Research in acoustics, combustion and fluid-structure interactions share a common theme of mathematical modeling and numerical simulation of problems in fluid and solid mechanics. Applications in these areas often involve wave propagation in complex constitutive materials, such as acoustic wave propagation in non-uniform media, detonation in heterogeneous explosives, or nonlinear deformation in elastic solids, among others.

Electromagnetism is a fundamental branch of physics and a key component of many natural and engineered systems. Since its inception, electromagnetism has been a rich source of fundamental mathematical problems, especially in the theory and numerical computation of solutions to partial differential equations.

Understanding how human activities impact the environment and ecosystems involves a web of interconnecting biological, chemical, and physical components. In addition to the disciplinary expertise required for each of these elements, mathematics plays a strong role in effectively analyzing and computing how the key drivers and parameters influence the various metrics of health of the ecosystem and environment.

Research in high performance computing and numerical analysis involves the development of algorithms designed to compute solutions of difficult, often nonlinear, mathematical problems spanning a wide range of applications. An essential element of the research concerns the analysis of the algorithms, which seeks to uncover important properties of the methods, such as stability and convergence, so that the algorithms can be employed with a clear understanding of their behavior and accuracy.

A common situation in modern scientific research is that the number of factors affecting quantities of interest (such as the shape of a biomolecule, changes in regional climate, or the biodiversity of the ecosystem in a certain lake) is so large that a model comprising all of them would be analytically and/or computationally intractable. A typical way of modeling such systems is to include a manageable number of explicit dynamical variables, and represent the others by some suitable statistical or stochastic terms.

Biological sciences have undergone a great expansion, beginning in about the middle of the last century, and introduced a wealth of new areas studying systems ranging in size from the molecular to that of ecosystems. The tools of investigation in many areas of modern biology have grown to be increasingly quantitative and reliant on other sciences, particularly mathematics. The biosciences have thus become a rich source of mathematical problems, inspiring advances in modeling, analysis, and computational methods.