Numerous complex systems in nature and in technology defy concise characterization because they exhibit strongly nonlinear behaviors that lack all symmetries and are highly non-periodic on a wide range of spatial and temporal scales. Characterization by detailed measurement (in lab experiments or direct numerical simulations) is now possible in many cases using modern measurement technologies or computational techniques. However, the resulting deluge of data often leads to little insight; in particular, there is frequently no good way to connect quantitatively experimental measurements of a particular complex system with the output from simulations/models of the same system. New, computationally-based, mathematical tools from algebraic topology have the potential to bridge the gap between measurements and models; the proposed research will explore the use of algebraic topology to link numerical simulations and laboratory experiments in situations where complexity arises because the system under study is driven out of thermodynamic equilibrium. The research focuses on an outstanding paradigm for nonequilibrium complexity: fluid flow driven by temperature gradients (thermal convection). The planned work brings three unique capabilities together in a single effort: (1) the experimental ability both to measure and to manipulate precisely complex, convective flows; (2) efficient methods for state-of-the-art, large scale, high-resolution numerical simulations of convective flow; (3) open source, general purpose, and efficient computational algorithms and software for computing algebraic topological invariants on large data sets. Topological tools will be developed both to characterize and to minimize model error as well as to compare and to quantify dynamical properties including Lyapunov exponents, dimensionality and bifurcations between complex spatiotemporal flow states. This effort should ultimately identify ways in which homology-based metrics can be used for building reduced order models that permit prediction and, perhaps, control of convective flow. More generally, we expect the metrics developed for convection should find broad application to PDE-modeled problems ranging from the control of cardiac arrythmias to the prediction of weather and climate.
The behaviors of complex systems in the world around us can now both be measured with high fidelity using advanced sensing technologies and simulated with great realism using modern computer techniques. However, the enormous data sets typically produced in these cases are often difficult to interpret because there exist few good mathematical tools to connect quantitatively the experimental measurements of a given complex system with the output of computer simulations of that same system. The proposed research explores the use of the mathematics of topology to relate lab measurements to computer outputs in a particular complex system, thermal convection. The results of this work should lead to new ways to understand, to predict, and, perhaps, to control convective flow, which plays a direct role in natural processes (e.g., volcanism, earthquake dynamics, continential drift) and industrial applications (e.g., thermal regulation of many devices, the growth of semiconductor materials). Moreover, the topological tools developed for thermal convection should apply more generally to a wide variety of other problems involving complex systems including the forecasting of weather and climate; the dynamics of the biomass in the oceans; the onset of turbulence; the evolution of reagent patterns on a catalytic metal surface; and ventricular fibrillation in a human heart.
