Discrete Quantum Mechanics: Can we hear the shape of a graph?

Can we hear the shape of a drum? This question has prompted a fundamental area of research in modern mathematics. One of the main characters in this story is the Laplace operator (aka the Laplacian), that can be defined using basic tools of calculus.

In this talk we will explore the graph Laplacian, a discrete version of the Laplace operator, which has broad applications in graph theory, combinatorics and mathematical physics. 

In particular, we will present a discrete version of the Schrödinger and Dirac equations and how it connects with some of the properties of a graph (connectivity, cycles/trees and walks) that can be extracted from the graph Laplacian, as well as recent work with undergraduate students on Dirac and Laplace operators on graphs. This will give us an opportunity towards the end of the talk to discuss different ways in which students can get involved with undergraduate research in mathematical physics.