Geometric analysis and applications to microbe ecology

Research Project, 2019 – 2022

My research team will apply the tools of geometric, microlocal, and harmonic analysis to the following categories of long-standing open problems: 1) geometric inverse problems, particularly inverse spectral geometry problems; 2) locality principles for solutions to the heat equation in non-smooth settings; 3) finite propagation speed of the wave equation in singular geometric settings; 4) understanding the Poisson relation in singular settings. From a purely theoretical mathematical perspective, we will simultaneously tackle long-standing problems in microbe ecology. We will implement a novel game theoretic approach to microbe ecology we have been developing since 2012. The connection between geometric analysis and game theory may not be obvious, but it is profound. Viewing non-cooperative games from a geometric analytic perspective is the key ingredient in the proof of Nash´s prized theorem on the existence of equilibrium strategies in non-cooperative games. Using geometric analysis, we classified the level sets of payoff functions for non-cooperative games. Combining geometric analysis with non-cooperative game theory we aim to rigorously mathematically explain two ecological phenomena of microbes: (A) tremendous phenotypic variability within species and (B) tremendous taxonomic diversity. We will simultaneously use these same mathematical tools to (C) revolutionise the mathematics which describes the motility patterns and spatial distribution of marine microbes.