Research
To paraphrase Arthur C. Clarke: Any sufficiently advanced engineering is indistinguishable from Mathematics.
My research interests lie in the field of dynamical systems, ergodic theory and computation.
I am working as a member of the research group led by Dr. Igor Mezić. Our group is pretty diverse with people working on topics ranging from nanosystems, to large-scale networks.
In my research, I take an operator-theoretic approach to analysis of dynamical systems. My project focuses on designing and implementing algorithms for analyzing asymptotic dynamics of dynamical systems. In particular, we use spectral methods to approximate the ergodic partition of measure preserving systems.
Operator-theoretic approach in dynamical system theory is a less prevalent method for studying dynamical systems. Whereas geometric approach studies trajectories of systems as main objects, the operator theoretic approach studies dynamical systems as operators, transforming observables, i.e. functions on the state space, into other observables. Such analysis conceptually relates to analysis of Markov chains using Perron-Frobenius theory of matrices.
An interesting object to study using this formalism is the ergodic partition of a measure preserving system, i.e. the partitioning of the state space into ergodic sets. Ergodic sets are minimal sets from which trajectories cannot escape, and into which trajectories started outside cannot enter. Using the Koopman operator, and looking at its spectrum and eigenfunctions, one can compute an approximation of such a partition, effectively getting a catalogue of different dynamical behaviors that a system can exhibit.
Even though my undergraduate degree is in Electrical Engineering and my graduate studies are in Mechanical Engineering, an impartial observer would probably conclude that I mostly do mathematics. My work consists of learning about relevant mathematical theories and results and then trying to use them in a way that was either not considered before or that was not computationally tractable in the past. If you would like to know more about my research, feel free to contact me.
(The figure shows almost-ergodic partition of dynamical system called “standard map” which describes a pendulum being perturbed by a small periodic force.)
Teaching
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Coursework
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