Symplectic Geometry

Prerequisite path Preparation & skip guidance · Differential & Riemannian Geometry · Algebraic Topology & Homotopy Theory

Studying Symplectic Geometry

A symplectic form pairs tangent directions in a way that leads naturally to Hamiltonian motion and Lagrangian submanifolds. Cannas da Silva's shorter IMPA course provides a route through local normal forms, cotangent bundles and Hamiltonian fields. Calegari's notes emphasize a topological continuation toward holomorphic curves and an outline of Floer homology. The latter needs geometric and analytic preparation beyond understanding Hamilton's equations in coordinates.

What to work on

  • Test closedness and nondegeneracy of a two-form.
  • Use Darboux's theorem with its local scope.
  • Compute Hamiltonian vector fields and identify their conserved energy.
  • Check when a graph in a cotangent bundle is Lagrangian.

A starting text

Introduction to Symplectic and Hamiltonian Geometry — Ana Cannas da Silva. Read 1.1-1.4 for linear algebra and symplectic manifolds, 1.7-1.9 for Moser's method and Darboux, 2.1-2.4 for cotangent bundles and Lagrangians, and 4.1-4.3 for Hamiltonian fields, Hamilton equations and brackets. Sections 4.6-4.7 and 5.1-5.3 continue to moment maps and reduction.

Choosing what to read

Compute the Hamiltonian vector field of a quadratic Hamiltonian using the sign convention of the chosen text. Check that the flow preserves the symplectic form, then test when the graph of a one-form is Lagrangian. Read Moser's argument by tracking the time-dependent vector field and the equation it must solve. Before following holomorphic curves or Floer theory, list the required compactness and regularity inputs; a course outline signals these constructions without supplying every foundation.

Introduction to Symplectic and Hamiltonian Geometry

Ana Cannas da Silva

Learn Darboux, cotangent-bundle and Hamiltonian calculations before advanced invariants.

How it reads
Short course from local symplectic geometry to Hamiltonian reduction.
What it expects
Smooth manifolds, differential forms and the relevant cohomology language.
What to keep in mind
This shorter IMPA work is distinct from the author's longer Lectures; neither selected route establishes Floer foundations.
Practice
Numbered public exercises verified within the notes; public solutions not verified. Some exercises include guiding explanations. Work through the exercises embedded in lectures 1-2 and 4; Exercises 17-18 in section 4.3 concern contraction identities and the Poisson-bracket Leibniz rule.

Complete public PDF.

Edition, reading range and source

Read 1.1-1.4 for linear algebra and symplectic manifolds, 1.7-1.9 for Moser's method and Darboux, 2.1-2.4 for cotangent bundles and Lagrangians, and 4.1-4.3 for Hamiltonian fields, Hamilton equations and brackets. Sections 4.6-4.7 and 5.1-5.3 continue to moment maps and reduction.

Source · Checked 2026-10-16

Notes on Symplectic Topology

Danny Calegari

See which constructions connect introductory symplectic geometry to its topological applications.

How it reads
Topological route toward holomorphic curves and an outline of Floer homology.
What it expects
Smooth manifolds, differential forms and algebraic topology.
What to keep in mind
An outline of Floer homology needs a separate analytic and foundational treatment before technical use.
Practice
Public worked solutions were not verified in the existing record; compare a Hamiltonian example across the two texts as an editorial task.

Complete author PDF linked from the University of Chicago course page.

Edition, reading range and source

Follow the symplectic-linear-algebra and Hamiltonian portions before the holomorphic-curve and Floer outline; exact section numbers were not rechecked.

Source · Checked 2026-10-13

Practice

Symplectic and Hamiltonian exercises. Work through the exercises embedded in lectures 1-2 and 4; Exercises 17-18 in section 4.3 concern contraction identities and the Poisson-bracket Leibniz rule.. Numbered public exercises verified within the notes; public solutions not verified. Some exercises include guiding explanations.

Try these before moving on

Try a proof or calculation before opening the hint. These solutions address the checks below, not the exercises in the linked textbooks.

  1. On R squared with omega=dq wedge dp, use the convention contraction of X_H with omega equals dH to compute X_H for H=(q squared+p squared)/2. Solve its flow and verify that H is conserved.

