Math/MCAIM Colloquium | The Theory and Application of Networks: From Mathematical Machine Learning to Simplicial Complexes

Abstract: Networks are ubiquitous in nature and appropriate for mathematical investigation of various systems. In this talk I will discuss some aspects at the intersection of mathematics, machine learning, and networks to introduce interdisciplinary methods with wide application. First, I will discuss some recent advances in mathematical machine learning for prediction on graphs. Machine learning is often a black box. Here I will present some exact theoretical results on the dynamics of weights while training graph neural networks using graphons - a limiting function of a graph with infinitely many nodes. Next, I will use these ideas to present a new method for early prediction of disease subtype, characterized by dynamic co-evolution of multiple variables, with remarkable success in prediction of Parkinson's subtype five years in advance. Then, I will discuss some work on higher-order models of graphs: simplicial complexes - that can capture simultaneous many-body interactions. I will present some results on spectral theory of simplicial complexes, as well as introduce a mathematical framework for studying the topology and dynamics of multilayer simplicial complexes using Hodge theory, applied to brain connectome data. Finally, I will discuss applications of such interdisciplinary methods to studying bias in society, opinion dynamics, hate speech propagation in social media, and extreme mountaineering.

Talk will be in-person and on Zoom: https://umich.zoom.us/j/98734707290