Floer homology and low-dimensional topology
This course will aim to bring graduate students to the forefront of recent research and developments in low-dimensional topology using Floer homology. We will start with a mild introduction to Morse theory and Morse homology. We will then introduce Heegaard Floer homology, an invariant for closed 3-manifolds and knot Floer homology, an invariant for knots, both defined by Ozsvath and Szabo. Afterwards the course will venture into several technical aspects of the theory such as invariance, functoriality and surgery formulas. This will be followed by various applications to low-dimensional topology pertaining to exotic smooth structures on 4-manifolds, homology cobordism in dimension 3, knot concordance and contact geometry. Discussing these applications will take the center stage of this course.
If time permits, we will also discuss some variants of Heegaard Floer homology such as bordered Heegaard Floer homology and involutive Heegaard Floer homology. The former is an invariant of 3-manifolds with boundary while the latter is another invariant of closed 3-manifold defined in the presence of a certain symmetry.
Some familiarity with Morse theory and Symplectic geometry will be helpful.