    A hint

    For X=a partial_q+b partial_p, contraction with dq wedge dp equals a dp-b dq. Keep this sign convention throughout.

    Work through the solution

    Write XH=a∂q+b∂pX_H=a\partial_q+b\partial_p. With ω=dq∧dp\omega=dq\wedge dp, ιXHω=a dp−b dq\iota_{X_H}\omega=a\,dp-b\,dq. Equating this to dH=q dq+p dpdH=q\,dq+p\,dp gives a=pa=p, b=−qb=-q, so XH=p∂q−q∂pX_H=p\partial_q-q\partial_p. The equations are q′=pq'=p, p′=−qp'=-q.

    For initial data (q0,p0)(q_0,p_0), the global solution is q(t)=q0cos⁡t+p0sin⁡tq(t)=q_0\cos t+p_0\sin t and p(t)=−q0sin⁡t+p0cos⁡tp(t)=-q_0\sin t+p_0\cos t. Differentiation verifies both equations and evaluation at t=0t=0 verifies the initial data. The flow is clockwise rotation in the (q,p)(q,p) drawing plane. Along it, dH/dt=qq′+pp′=qp−pq=0dH/dt=qq'+pp'=qp-pq=0, so HH is constant. Choosing the opposite contraction convention would reverse the direction, which is why the convention is part of the question.

    If this was the sticking point:

Already comfortable with this? Skip the first lecture only if differential forms and symplectic linear algebra are usable in proofs. Skip the selection if both checks are routine and you can explain why Darboux gives local equivalence without implying global symplectomorphism.

Review the prerequisites · Compare the full subject collection

Where this can lead

Reading-source dates
  • Introduction to Symplectic and Hamiltonian Geometry · 2026-10-16

Reading choices revised 2026-10-17.

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Lectures on Symplectic Geometry

PDF

Ana Cannas da Silva · Lecture text · Introductory

A graduate introduction to symplectic forms, Hamiltonian geometry and reduction. Efficient for readers with manifold foundations; not an elementary introduction to geometry.

Preparation: Smooth manifolds; differential forms; de Rham cohomology.

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Additional reading

Search the complete catalogue.

Graduate reading

Introduction to Symplectic Topology

Dusa McDuff and Dietmar Salamon · Textbook · Graduate · English · Purchase / institutional access

Purchase or institutional access required; previews are not the complete book.

A broad introduction to symplectic manifolds, Hamiltonian geometry and global topology. Requires smooth geometry and some topology.

Preparation: Smooth manifolds, differential forms and algebraic topology.

Official page / access details
Edition and sources

Version: Third edition, 2017 · Checked 2026-10-11

An Introduction to Contact Topology

Hansjörg Geiges · Textbook · Graduate · English · Purchase / institutional access

Purchase or institutional access required; previews are not the complete book.

Develops contact manifolds, three-dimensional structures, fillings and surgery. A companion direction to symplectic geometry, not the same subject.

Preparation: Smooth manifolds, differential forms and algebraic topology.

Official page / access details
Edition and sources

Version: Print 2008; electronic publication 2009 · Checked 2026-10-11

Morse Theory and Floer Homology

Michèle Audin and Mihai Damian · Textbook · Graduate · English · Purchase / institutional access

Purchase or institutional access required; previews are not the complete book.

Connects Morse-theoretic foundations with Floer homology and Hamiltonian applications. Available in English and French.

Preparation: Smooth manifolds, differential forms and algebraic topology.

Official page / access details
Edition and sources

Version: English edition, 2014 · Checked 2026-10-11

Hamiltonian Group Actions and Equivariant Cohomology

Shubham Dwivedi, Jonathan Herman, Lisa C. Jeffrey and Theo van den Hurk · Textbook · Graduate · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

Connects moment maps, reduction and equivariant cohomology. A branch into symmetry after the basic symplectic form and manifold constructions.

Preparation: Differential forms; symplectic manifolds; algebraic topology.

Official page / access details
Edition and sources

Version: 2019 · Checked 2026-10-11

Introduction to Symplectic and Hamiltonian Geometry

Ana Cannas da Silva · Long lecture notes · Graduate · English · PDF

Complete author-hosted course notes; reading access is not permission to redistribute.

