Symplectic Geometry and Floer Theory
(Fall 2017)

This course is an introduction to Floer theory, in particular Lagrangian Floer homology in symplectic geometry. Floer theory studies certain differentiable functions on infinite-dimensional manifolds. The motivation for doing so comes from situations where critical points of a certain function correspond to geometric objects (e.g. closed geodesics, intersections of Lagrangian submanifolds in a symplectic manifold, flat connections in a principal bundle). This can be seen as a generalization of Morse theory, the study of critical points of differentiable functions on finite-dimensional manifolds.

In more detail, for a certain type of function on a finite-dimensional manifold there is a chain complex generated by its critical points whose differential counts solutions to an ordinary differential equation, the gradient flow. The cohomology of this complex corresponds to the singular cohomology of the underlying manifold. Attempting to generalize Morse theory to the infinite-dimensional case requires solving the analogue of the gradient flow equation, which amounts to solving an elliptic PDE. It turns out that the resulting cohomology theories do not correspond to the cohomology of the underlying space.

Setting up this theory will require some technical input, in the form of functional analysis and Fredholm differential topology. My goal is to introduce the relevant ideas via examples to students with background in differential topology and some knowledge of analysis.

We'll begin with a discussion of Morse theory with an approach that highlights the structures which persist in the infinite-dimensional case (along with some interesting asides). I'll also introduce some of the analytic foundations, and discuss some extra topics in Morse theory which are also relevant to the infinite-dimensional case.

The next part of the course is the introduction to Lagrangian Floer theory. I'll review the basic concepts of symplectic geometry that we'll need, along with some results on elliptic PDE.

After constructing Lagrangian Floer theory I'll describe some applications in symplectic geometry and low-dimensional topology.

Since each class period is an unreasonably long 1 hour and 35 minutes, I plan to split the time in two and give two lectures with a short break in between.

Partial Schedule(subject to change)
Class will be in McHenry 1270 on Tuesdays and Thursdays, 11:40AM-1:15PM. I'll post notes after lectures.

Introduction and summary of course (9/28)
Intro, Action Functional, Chern-Simons. PDF

Morse Functions, Gradient Flow, Handles (10/3)
Morse functions, the Morse Lemma, handle decompositions of smooth manifolds with examples. PDF
Defining Morse Homology, Flowlines (10/5)
Definition of Morse complex, ascending and descending manifolds, Morse-Smale pairs, gradient flowlines. PDF

Elliptic Equations and Sobolev Spaces, Regularity (10/10)
Analysis background: elliptic equations, weak derivatives, Sobolev-Morrey embedding, Fredholm operators. PDF
Compactness of Morse Trajectories (10/12)
Construction of the moduli space of flowlines, compactness. PDF

Invariance of Morse homology, Genericity, Cerf Theory (10/17)
Independence of Morse homology from choice of metric, Morse function. Genericity of Morse functions. 1-parameter families of Morse functions. Thanks to Erman and Elijah for talking through a mistake in these notes after lecture. PDF
Morse-Bott Functions on Grassmannians (10/19)
Grassmannians, embedding in Lie algebra of U(n), Morse functions, Schubert cells. PDF

Symplectic Linear Algebra(10/24)
Subspaces of symplectic vector spaces, the symplectic group, the Lagrangian Grassmannian, the Maslov index. PDF
Symplectic Manifolds (10/26)
Submanifolds of symplectic manifolds, Moser's Theorem, locally Hamiltonian fibrations. PDF

Symplectic Reduction (10/31)
Lie Group Actions, moment maps, symplectic reduction. PDF
Almost Complex Structures, Pseudoholomorphic Curves (11/2)
Tame/compatible almost complex structures, local/global properties of pseudoholomorphic curves. PDF

Banach Manifolds, The Action Functional (11/7)
Banach manifolds and smooth maps, computation of first and second variation of the action functional. PDF

L^2 Gradient of the Action, Index Theorems(11/9)
Gradient of the action functional, Riemann-Roch and index theorems. PDF

Gromov Compactness (11/14)
Rescaling, bubbling, examples.
NO CLASS (11/16)

NO CLASS (11/21)
Thanksgiving (11/23)

Construction of Lagrangian Floer Homology, Invariance, The Cotangent Bundle(11/28)
Definition of HF, continuation maps, isomorphism with singular cohomology in the case of the cotangent bundle.
Classical results: Lagrangians in C^n(11/30)
No exact Lagrangians in C^n, Non-squeezing, isotopy classes of Lagrangian tori in C^n. A closer look at Dehn twists and Lagrangian spheres.

Applications of HF: Graded Lagrangians, Symplectically Knotted Spheres (12/5)
Covers of the Lagrangian Grassmannian, gradings.
Applications of HF: Heegaard Floer Homology(12/7)
Introduction to Heegaard Floer homology. Symmetric Products, branched covers of discs. Examples.


References
Introduction to Symplectic Geometry, D. McDuff and D. Salamon.
Canonical textbook in symplectic geometry and topology.

Morse Theory, J. Milnor.
One of several possible introductions to Morse theory. For more detail also see Lectures on the h-Cobordism Theorem by the same author.

J-Holomorphic Curves and Symplectic Topology, D. McDuff and D. Salamon.
A comprehensive introduction to pseudo-holomorphic curve theory.

Symplectic Topology and Floer Homology, Y.-G. Oh.
Thorough explanation of analytical details and applications of Lagrangian Floer homology, available online here. Floer's original papers are also a good reference.

Uhlenbeck Compactness, K. Wehrheim.
Introduction to gauge theory aimed at graduate students. Covers principal bundles, connections, curvature, Uhlenbeck compactness etc. with analytical details.

Morse Homology, M. Schwarz.
A treatment of Morse theory from the functional analytic viewpoint, as one would set things up in Floer theory.

Lecture Notes on Elliptic Partial Differential Equations, L. Ambrosio.
Notes from a course on elliptic PDE, available online here.

Morse Theory, D. Gay.
YouTube videos of a 2012 course on Morse theory, from the standpoint of low-dimensional topology. Awesome pictures.

Morse Theory, Jesse Freeman, Jake Rasmussen.
Notes by Jesse Freeman from a course on Morse theory. Also has nice pictures.

Lectures on Morse Homology, A. Banyaga, D. Hurtubise.
Excellent treatment of (among other things) Morse homology on Grassmannians.

A Student's Guide to Symplectic Spaces, Grassmannians, and Maslov Index, P. Piccione, D. Tausk.
Thorough treatment of the Maslov index and Grassmannians aimed at graduate students. Does symplectic linear algebra in gory detail. Available online here.

Torus actions on Symplectic Manifolds, M. Audin.
Moment maps, Hamiltonian actions, examples.

Introduction to Global Analysis, J.D. Moore.
Excellent treatment of Banach manifolds and Morse theory of functionals on them, from the perspective of geodesics and harmonic maps. Available online here.