analysis and topology in analytic combinatorics

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-GĂ„rding theory of fundamental solutions to hyperbolic linear PDEs; applications to generating functions.
  • Lacunas and discontinuities of the coefficient growth exponents.

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