You should be familiar with the basics of topology (and ideally a little bit about knot theory) and algebra (such as notions of chain complexes). Anything else that we need, we will cover them in class; so this will be a good learning opportunity for many modern mathematical techniques. Since this a very new subject, there are no good books written on the topic, so the lectures will follow papers instead. Here is a tentative list of papers (a subset of) which we plan to cover.

Paper | Description |

A categorification of the Jones polynomial | The original paper by Khovanov. |

On Khovanov's categorification of the Jones polynomial | A very user-friendly introduction to Khovanov homology. |

Patterns in knot cohomology I | Reduced Khovanov homology. |

An endomorphism of the Khovanov invariant | Lee's deformation of Khovanov homology. |

Link homology and Frobenius extensions | Khovanov homology with respect to other Frobenius algebras. |

Khovanov homology and the slice genus | Rasmussen's computation of 4-ball genus of torus knots. |

Khovanov's homology for tangles and cobordisms | Bar-Natan's tangle invariant. |

The universal Khovanov link homology theory | Information content of Bar-Natan's tangle invariant. |

A functor-valued invariant of tangles | Khovanov's tangle invariant. |

An invariant of link cobordisms from Khovanov homology | Knot cobordism maps in Khovanov homology. |

A Khovanov homotopy type | A Khovanov stable homotopy type. |

Khovanov homotopy type, Burnside category, and products | Another construction of Khovanov stable homotopy type. |

A Steenrod square on Khovanov homology | Stable cohomological operations on Khovanov homology. |

A refinement of Rasmussen's s-invariant | New invariants from the stable homotopy type. |

Matrix factorizations and link homology | Khovanov-Rozansky's sl(n) homologies. |

Matrix factorizations and link homology II | Khovanov-Rozansky's HOMFLY-PT homology. |

Some differentials on Khovanov-Rozansky homology | Spectral sequences connecting the sl(n) homologies. |