This course is put on by the Taught Course Centre, and is available via MS Teams to graduate students at the Universities of Bath, Bristol, Oxford, Warwick and Imperial College London. I'm sorry, but random people from elsewhere can't come. If you are eligible, to sign up you should email gradstud@maths.ox.ac.uk with your MS Teams email address.

Lectures 1-8 will introduce elementary symplectic geometry: symplectic manifold, Lagrangian submanifolds, examples, Darboux' Theorem, symmetries and the symplectic quotient, contact manifolds and Legendrian submanifolds.Lectures 9-16 will focus on symplectic geometry and *J*-holomorphic
curves. We give overviews of Gromov-Witten invariants, Lagrangian Floer
homology, and the Arnold Conjecture, and a brief sketch of Gromov-Witten
invariants in Mirror Symmetry.

R. Berndt, *An Introduction to Symplectic Geometry*, Graduate Studies
in Mathematics 26, A.M.S., 2001.

A. Cannas da Silva, *Lectures on Symplectic Geometry*, Lecture Notes in
Mathematics 1764, Springer, 2001.

D.A. Cox and S. Katz, *Mirror Symmetry and Algebraic Geometry*,
Mathematical Surveys and Monographs 68, A.M.S., 1999.

M. Kontsevich and Y. Manin, *Gromov-Witten classes, quantum cohomology,
and enumerative geometry*, Comm. Math. Phys. 164 (1994), 525-562, reprinted
in B. Greene and S.-T. Yau, editors, *Mirror Symmetry II*, A.M.S., 1997,
pages 607-653. hep-th/9402147.

D. McDuff and D. Salamon, *J-holomorphic curves and Quantum
Cohomology*, University Lecture Series 6, A.M.S., 1994.

D. McDuff and D. Salamon, *Introduction to Symplectic Topology*, second
edition, Oxford University Press, 1998.

A. Weinstein, *Lectures on Symplectic manifolds*, CBMS Regional
Conference Series 29, A.M.S., 1977.

I recommend Cannas da Silva for lectures 1-8, and McDuff and Salamon 1994 and Cox and Katz for lectures 9-16.

**Lecture 1:** Symplectic Linear Algebra. Nondegenerate 2-forms on R^{2n}. Different
kinds of vector subspaces: symplectic, isotropic, coisotropic, and Lagrangian.
The symplectic group, the Lagrangian Grassmannian. Relation with complex
geometry: Hermitian metrics on C^{n}, Kähler forms
and complex structures, the unitary group.

**Lecture 2:** Symplectic manifolds. Different kinds of submanifolds:
symplectic, isotropic, coisotropic, and Lagrangian. Examples of symplectic
submanifolds: R^{2n}, cotangent
bundles, Kähler manifolds.

**Lecture 3:** Symplectomorphisms and Hamiltonian flow. The automorphism
group of a symplectic manifold is infinite-dimensional. The Poisson bracket.
Sketch of symplectic geometry in classical and quantum mechanics. Darboux'
Theorem. Discussion: comparison of symplectic geometry, and complex geometry,
and Riemannian geometry. Local triviality of symplectic and complex structures.
Local and global geometry.

**Lecture 4:** Moser's Theorem and deformations of symplectic manifolds.
Proof of Darboux' Theorem. The Lagrangian Neighbourhood Theorem.

**Lecture 5:** Symmetries of symplectic manifolds. Moment maps. Moment
maps are constant on invariant Lagrangians. Homogeneous symplectic manifolds
and coadjoint orbits.

**Lecture 6:** The symplectic quotient and the Kähler quotient. Moment
maps of torus actions. Toric symplectic manifolds.

**Lecture 7:** Contact geometry, an odd-dimensional analogue of
symplectic geometry. Legendrian submanifolds. Examples in R^{2n+1}. Relations
between symplectic and contact geometry: symplectic cones on contact manifolds,
contact manifolds as boundaries of contact manifolds, contact structures on
R- or U(1)-bundles over
symplectic manifolds.

**Lecture 8:** More on contact geometry. Reeb vector fields.
Contactomorphisms, contact vector fields. The automorphism group of a contact
manifold is infinite-dimensional. The contact analogue of Darboux' Theorem.
Gray's Stability Theorem. Contact structures on 1-jet bundles. The Legendrian
Neighbourhood Theorem.

**Lecture 9:** Counting invariants. Introduction to the general framework
of *invariants* counting moduli spaces of solutions of nonlinear elliptic
p.d.e.s, yielding answers independent of large chunks of the background
geometric data. Examples: Gromov-Witten invariants, Donaldson invariants,
Seiberg-Witten invariants, Donaldson-Thomas invariants. Using similar ideas one
can study *homology theories* which are independent of parts of the
background data, up to isomorphism. Examples: contact homology, Lagrangian
Floer homology, instanton Floer homology, Seiberg-Witten Floer homology.

**Lecture 10:** Moduli spaces of *J*-holomorphic curves. Marked
points. Moduli spaces of nonsingular curves with marked points, their
compactification by Deligne--Mumford stable curves. The moduli spaces
*M*_{g,m}, as compact complex orbifolds. Moduli
spaces *M*_{g,m}(*M,J,β*) of stable *J*-holomorphic maps
into a compact symplectic manifold (*M*,ω).

**Lecture 11:** Gromov-Witten invariants. Kuranishi spaces, virtual
classes. Gromov-Witten invariants and classes. Algebraic versus symplectic
Gromov-Witten invariants. Axioms for Gromov-Witten invariants.

**Lecture 12:** Quantum cohomology. Three point Gromov-Witten equations,
and relations on them. Small quantum cohomology, with its associative product.
The Gromov-Witten potential and the WDVV equation. Big quantum cohomology.

**Lecture 13:** (Closed) String Theory and Mirror Symmetry. Outline of
String Theory and Mirror Symmetry. More detail on Mirror Symmetry: B-fields,
SCFT's and their moduli space. Large radius and complex structure limits.
Closed String Mirror Symmetry, predicting genus 0 Gromov--Witten invariants of
*Y* in terms of variation of Hodge structure of *Ŷ*. Relation to quantum cohomology.

**Lecture 14:** Open Strings and Homological Mirror Symmetry. Open String
Theory, boundary conditions for strings. A-branes (Lagrangians) and B-branes
(coherent sheaves). Kontsevich's Homological Mirror Symmetry proposal. Physical
branes and stability conditions.

**Lecture 15:** Lagrangian Floer cohomology, an introduction. Morse
homology for finite-dimensional manifolds. The space of paths between two
Lagrangians *L*_{0},*L*_{1}, as an
infinite-dimensional manifold, with a 'Morse function' defined using symplectic
area. Lagrangian Floer homology
*HF*^{*}(*L*_{0},*L*_{1}) as the Morse
homology of the path space using this 'Morse function'. Moduli spaces of
*J*-holomorphic discs with boundary in *L*_{0} ∪
*L*_{1}. A simple version of
*HF*^{*}(*L*_{0},*L*_{1}) .

**Lecture 16**: Lagrangian Floer cohomology and applications. Defining
Lagrangian Floer cohomology, assumptions ensuring it is well-defined.
Independence of *J*. Independence under Hamiltonian equivalence.
Nonexistence of compact, embedded, exact Lagrangians in R^{2n}. Arnold's
Conjecture for cotangent bundles. Arnold's Conjecture for Hamiltonian flows.
Products on Lagrangian Floer homology. Bounding cochains and the Fukaya
category.