NSF Award
1515851
The objective of this research is the analysis of mathematical mechanisms that explain how small actions exerted on dynamical systems can lead to large effects, and applications of these mechanisms to real-world systems, such as in astrodynamics, molecular biology and bioengineering, and climate science. In astrodynamics, small forces resulting from the intertwining of the natural gravity of Moon, Sun, and planets, can be exploited to design spacecraft trajectories to distant locations within the Solar System, or to correct the orbits of artificial satellites near the Earth, with significant fuel savings. In climate science, small changes in the system parameters, such as in the concentration of greenhouse gases, can cause sharp transitions of the climate from one regime to another, and the mathematical analysis of climate data can detect early warning signs of such transitions. In molecular biology, small changes in the structure of molecules involved in the process of protein folding can have significant functional effects; such changes can be detected via topological data analysis of electron microscopy imaging. <br/><br/>This project seeks to discover explicit mechanisms that produce large effects in general mathematical models of various real-world systems, as well as to infer such mechanisms directly from data. Data is often used in the case of complex systems where explicit models are difficult to conceive from fundamental principles. The first research direction is devoted to large effects in mechanical Hamiltonian systems subject to small perturbations, deterministic or random. The goal is to describe mechanisms that yield trajectories that travel a large distance in the phase space, as well as trajectories with prescribed itineraries. Applications of these mechanisms include astrodynamics and celestial mechanics. The second research direction is devoted to critical transitions in complex systems. The goal is to develop methods based on persistent homology that detect critical transitions in noisy dynamical systems with slowly evolving parameters, and to study topological changes in macromolecular processes. These methods are applied to paleoclimate data and GroEL/ES chaperonin systems involved in protein folding.
