Sandpiles and walks

What do sand dunes, earthquakes, population growth, and even quantum walks have in common? They are processes that evolve in time and often behave unpredictably. This project explores such processes from the mathematical point of view, with a focus on sandpile models, random walks, and branching random walks. Imagine a pile of sand where adding one more grain might trigger an avalanche. Can we predict when it will stop or how far the avalanche will spread? Similar questions arise in naturefor example, how far the effects of an earthquake or an epidemic might reach. While scientists have simulations and data for these models, this project aims to give mathematical proofs because a simulation/experiment shows how a model behaves but a mathematical proof reveals why. One exciting part of the project involves quantum random walks, which combine randomness with the principles of quantum physics. By linking these ideas together, the project aims to uncover new insights into how complex systems behave, and to build bridges between mathematics, physics, computer science, and biology. Even though the models are abstract, the potential applicationsfrom natural disasters to disease spread to quantum computingare very real.