There is long history of use of mathematical decompositions to describe complex phenomena using simpler ingredients. One example is the decomposition of string vibrations into its primary, secondary, and higher modes. Recently, a spectral decomposition relying on Koopman operator theory has attracted interest in science and engineering communities. The spectral decomposition is based on an extension of the Koopman-von Neumann formalism to dissipative, possibly infinite-dimensional systems, including those describing flow of viscous fluids at the fundamental level, but also thermal flows in buildings, and power grid dynamics, at a more applied level. At its mathematical foundations, it is a spectral theory of composition operators. We will present the foundations of the theory, the numerical analysis approach, and its applications in the variety of applied contexts.
Igor Mezic is currently a Professor and Director at the Center for Energy-Efficient Design and Head of Buildings and Design Solutions Group of the Institute for Energy Efficiency at the University of California, Santa Barbara. He received an M.S. degree in Mechanical Engineering from the University of Rijeka, Croatia in 1990 and a Ph.D. in Applied Mechanics from the California Institute of Technology in 1994. Before coming to UC Santa Barbara in 1995, he was a Postdoctoral Research Fellow at the Mathematics Institute at the University of Warwick, UK. From 2000-2001, he also served as an Associate Professor in the Division of Engineering and Applied Science at Harvard University. Igor Mezic’s current research interests include dynamical systems theory of complex systems, including large-scale social systems. He was awarded the National Science Foundation CAREER Award for research on Nonlinear Dynamics and Control from Microscale to Macroscale (1999), as well as a Sloan Foundation Fellowship in Mathematics (1999) and the Axelby Outstanding Paper Award (2000). For his technology contributions, he was awarded the United Technologies Senior Vice Presidents Special Award (2007), and gave a number of plenary lectures. In addition to contributing his time and expertise to a significant number journals, panels, workshops, and conferences, Mezic has over 150 journal publications, has edited or co-written three books and has received numerous grants and industrial contracts. Mezic is a Fellow of the Society for Industrial and Applied Mathematics (SIAM) and the American Physical Society.