CAREER: Connecting Mathematical Models Across Scales

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NONTECHNICAL SUMMARYThis CAREER award supports theoretical and computational research aimed at developing simple models of complex systems. Complex systems are made up of a large number of smaller components that interact in ways that can lead to interesting collective behaviors where the components act in concert. Examples of such systems include materials, biological pathways, and neural networks as well as many engineered systems and social networks. Complex systems are difficult to model. Usually, mathematical models are most useful when they are sufficiently simple to capture the relevant parts of the phenomenon of interest while ignoring irrelevant details. For complex systems, it is difficult to know from the start which components are relevant and which are irrelevant. Consequently, models of real-world systems tend to be overly complicated, difficult to work with, and have limited predictive power compared to more parsimonious representations.This project leverages recent advances that combine the mathematical areas of information theory with differential geometry and topology. The basic idea utilizes a systematic method for pruning irrelevant complications from a model of a complex system until a sufficiently simple model is obtained. By enumerating all of the resulting approximations, the scientist has a roadmap from the complicated, detailed representation of the physical system, through various types of approximations and the system behaviors they describe. Put another way, the process acts as a mathematical bridge across scales: connecting microscopic mechanisms to systems-level phenomena. In this project, the PI will apply these new mathematical and associated computational tools to three target application areas: crystal structures of alloys, biochemical kinetics of developmental pathways, and networks of neurons. By better understanding how mathematical models reflect relevant physical details, models will be better at predicting the behavior of complex systems and enable more sophisticated design and control of complicated materials and processes. This work also includes an educational component that will develop a pedagogy of interdisciplinary science. This pedagogy will focus on university faculty in the form of a seminar series; university students in the form of a multi-department, special topics course; and high school teachers in the form of teaching workshops and web resources for high school science teachers.TECHNICAL SUMMARY This CAREER award supports theoretical and computational research aimed at developing simple models of complex systems. Simple mathematical models have always played an important role in scientific inquiry. For systems with separated scales or symmetries, there are standard techniques for constructing parsimonious representations from complicated, mechanistic descriptions. Recent advances combining information theory and differential geometry and topology suggest new methods for reasoning about the relationship between mechanisms and phenomena in models of complex physical systems. For this research, the PI takes the approach that the predictions of a multi-parameter model form a manifold embedded in an abstract data space. It has been observed that typical model manifolds are bounded and that simple, approximate models reside on the boundary of the model manifold. The Manifold Boundary Approximation Method constructs a sequence of limiting approximations, that is, "small parameters" that corresponds to a sequence of simple, approximate models. This research will extend these results to three target application areas: alloy crystal structure, biochemical kinetics of developmental pathways, and networks of neurons. Within these target application areas, the PI will develop a new approach to modeling based on the concept of a "supremum model", that is, the minimal model that simultaneously explains several behaviors. When combined with the Manifold Boundary Approximation Method, the supremum model concept enables one to bootstrap from several models of overlapping system components to models of much larger systems, the direct exploration of which would be computationally challenging or infeasible.This work also includes an educational component that will develop a pedagogy of interdisciplinary science. This pedagogy will target university faculty in the form of a seminar series; university students in the form of a multi-department, special topics course; and high school teachers in the form of teaching workshops and web resources for high school science teachers.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.