Research Themes

Mathematical and statistical analyses form the backbone of theoretical and empirical science. The analysis of complex phenomena is a crucial component to modern scientific inquiry. Mathematical analysis includes the approximation and optimization of dynamical equations, and is crucial in describing and understanding the uncertainty and complexity of algorithms used to solve problems. Statistical analysis refers to the collection, study, interpretation, explanation, and presentation of data, and plays a central role in connecting theory with observation.

If mathematics and statistics are the backbone of scientific research, than computation provides the muscle. Computational challenges are usually associated with the complexity of the problem. Indeed many algorithms are "good" in the sense that they will produce quality results and yet are impractical to use because they do not scale well in high-dimensional settings due to constraints in spatial and/or temporal complexity. This "curse of dimensionality" requires new algorithms that are usually weaker approximations than the original, but that are more readily computable. This interplay between uncertainty and complexity is a fundamental trade off in the theoretical and empirical sciences, and needs to be studied and understood by students.

In theoretical science, the ultimate goal is for the mathematical descriptions of behavior to be commensurate with the corresponding physical accounts. This allows for analysis and computation to supplement physical observation, thus avoiding or reducing many of the practical and economical constraints of experiment. Moreover it also provides an opportunity for the theoretician to participate in the process of phenomenological and scientific discovery.

The scientific method represents a four-stage process of characterization, modeling, prediction, and validation. Characterization refers to the gathering and organization of data and information in the system. Then the problem is formulated and the variables defined. The next stage deals with describing observed phenomena in a functional form called a model. Sometimes one presents a set or class of models and then optimizes the choice of model by selecting the one that best represents the data. Once a specific model has been hypothesized, one can make predictions on the outputs of future inputs, as well as making decisions based on those predictions. Sometimes the decisions made feedback into the dynamics of the system itself, and so the effects of decisions made need to be carefully considered during the modeling stage. At the final stage, validation, one needs to check that the predictions correctly represent observation.