Geometry and algebra of pseudo-holomorphic curves


Instructor: John Pardon
phone: 609-258-5160
office: Fine 906
TA: Mohan Swaminathan

Abstract:

Pseudo-holomorphic curves were introduced into symplectic geometry by Gromov in 1985, and they quickly emerged as a (perhaps the) central tool in the field. This course will start with a discussion of the moduli properties of pseudo-holomorphic curves (e.g. Gromov compactness, gluing, transversality) and will continue with some of their spectacular applications from Gromov's first paper on the subject (e.g. Gromov non-squeezing, homotopy type of symplectomorphism groups of CP^2 and S^2xS^2). If time permits, we will conclude with a brief preview of recent applications of pseudo-holomorphic curves.

Reading:

  • Chris Wendl, Lectures on Holomorphic Curves in Symplectic and Contact Geometry.
  • Dusa McDuff and Dietmar Salamon, J-holomorphic Curves and Symplectic Topology.
  • Dusa McDuff and Dietmar Salamon, J-holomorphic Curves and Quantum Cohomology.
  • Misha Gromov, Pseudo holomorphic curves in symplectic manifolds.
  • Kenji Fukaya, Paul Seidel, and Ivan Smith, The symplectic geometry of cotangent bundles from a categorical viewpoint.


    • Lecture 1: Gromov non-squeezing

    We start with some basics of symplectic geometry: Darboux's theorem, Gromov alternative, almost complex structures, and pseudo-holomorphic curves. We then "prove" Gromov non-squeezing, assuming many nontrivial results about holomorphic curves which will be addressed in later lectures. We will prove the monotonicity inequality needed in the proof of non-squeezing.

    Lecture 2: Non-linear Fredholm theory and generic transversality

    We examine the foundational local theory of moduli spaces of pseudo-holomorphic curves, specifically the formalism of Banach manifolds and linearized operators. We will discuss under what conditions such moduli spaces can be ensured to be cut out transversally (and thus be finite-dimensional manifolds). We define the Gromov topology and state Gromov compactness.

    Lecture 3: Symplectomorphism group of S^2xS^2 and closed Reeb orbits on S^3

    We discuss Gromov's determination of the homotopy type of Symp(S^2xS^2) as a consequence of automatic transversality. We discuss a neck stretching argument to show that every Reeb vector field on S^3 has a closed orbit based on Hofer's compactness theorem and the result that for any compatible almost complex structure on CP^2, there is exactly one CP^1 of degree one passing through any pair of points.

    Lecture 4: Fukaya categories

    We introduce the basics of Fukaya categories. We discuss the wrapped Fukaya category of cotangent bundles, in particular how it is applied by Fukaya--Seidel--Smith--Nadler--Zaslow--Abouzaid--Kragh to obtain results towards the nearby Lagrangian conjecture.