Critical Orientation of Mathematics to Produce Advancements in Science and Security (COMPASS)

Mathematics is a pillar of national security. A decision-maker’s ability to synchronize military activities across five domains (i.e., air, land, maritime, space, and cyberspace), and adapt to rapidly changing threat landscapes hinges on robust mathematical frameworks and effective problem formulations that fully encapsulate the complexities of real-world operational environments. Unfortunately, mathematical approaches in Defense often rely on “good-enough” approximations, resulting in fragile solutions that severely limit our nation’s ability to address these evolving challenges in future conflicts. In contrast, establishing robust mathematical frameworks and properly formulating problems can yield profound and wide-reaching results.

For instance, the Wiener filter1 was developed during World War II to help the U.S. military discern threats in the air domain from noisy radar observations. However, the technology’s effectiveness was limited due to its strong assumption of signal stationarity, a condition rarely satisfied in operational settings. By leveraging a dynamical systems approach, in 1960 Rudolf Kalman reformulated the filtering problem in a more robust state-space framework that inherently addressed non-stationarity.2 Sixty years later, the Kalman filter remains a pillar of modern control theory, supporting military decisions in autonomous navigation, flight control systems, sensor fusion, wireless communications and much more. The combination of a robust mathematical framework with the right problem formulation enables transformative Defense capabilities. Achieving this, however, requires deep mathematical insight to properly formulate the problem within the context of the specific Defense challenge at hand.

To excel in increasingly complex, dynamic, and uncertain operational environments, military decision-makers need richer mathematical frameworks that fully capture the intricacies of these challenges. Emerging fields in mathematics offer the potential to provide these frameworks, but realizing their full potential requires innovative problem formulations.

This ARC opportunity is soliciting ideas to explore the question: How can new mathematical frameworks enable paradigm shifting problem formulations that better characterize complex systems, stochastic processes, and random geometric structures?