Andrea Braides (University of Rome Tor Vergata)
Variational Problems on Networks

A PhD course at SISSA, Trieste (10 lectures, 2 hours each)
Room 134 at SISSA and online


Tuesday January 19 from 16:00-18:00
Wednesday January 20 from 16:00-18:00
Friday January 22 from 14:00-16:00

Tuesday February 2 from 16:00-18:00
February 3 from 16:00-18:00
February 5 from 16:00-18:00

Tuesday February 9 from 16:00-18:00
February 10 from 16:00-18:00

Thursday February 25 at 14:45
Friday February 26 at 13:45


The subject of the course is the description of variational problems defined on networks or graphs and depending on a “scalar spin function” (a variable taking only two values). In the terminology of Statistical Mechanics these problems are linked to Ising systems (at zero temperature). We will consider problems defined on Bravais lattices, on random lattices or with random interactions, and on dense graphs. The course will be an occasion to use techniques of Gamma-convergence in a Geometric Measure Theory setting, introduce some issues in Homogenization and from Percolation Theory, and (
in the case of dense graphs) some combinatoric concepts.


Lecture 1 (Jan 19).
Model problems and energies on graphs. Scaling of graphs and emergence of surface energies. A brief introduction to sets of finite perimeter. Discrete-to-continuum Gamma-convergence. A prototypical example: ferromagnetic nearest-neighbour interactions.

Lecture 2 (Jan 20). Gamma-limit of homogeneous pair-interaction systems in a cubic lattice. A superposition argument. Generalization to an arbitrary Bravais lattice.

Lecture 3 (Jan 21). Homogenization of periodic pair-interaction systems. The blow-up method by Fonseca and Müller. Fixing of boundary data. An asymptotic homogenization formula. Use of the formula to construct a recovery sequence.

Lecture 4 (Feb 2) Homogenization of Quasicrystals/Penrose tilings. The Discrete-to-Continuum Localization Method. Remarks and extensions. Coarse graining.

Lecture 5 (Feb 3) Design of networks. Optimal bounds for mixtures.

Lecture 6 (Feb 5) Homogenization of random mixtures. The coercive case. The rigid case.

Lecture 7 (Feb 9) Homogenization of random mixtures. The dilute case. Mixtures of ferromagnetic and antiferromagnetic interactions.

Lecture 8 (Feb 10) Homogenization on Poisson clouds

Lecture 9 (Feb 25) An example of a spin system with diffuse interfaces. Problems on abstract graphs. Topological convergence of graphs. Dense graphs

Lecture 10 (Feb 26) Graphons. The cut norm. Compactness of sequences of graphs up to reparametrization. Gamma-convergence of dense graphs. Convergence of minimal-cut problems