Gorsky, EugeneKivinen, OscarSimental, Jose2023-01-162023-01-162023-01-162022-12-2410.1112/blms.12761https://infoscience.epfl.ch/handle/20.500.14299/193827WOS:000903661700001These notes cover the lectures of the first named author at 2021 IHES Summer School on "Enumerative Geometry, Physics and Representation Theory" with additional details and references. They cover the definition of Khovanov-Rozansky triply graded homology, its basic properties and recent advances, as well as three algebro-geometric models for link homology: braid varieties, Hilbert schemes of singular curves and affine Springer fibers, and Hilbert schemes of points on the plane.Mathematicsaffine springer fibersquiver gauge-theoriest-catalan numbershilbert schemescompactified jacobianscoulomb branchesmatrix factorizationshochschild homologyvarietiestopologytext::journal::journal article::research article