The course will cover a collection of topics around complex-analytic tools useful for combinatorics.
We will rely heavily on Melszer, Pemantle and Wilson books, but also on material beyond those.
The course will be graded on 3-4 homeworks (50%) and a project (50%).
Topics to be covered:
- Introduction into generating functions, and how to generate them.
- Generating functions and their coefficients. Cauchy formula. One function, many series.
- Dependence on parameters and basic formalism of statistical mechanics. Rudiments of large deviation theory.
- Many variables: amoebas, their properties.
- Theory of residues in higher dimensions. Iterated residues.
- Main results for rational generating function: smooth pole.
- Asymptotics of oscillating integrals.
- Lattice quantum random walks.
- Algebraic generating functions.
- Frozen regions in dimer models and rational generating functions with singular poles.
- Petrovsky-Atiyah-Bott-Gårding theory of fundamental solutions to hyperbolic linear PDEs; applications to generating functions.
- Lacunas and discontinuities of the coefficient growth exponents.