A shorter course on symplectic structures and Hamiltonian geometry, distinct from the longer Lectures on Symplectic Geometry. Useful for orientation and selective review.

Preparation: Smooth manifolds and differential forms.

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Edition and sources

Version: IMPA short course 2002; revised November 2007 · Checked 2026-10-11

Introduction to Symplectic Geometry

Jean-Louis Koszul; translated by Yiming Zou · Textbook · Graduate · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

A lecture-based development including symplectic group actions. Compare its conventions with modern differential-form treatments.

Preparation: Differential geometry; Lie groups.

Official page / access details
Edition and sources

Version: English edition, 2019 · Checked 2026-10-11

Notes on Symplectic Topology

Danny Calegari · Course notes · Graduate · English · PDF

The author’s university page links this freely readable PDF.

A topological route from symplectic linear algebra and Hamiltonian flows to holomorphic curves and the outline of Floer homology.

Preparation: Smooth manifolds; differential forms; algebraic topology.

Use: Prepare for analytic and Floer-theoretic treatments.

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Edition and sources

Version: March 5, 2025 revision; Winter 2022 course · Checked 2026-10-13

Introductory Lectures on Contact Geometry

John B. Etnyre · Lecture notes · Graduate · English · PDF

The complete author-deposited PDF opens in the arXiv archive.

Introduces contact structures and their local models, characteristic foliations, tightness and Legendrian knots.

Preparation: Smooth manifolds; differential forms; elementary three-manifold topology.

Use: Move from symplectic basics into three-dimensional contact topology.

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Edition and sources

Version: arXiv v2, November 11, 2002; published 2003 · Checked 2026-10-13

A Beginner's Introduction to Fukaya Categories

Denis Auroux · Lecture notes · Graduate · English · PDF

The complete PDF opens on the author’s or university’s public site.

Reviews Lagrangian Floer cohomology before introducing Fukaya categories, generators and wrapped variants. The exposition intentionally omits some technical foundations.

Preparation: Symplectic geometry; exposure to holomorphic curves; homological algebra helpful.

Use: Bridge a first Floer course to categorical formulations and mirror symmetry.

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Edition and sources

Version: June 2011 Nantes lectures; author PDF revision date not stated · Checked 2026-10-13

Lectures on Holomorphic Curves in Symplectic and Contact Geometry

Chris Wendl · Book draft / lecture notes · Graduate · English · PDF

The complete author-deposited PDF opens in the arXiv archive.

Develops analytic foundations for holomorphic curves, including elliptic regularity, Fredholm theory, moduli spaces and compactness.

Preparation: Smooth manifolds; differential forms; complex analysis; functional analysis.

Use: Build the analytic background needed for contact and symplectic research.

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Edition and sources

Version: Work-in-progress version 3.2, May 28, 2014; arXiv v2 · Checked 2026-10-13

Lecture Notes on Equivariant Cohomology

Matvei Libine · Long lecture notes · Graduate · English · PDF

Public full text; checked against the title and contents.

Introduces equivariant cohomology, its geometric models and localization methods in a graduate-course format.

Preparation: Differential forms; Lie groups; cohomology.

Use: Develop the equivariant models before using the localization formulas.

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Found via — References: Lecture Notes on Equivariant Cohomology

Edition and sources

Version: arXiv 0709.3615; inspected electronic version · Checked 2026-11-03

Research reading

Kuranishi Spaces and Symplectic Geometry, I–II

Dominic Joyce · Book draft · Research · English · PDF

Develops Kuranishi foundations for symplectic applications. Both preliminary volumes are available; neither is represented as a final published book.

Preparation: Symplectic geometry; moduli; derived differential geometry.

Use: Advanced foundation for virtual geometric constructions.

Open full PDFSource / termsCompanion volume PDF
Edition and sources

Version: 2017 preliminary two-volume manuscript; counted once · Checked 2026-10-11

J-Holomorphic Curves and Symplectic Topology

Dusa McDuff and Dietmar Salamon · Textbook · Research · English · Purchase / institutional access

Purchase or institutional access required; previews are not the complete book.

