My research focuses on the study of dynamical systems, that is, systems that change in time. This is a very broad research field which uses ideas and techniques from many other mathematical domains.
The main research direction I have pursued is the topology and geometry of integrable Hamiltonian systems where I use ideas from differential geometry, group theory, algebraic topology, singularity theory, and more, to understand the interplay between dynamics and geometry in such systems.
A second research direction is the study of collective dynamics in systems of coupled oscillators. Such systems are being used to model many types of interactions that occur in nature — pacer cells in the heart, cells in the brain’s suprachiasmatic nucleus, electric power grids, and more. Research in this direction combines dynamical systems theory with extensive numerical computations.
At DKU, I am interested in mentoring research and Signature Work projects in topics related to integrable Hamiltonian systems, collective dynamics, and other topics broadly related to dynamical systems, geometry and topology, mathematical physics, and numerical methods. If you are interested in any of the topics contact me through email.
More information on specific topics can be found below.
When simple systems interact, they can exhibit a wide variety of fascinating dynamical behaviors. For example, the systems can synchronize to each other, much like, people in a theater who at the end of a play clap in a synchronized manner even though there is no one controlling them. Or they can behave in a completely unpredictable manner. Efforts to understand such phenomena start with Christiaan Huygens in the 17th century, but it is only in the past half century, with the development of modern dynamical systems theory and the introduction of statistical physics methods that synchronization begins to be properly understood.