Vertex algebras, chiral algebras and applications
chiral algebras after Beilinson and Drinfeld, relation to vertex algebras and OPE, conformal blocks, BRST reduction.
Kac-Moody and W-algebras, lattice algebras, chiral differential operators. Modules over vertex algebras, fusion product.
Kazhdan-Lusztig equivalence (between representation of affine algebras and quantum groups).
Topological field theories and factorization algebras
the notion of extended topological field theory, cobordism conjecture (with some background from higher category theory), En and factorization algebras, factorization homology)
Universality for lozenge tiling local statistics
height functions and Gibbs measure for lozenge tilings, non-intersecting random walks and coupling techniques, effective global laws of large numbers, local law and multi-scale analysis
Gibbs measures (GM) in statistical mechanics
existence, uniqueness, and non-uniqueness of GM; the role of symmetry group (discrete – continuous – compact — non-commutative) and interaction smoothness; Gaussian random fields as GM; Ising crystal and its relation to the lozenge tilings
Exact results about planar dimers: a tutorial
Kasteleyn theorem, transfer matrices, surface tension, special domains