Differential Geometry

Differential geometry investigates the properties of smooth manifolds and curves using calculus and algebra. Researchers explore concepts such as curvature, geodesics, and topological invariants. This field provides a framework for understanding geometric structures, including their local and global properties. Specific areas of study involve the analysis of differential equations on manifolds, the application of Lie groups and algebras to geometric problems, and the investigation of spectral properties of differential operators, such as Steklov eigenvalue problems.

In Arkansas, the principles of differential geometry inform research in areas relevant to the state's economy and environment. For example, understanding the geometry of complex systems can aid in modeling population dynamics and network behavior, which are crucial for managing natural resources like forests and agricultural lands, and for analyzing the spread of infectious diseases within the state. The mathematical tools developed in differential geometry can also be applied to the design and analysis of complex engineered systems.

This research area engages with numerous interdisciplinary fields, including the analysis of differential equations, mathematical biology, population dynamics modeling, and network behavior analysis. Across Arkansas institutions, faculty members contribute to and benefit from these diverse connections, fostering a broad engagement with fundamental mathematical inquiry and its applications.