INSTRUCTOR: Christian Borgs
TIME: Tuesday, Thursday 12:30-2:00
PLACE: Soda 310
OFFICE HOURS: Wednesday, 1:30-2:30, Thursday, 2:15 - 3:15
- (8/26/2021) Welcome to the course! Homework sets will be posted below, while Scribes and solutions to homework sets will be posted on b-courses
- (9/01/2021) Scribe 1 was updated to include some literature, see
Phase transitions are ubiquitous in physics and many other real-world systems, but can also be found in many areas of computer science and combinatorics,
from the classical phase transition involving the appearance of a giant component in random graphs,
to transitions from fast to slow convergence of randomized (e.g., Monte Carlo) algorithms,
to phase transitions in random satisfiability formulas,
to thresholds for learning models from sparse data.
In this class, we will study these phase transitions using methods from mathematical physics, combinatorics, and computer science. In addition, I will also cover the topic of convergence for sparse graphs, known as local weak convergence and its relation to graphical models on such sequences.
Since several of the topics covered in this class are drawn from the recent research literature, the is no single book covering the material of this class.
For some of the earlier topics involving branching processes, random graphs, and SIR epidemics,
I will roughly follow Chapters 2 and 3 of Durrett’s book Random Graph Dynamics,
while the methods from statistical physics used in later classes are based on a CBMS lecture series I gave in Memphis many years ago
(I will make chapters of my yet to be finished book on this subject available during the class).
Additional literature, including original research papers, will be provided during the course.
The course is open to graduate students with a strong mathematical background / good level of mathematical maturity
(similar to that for graduate classes in mathematics, statistics, or theoretical computer science).
Tentative List of Topics
A first look at Phase Transitions:
Expansion Methods 1:
- Survival vs Extinction in Branching Processes
- Emergence of the Giant Component in Random Graphs
- Epidemics and R_0
- Phase Transition in the Ising Magnet
- Fast vs Slow Mixing
Expansion Methods 2:
- Abstract Polymer Systems and Lovasz Local Lemma
- Zeros of Chromatic Polynomials
- Convergence for Sparse Graphs and Graphical Models on them s
- Polynomial Time Approximation Schemes
Computer Science and Learning:
- The Potts model and the Swendsen-Wang Algorithm
- Pirogov Sinai Theory
- Slow Mixing for the Swendsen Wang Algorithm
- Algorithmic Pirogov Sinai Theory
- Random K-SAT
- Number Partitioning
- Learning Stochastic Block Models
ATTENDANCE, ASSIGNMENTS, ETC.:
- The course will be in person and not be recorded, and I hope for active participation of everyone enrolled.
Therefore attendance is mandatory.
- I will give problem assignments every two weeks, for which I encourage collaboration of 2-3 students,
but expect separately written up solutions from every participant.
- While some scribes will be provided by the lecturer, most are written by participants and are part of the class requirement.
- There will be no exams, but depending on the class size, there may be final projects.
SYLLABUS (First few lectures)
- Introduction (Lectures 1)
- Branching Process (Lectures 2)
- Erdos-Renyi random graphs (Lectures 3-4)
- SIR Model (Lecture 5)
- Phase Transtion for the Ising Model (Lecture 6-7)