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Discrete complex analysis

Lecturer

Konstantin Izyurov

Type

Advanced studies

Scope

5 sp

Lectures

Friday 14-16 at CK107, starting on January 16. Note: no lecture 06.03

Exercises

Friday 12-14 at CK108, starting on January 23

Course description

Discrete complex analysis is a theory of "analytic" functions defined on lattices in the complex plane (or, more generally, on planar graphs). It turns out that many results in the classical complex analysis have their counterparts in the discrete setting. The theory has recently found a number of spectacular applications, ranging from analysis and mathematical physics to image processing and architecture.

The course will cover basic theory of discrete analytic and discrete harmonic functions, including the discrete analogs of the Cauchy theorem and Green's formula, the Harnack estimate, Montel theorem, Beurling estimate, and convergence of discrete analytic/harmonic functions. Time permitting, we will discuss circle packings, a way to discretize conformal maps going back to Koebe and rediscovered by Thurston.

I will also try to include as many fun applications as possible. Depending on preferences of the audience, these may include e. g. "discrete" proofs of uniformization theorems, Cardy-Smirnov formula for critical percolation, local statistics of domino tilings, some glimpse of conformal invariance in the Ising model, Kesten's bound on the growth of DLA.

Prerequisites

Very basic complex analysis (on the level of the Cauchy integral formula) and very basic probability.

Bibliography

Exercises. Exercise assignment for 24.04: Section "The Ising model".

Additional reading

  1. Discrete holomorphic and harmonic functions: Conformal invariance of lattice models by H. Duminil-Copin and S. Smirnov; Discrete complex analysis on isoradial graphs by D. Chelkak and S. Smirnov.
  2. Dimer model: Conformal invariance of Domino tilings, Lectures on dimers, Dominos and the Gaussian Free field by R. Kenyon.
  3. Cardy-Smirnov formula: the original article by S. Smirnov, an easy way by V. Beffara, Lectures on 2D critical percolation by W. Werner.
  4. Discrete Beurling estimate with optimal exponent: Random walk: a modern introduction by G. Lawler and V. Limic.
  5. Kesten's bound on the growth of DLA: How long are the arms in DLA by H. Kesten
  6. Ising model: slides

 

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