Projects

My research focuses on the study of dynamical systems, that is, mathematical models of systems that evolve in time. This is a broad area of research that draws on ideas and techniques from many branches of mathematics.

The main research direction I have pursued concerns the geometry and topology of integrable Hamiltonian systems. In this work, I use tools from differential geometry, group theory, algebraic topology, singularity theory, and related areas to understand how geometric and topological structure shapes the dynamics of such systems, and how global features emerge from locally simple behavior.

A second research direction involves the study of collective dynamics in systems of coupled oscillators. These systems are used to model a wide range of interacting phenomena in nature, including pacemaker cells in the heart, cells in the brain’s suprachiasmatic nucleus, electric power grids, and other networked systems. Research in this area combines dynamical systems theory with extensive numerical computation.

At DKU, I am interested in mentoring research and Signature Work projects on topics related to integrable Hamiltonian systems, collective dynamics, and more broadly to dynamical systems, geometry and topology, mathematical physics, and numerical methods. If you are interested in any of these topics, please contact me by email.

More information can be found by following the links below and by checking out my publications.

Integrable Hamiltonian Systems

My research studies mathematical models from classical mechanics known as integrable Hamiltonian systems. These systems are special because they have enough integrals, that is, quantities that stay constant over time, to make their motion highly structured and, in principle, exactly solvable. Locally, the behavior of such systems is very simple: motion takes place on invariant tori — smooth, torus-shaped surfaces in an abstract space that records position and momentum — and this motion can be described using special coordinates called action–angle variables. The main themes here are global geometry and topology of invariant tori, monodromy, Hamiltonian bifurcations, and applications to physical examples.

Synchronization and Collective Dynamics

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.