Research

About | Research | Teaching | Curriculum Vitae

Overview

My main research interests are in Functional Analysis, Partial Differential Equations, and Mathematical Physics. As an undergraduate, I did research on fluid dynamics, and in particular, shell models for turbulence under the mentorship of Vincent R. Martinez. For my honors thesis at Emory University, I worked on conformal mapping under the advisement of Milivoje Lukić. Most recently, I have been working on Koopman-von Neumann mechanics under the supervision of Patrick Gelß at the Zuse Institute in Berlin. See details and documents of my research below, or visit my Curriculum Vitae for a formal presentation of my research experience.

Koopman-von Neumann Mechanics on a Baroclinic Annulus

Koopman-von Neumann mechanics provides a functional analytic framework for translating problems in nonlinear dynamical systems into the language of linear operators on an infinite-dimensional Hilbert space. The reason we are interested in this approach is because this translates nonlinear problems into the language of quantum computers, which require linearity and unitarity.

Above is a simulation of the fluid density evolution on an annulus that I approximated and generated using Koopman-von Neumann mechanics. To see more about my project on Koopman-von Neumann mechanics, see my presentation that I gave at a seminar at the Zuse Institute in Berlin below.

Schwarz-Christoffel Mappings onto Periodic Comb Domains

For my undergraduate honors thesis, I developed a new method to compute Schwarz-Christoffel mappings onto the particular class of periodic comb domains. An example is displayed above. A Schwarz-Christoffel map is a of conformal map from the upper half plane to a polygonal domain.

This projected consisted of deriving a new Schwarz-Christoffel formula particular to periodic combs, and numerically solving this formula to compute the associated conformal map. See my published thesis on the Emory Theses and Dissertations repository, and my thesis defense presentation below.

Acknowledgments

I'd like to acknowledge my previous advisors, mentors, and collaborators who have helped with or inspired some of my past work.

• Vincent R. Martinez - fluid dynamics research
• Milivoje Lukić - conformal mapping research
• Patrick Gelß - Koopman-von Neumann mechanics research
• Yizhou (Joey) Chen - classmate at Emory
• Olena Kovalenko - collaborated on fluid dynamics research
• Dexter Thiesen - collaborated on fluid dynamics research
• Bart Rosenzweig - collaborated on fluid dynamics research
• Jon Legasa - collaborated on PINNs and fluids research

This is in no order and not a formal list, just people I appreciate for helping me develop and grow as a researcher.