A research-oriented development of pseudoholomorphic curves and symplectic applications. The DOI identifies the complete book, not a free PDF.

Preparation: Smooth manifolds, differential forms and algebraic topology.

Official page / access details
Edition and sources

Version: Linked first edition, 2004; later editions are not counted separately · Checked 2026-10-11

Kuranishi Structures and Virtual Fundamental Chains

Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta and Kaoru Ono · Textbook · Research · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

A technical reference for virtual-chain constructions in holomorphic-curve theories. Different from Joyce’s Kuranishi-space approach; neither is a beginner’s first Floer course.

Preparation: Symplectic topology, moduli of holomorphic curves and Fredholm analysis.

Official page / access details
Edition and sources

Version: 2020 · Checked 2026-10-11

The Geometry of the Group of Symplectic Diffeomorphisms

Leonid Polterovich · Textbook · Research · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

Studies the geometric structure of symplectic and Hamiltonian diffeomorphism groups. A specialist branch beyond a first symplectic course.

Preparation: Symplectic geometry; Hamiltonian dynamics; Floer-theoretic background helpful.

Official page / access details
Edition and sources

Version: 2001 · Checked 2026-10-11

Fukaya Categories and Picard–Lefschetz Theory

Paul Seidel · Research monograph · Research · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

Develops Fukaya categories and the invariants of Lagrangian submanifolds through Lefschetz-fibration methods. This is a technical research monograph, not a first category-theory course.

Preparation: Symplectic topology; Floer theory; A-infinity algebra.

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Edition and sources

Version: 2008 · Checked 2026-10-11

Symplectic Invariants and Hamiltonian Dynamics

Helmut Hofer and Eduard Zehnder · Textbook · Research · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

Connects capacities, rigidity, periodic orbits and symplectic fixed-point questions. Read after symplectic forms and Hamiltonian flows.

Preparation: Symplectic geometry; variational methods; dynamical systems.

Official page / access details
Edition and sources

Version: 2011 reprint; original 1994 · Checked 2026-10-11

Lectures on Open Book Decompositions and Contact Structures

John B. Etnyre · Lecture notes · Research · English · PDF

The complete PDF opens on the author’s or university’s public site.

Explains Giroux’s correspondence between contact structures and open books and develops applications to fillings and low-dimensional topology.

Preparation: Contact geometry; Dehn surgery; mapping tori; algebraic topology.

Use: Read after introductory contact notes to connect contact structures with surface monodromy.

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Edition and sources

Version: 2004 Clay summer-school lectures; published 2006 · Checked 2026-10-13

Lecture Notes on Embedded Contact Homology

Michael Hutchings · Lecture notes · Research · English · PDF

The complete PDF opens on the author’s or university’s public site.

Develops embedded contact homology, its index and differential, computations and capacities for symplectic embedding problems.

Preparation: Contact geometry; holomorphic curves; Floer homology background.

Use: Study ECH constructions and their four-dimensional applications.

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Edition and sources

Version: Notes based on 2012 lectures; PDF revision date not stated · Checked 2026-10-13

A Biased View of Symplectic Cohomology

Paul Seidel · Research survey · Research · English · PDF

The complete author-deposited PDF opens in the arXiv archive.

Explores Liouville domains, symplectic cohomology, functoriality and algebraic structures, emphasizing classification problems and geometric examples.

Preparation: Hamiltonian Floer theory; contact-type boundaries; holomorphic curves.

Use: Explore symplectic cohomology after introductory Floer material.

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Edition and sources

Version: arXiv v6, February 15, 2010; manuscript originally dated April 13, 2007 · Checked 2026-10-13

Family Floer Cohomology and Mirror Symmetry

Mohammed Abouzaid · Research survey · Research · English · PDF

The complete author-deposited PDF opens in the arXiv archive.

Explains the family-Floer construction of mirrors from nonsingular Lagrangian torus fibrations and associated coherent sheaves.

Preparation: Lagrangian Floer theory; Fukaya categories; basics of mirror symmetry.

Use: Advance from categorical Floer introductions to geometric mirror constructions.

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Edition and sources

Version: arXiv v1, April 10, 2014 · Checked 2026-10-13