Supervisor: Dr Anna Maltsev
Project description:
The main themes of this project are geometric graphs and their direct applications in biophysics for a better understanding of heartbeat initiation. We will take a cohesive approach to theory and applications: we let our theoretical problems be motivated by applications, and our conclusions further knowledge in applied fields. In particular, we aim to establish the effect of the underlying geometry of a graph on its spectral properties and explicate its biological significance in a subcellular system of interacting ion channels described by a similar geometric model.
This project will be divided into two parts: Part 1 will be an investigation into a subcellular calcium signalling system and related epidemiological problems using numerical simulations and existing mathematical theory; Part 2 will be an expansion of mathematical theory in the direction motivated by these novel applications, particularly in the direction of eigenfunctions on metric perturbed lattice graphs and spectral theory of random geometric graphs. In both parts, we will take a threefold holistic approach with numerical simulations, mathematical theory, and applications.
Further information:
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Fees and funding
Quantum graphs are very important as they model various physical systems such as road and electrical networks, networks of neurons, and water and gas pipes. In our recent studies we have found the precise links between spectral properties of a weighted random matrix and a quantum graph. There are a lot of powerful methods that are developed for random matrices that can now be transferred to random graphs to solve famous open problems such as special cases of the Bohigas-Giannoni-Schmit conjecture, which links chaotic dynamics and spectral statistics.