Includes supplementary material: [...]

A presentation of the deformed W1+1 algebra.- Generating series of the Poincaré polynomials of quasihomogeneous Hilbert schemes.- PBW filtration over Z and compatible bases for VZ(_) in type An and Cn.- On the subgeneric restricted blocks of affine category O at the critical level.- Slavnov determinants, Yang-Mills structure constants, and discrete KP.- Monodromy of partial KZ functors for rational Cherednik algebras.- Category of finite dimensional modules over an orthosymplectic Lie superalgebra: small rank examples.- Monoidal categorifications of cluster algebras of type A and D.- A classification of roots of symmetric Kac-Moody root systems and its application.- Fermions acting on quasi-local operators in the XXZ model.- The Romance of the Ising Model.- A(1) n -Geometric Crystal corresponding to Dynkin index i = 2 and its ultra-discretization.- A Z3-orbifold theory of lattice vertex operator algebra and Z3-orbifold constructions.- Words, automata and Lie theory for tilings.- Toward Berenstein-Zelevinsky data in affine type A, part III: Proof of the connectedness.- Quiver varieties and tensor products, II.- Derivatives of Schur, Tau and Sigma Functions on Abel-Jacobi Images.- Padé interpolation for elliptic Painlevé equation.- Non-commutative harmonic oscillators.- The inversion formula of polylogarithms and the Riemann-Hilbert problem.- Some remarks on the Quantum Hall Effect.- Ordinary differential equations on rational elliptic surfaces.- On the spectral gap of the Kac walk and other binary collision processes on d-dimensional lattice.- A restricted sum formula for a q-analogue of multiple zeta values.- A trinity of the Borcherds _-function.- Sum rule for the eight-vertex model on its combinatorial line.