Graduate lecture course, 16 lectures, Spring Term 2021.

Professor Joyce

Symplectic Geometry

Tuesdays 2pm-4pm, starting in week 1 (19th January) and finishing in week 8 (9th March), online via MS Teams.

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.

Bibliography:

M. Audin and J. Lafontaine, editors, Holomorphic curves in symplectic geometry, Progress in Mathematics 117, Birkhäuser, 1994.

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.

Description of lectures:

Lecture 1: Symplectic Linear Algebra. Nondegenerate 2-forms on R2n. Different kinds of vector subspaces: symplectic, isotropic, coisotropic, and Lagrangian. The symplectic group, the Lagrangian Grassmannian. Relation with complex geometry: Hermitian metrics on Cn, 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: R2n, 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 R2n+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 Mg,m, as compact complex orbifolds. Moduli spaces Mg,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 L0,L1, as an infinite-dimensional manifold, with a 'Morse function' defined using symplectic area. Lagrangian Floer homology HF*(L0,L1) as the Morse homology of the path space using this 'Morse function'. Moduli spaces of J-holomorphic discs with boundary in L0L1. A simple version of HF*(L0,L1) .

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 R2n. Arnold's Conjecture for cotangent bundles. Arnold's Conjecture for Hamiltonian flows. Products on Lagrangian Floer homology. Bounding cochains and the Fukaya category.

PDF files to download:

Synopses

Slides of lectures 1 and 2

Slides of lectures 3 and 4

Slides of lectures 5 and 6

Slides of lectures 7 and 8

Slides of lectures 9 and 10

Slides of lectures 11 and 12

Slides of lectures 13 and 14

Slides of lectures 15 and 16

Miniprojects, for assessment over Christmas vacation