Abstract:

There has been growing interest to study the interaction between additional symmetries appearing in low-dimensional topology and homological invariants (Floer homology and Khovanov-type homology). It has led to more powerful invariants as well as deeper understanding of the topology of manifolds and knots.  In this short course, I’ll discuss several constructions in this spirit and their topological applications. Some familiarity with Floer theory will be assumed.

Tentative plan:
Lecture 1 Overview & Equivariant algebraic topology
Overview of the course. Equivariant cohomology. 

Lecture 2 Pin(2)-equivariant Seiberg-Witten theory
Pin(2)-symmetry in Seiberg-Witten theory. Disproof of the triangulation conjecture.

Lecture 3 Involutive Heegaard Floer homology
Motivation and construction of involutive Heegaard Floer homology. Applications to homology cobordism group, knot concordance group, and corks.

Lecture 4 Knot symmetries and Khovanov homology
Different types of knot symmetries. Lawson-Lipshitz-Sarkar‘s (re)construction of Khovanov stable homotopy types. Stoffregen-Zhang’s work. 

Lecture 5 Localization in symplectic geometry
Seidel-Smith’s localization theorem. Symplectic Khovanov homology. Reformulation of the Ozsvath-Szabo spectral sequence.
Rank inequalities in Heegaard Floer homology.

Lecture 6 Localization in Seiberg-Witten theory
Real Seiberg-Witten theory. TBD

Time:

September 4,6,9,11,13,16----15:00-17:00