Ko Honda

Higher-dimensional Heegaard Floer homology (week 1)

In the lecture series I will introduce higher-dimensional Heegaard Floer homology (HDHF), which simultaneously generalizes the usual Heegaard Floer homology (due to Ozsváth-Szabó) and a variant symplectic Khovanov homology (due to Seidel-Smith). I will then proceed to explain the relationship of HDHF with Hecke algebras (including the double affine Hecke algebra), skein theory, and string topology.

Copilots:

Mikhail Khovanov

KLR algebras and link homology (week 1)

In this lecture series we will review KLR algebras, which categorify positive halves of quantum groups. We then examine several ways in which they connect to link homology theories. One such construction is via foams and categorified Howe duality. The other is via Webster's categorification of tensor products of quantum group representations and Reshetikhin-Turaev invariants.

Copilots:

Tobias Ekholm

Skein valued curve counts (week 2)

The lectures will cover: (1) Wall crossing for holomorphic curves with Lagrangian boundary condition and the HOMFLYPT skein, (2) Recursion relations, (3) Applications to the Ooguri-Vafa conjecture and the topological vertex, (4) Skein trace maps.

Copilots:

Vivek Shende

Fukaya categories and higher representation theory (week 2)

In this course I will explain how certain expected structures in higher representation theory — in particular, a `tensor product' for categorified representations of quantum groups, and a compatible braid group action — arise naturally from structural considerations around Fukaya categories on certain monopole moduli spaces. Along the way I will mention how Khovanov homology appears in this context. This presents joint work with Mina Aganagic, Elise LePage, and Peng Zhou.