to give a geometric construction of all the trigonometric solutions to associative Yang-Baxter equation parametrized by the associative analogue of the Belavin-Drinfeld data. This is based on joint work with Polishchuk.
I will survey recent work devoted to singularities of Lagrangian skeleta, with a focus on applications to mirror symmetry. We consider the symplectic cohomology of the total space of a Lefschetz fibration. Under suitable assumptions, this can be equipped with a connection an operator of differentiation with respect to the Novikov variable. We will show that with respect to this operator, the Borman-Sheridan class satisfies a nonlinear first order differential equation a Riccati equation.
I will sketch an argument that the wrapped Fukaya category localizes to a cosheaf of categories on a Lagrangian skeleton, locally modeled on the cosheaf dual to the Kashiwara-Schapira sheaf. This is joint work with Sheel Ganatra and John Pardon. Complex Manifolds. Hamiltonian Vector Fields. Variational Principles. Legendre Transform. Hamiltonian Actions. The Marsden-Weinstein-Meyer Theorem. Moment Map in Gauge Theory. Existence and Uniqueness of Moment Maps.
Classification of Symplectic Toric Manifolds. Delzant Construction.
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Duistermaat-Heckman Theorems. Back Matter Pages About this book Introduction The goal of these notes is to provide a fast introduction to symplectic geometry for graduate students with some knowledge of differential geometry, de Rham theory and classical Lie groups.