Tilings: Algebraic & Probabilistic Expedition

Abstract

We will introduce the mathematics of tilings of the plane and higher dimensional shapes from both an algebraic and a probabilistic perspective. Tilings takes an interesting question that serves as a perfect meeting ground for geometry, group theory, combinatorics, representation theory, and probability. On the algebraic side, we study how symmetric groups classify periodic tilings and explore substitution tilings such as the Penrose tiling. On the probabilistic side, we investigate random tilings, domino models, and the striking emergence of global order from local randomness, such as the arctic circle phenomenon. Expect to start off by seeing a lot of beautiful structures, and slowly delve into it’s connections to algebra, probability, and, if time permits, topology.

Main References

• Tilings and Patterns – Branko Grünbaum, G. C. Shephard
• Lectures on Random Lozenge Tilings – Vadim Gorin
• The Tiling Book – Colin Adams

Overview

Day 1: Basic Introduction to Tilings, Related and Ad-Hoc Ideas

This was the very first session of the Tilings UDGRP. We used the time to get introduced to each other, everyone’s motivation, as well as give a brief introduction to the topic at hand to decide what path we will follow.

• Slides: Available here
• Topics Covered:
  • Basic Definitions
  • Generating Functions
  • Basic Graph Theoretic Definitions
  • Transfer Matrix and Transfer Graph
• Date: 25th November, 2025