Algebraic combinatorics: applications to statistical mechanics and complexity theory

We will give a brief overview of the classical topics, problems and results in Algebraic Combinatorics. Emerging from the representation theory of $S_n$ and $GL_n$, they took a life on their own via the theory of symmetric functions and Young Tableaux and found applications into new fields. In particular, these objects can describe integrable lattice models in statistical mechanics like dimer covers on the hexagonal grid, aka lozenge tilings. We will show how asymptotics of normalized Schur functions leads to limit behavior in the dimer model like GUE-eigenvalues distributions of observables near the boundary and limit surface of the height function. Time permitting, we will also discuss the connections of Algebraic Combinatorics to Geometric Complexity Theory and the permanent versus determinant (VP vs VNP) problem.