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Topics in Floer theory

University of Texas at Austin, Spring semester, 2011

  • Course number: M392C. Unique identifier: 56540.

  • Tues., Thurs. 2-3:30 pm, RLM 12.166

  • Instructor: Tim Perutz (Assistant Professor)

  • Office hours: Tuesday, Wednesday 4-5 p.m., RLM 10.136.

  • Email: perutz AT math DOT utexas DOT edu



    Course description

    One way to compute the cohomology of a manifold is to take the cohomology of a certain cochain complex generated by critical points of a Morse function. The differential for this complex is a matrix whose entries enumerate solutions to an ODE (the gradient flow equation).

    Andreas Floer discovered a way to carry out a similar procedure with certain special Morse functions on infinite-dimensional manifolds, notably the space of loops on a symplectic manifold. In Floer's examples the differential is again a matrix, but its entries can now be interpreted as solutions to an elliptic PDE (a version of the Cauchy-Riemann equation in the loop-space case). The resulting cohomology theory is not the same as the ordinary cohomology of the infinite-dimensional manifold. Such "Floer cohomology theories" have proliferated, and so have their applications in symplectic topology and in low-dimensional topology.

    The focus of this course will be the Floer cohomology theory called symplectic cohomology, a form of the loop-space Floer cohomology on non-compact symplectic manifolds with constrained geometry at infinity (Liouville manifolds). This theory was designed to tackle problems in Hamiltonian dynamics. Recently, exciting new applications have emerged, highlighting the difference between differential topology and symplectic topology. A sample result: there are uncountably many distinct Liouville manifold structures on high-dimensional Euclidean spaces, distinguished by their symplectic cohomology.

    Beginning with Morse theory, I'll move towards loop-space Floer cohomology, symplectic cohomology and applications. The idea is to give a concrete introduction to the geometry and analysis involved in Floer theory.

    Audience

    The course is aimed not just at those interested in symplectic or contact manifolds, but at geometric topologists (Heegaard Floer theory is based on very similar principles), mathematical physicists (Floer cohomology is closely related to quantum field theory and to Hamiltonian dynamics), analysts with an interest in elliptic PDE, and algebraic geometers (symplectic cohomology gives an invariant of smooth, complex, affine varieties).

    Prerequisites

    Basic differential topology (as in prelim course M392D). Basic algebraic topology, especially homology (as in the prelim course M382C).

    Those who attended Dr. Gompf's course on symplectic topology will find that useful, but it is not a prerequisite: nuts and bolts of symplectic geometry will be outlined as needed.



    Texts. No required textbooks. Relevant books:
    • Dusa McDuff and Dietmar Salamon, Introduction to symplectic topology (OUP).
    • Dusa McDuff and Dietmar Salamon, J-holomorphic curves in symplectic topology (AMS).
    • John Milnor, Morse theory, Princeton University Press.


    Bibliography of online papers:

    Syllabus (rough draft)

    Date

    Notices

    Lecture plan

    Morse cohomology

    Introduction.
    Cellular homology. Handle decompositions.
    Self-indexing Morse functions compute cellular homology.
    Gradient flows and the Morse complex, mod 2 (without analytic details).
    Orientations. Maps from correspondences; invariance of Morse homology.
    Linear analysis: cylindrical ends and Fredholm operators.
    Non-linear analysis (gluing and compactness).

    Constructing symplectic cohomology
    Liouville domains and Liouville manifolds. Weinstein handle attachment.
    Hamiltonian systems, Reeb orbits, the symplectic action functional. Conjectures of Arnold and Weinstein.
    Preview of symplectic cohomology. Examples of Euclidean spaces, loop-spaces, manifolds with periodic Reeb flow on the boundary.
    Floer's equation as the gradient flow of the symplectic action. Non-degeneracy and convergence.
    Fredholm property and linear gluing. Index.
    Non-linear gluing and Gromov-Floer compactness. Role of the convex boundary.
    Definition and invariance of Hamiltonian Floer cohomology.
    Continuation maps and symplectic cohomology. Invariance.

    Calculations and applications.

    Symplectic cohomology of Euclidean spaces; spaces with periodic Reeb flow on the boundary.
    Map from ordinary cohomology and the Weinstein conjecture.
    Cotangent bundles and undecidable problems in symplectic topology.
    Contractible affine varieties and exotic Euclidean spaces.
    Exact Lagrangian submanifolds.