Random Walks on Infinite Graphs and Coarse Geometry

Professor Introduction

Eddie Y | PhD in Mathematics

Home Institute：California Institute of Technology

[ Research Interests ] Quantum physics, topological quantum states, number theory
[ Teaching Experience ] Former instructor at Ross Mathematics Program
[ Additional Experience ] Extensive experience in mentoring high school students for research competitions
[ Publications ] Authored multiple papers in the Journal of Number Theory and the Journal of Mathematical Physics

Project Description

This Project is designed to provide an in-depth exploration of random walks on infinite graphs, utilizing the framework of coarse geometry. Random walks on graphs have been extensively studied across various fields such as computer science, physics, and mathematics, offering valuable insights into network structural properties and dynamics. Recently, there has been growing interest in understanding random walks on infinite graphs, with an infinite number of vertices and edges. Due to infinite graphs' unique challenges and complexities, traditional tools used for analyzing random walks on finite graphs are often inadequate. In this course, we aim to bridge this gap by examining the interplay between coarse geometry and random walks on infinite graphs. Coarse geometry is a mathematical framework that studies the large-scale properties of metric spaces. By leveraging techniques from coarse geometry, we can gain deeper insights into random walks on infinite graphs and develop new analytical methods. This research has the potential to advance the field of graph theory and provide valuable insights into the behavior of random walks in infinite settings.

Project Keywords

Project Outline

Part 1 : Introduction to Graph Theory and Random Walks
• Overview of graph theory fundamentals
• Introduction to random walks on finite and infinite graphs
• Key differences and challenges in studying infinite graphs

Part 2 : Coarse Geometry Fundamentals
• Introduction to coarse geometry
• Key concepts and techniques in coarse geometry
• Applications of coarse geometry to metric spaces

Part 3 : Random Walks on Infinite Graphs
• Detailed study of random walks on infinite graphs
• Analysis of structural properties and dynamics
• Case studies and examples

Part 4 : Interplay Between Coarse Geometry and Random Walks
• Exploring the connections between coarse geometry and random walks
• Developing new analytical methods using coarse geometry
• Practical applications and theoretical implications

Part 5 : Advanced Topics and Research Directions
• Current research trends in random walks on infinite graphs
• Open problems and future research directions
• Student research projects and presentations

Suitable for

• High School Students preparing for research competitions or pursuing advanced studies in related fields
• Undergraduate and graduate students in mathematics, computer science, and physics