I am a fifth-year PhD candidate in the Department of Mathematics at Tufts University. My doctoral advisor is Dr. Bruce Boghosian. My previous degrees are a BA in mathematics and physics from Middlebury College and a MS in mathematics from Tufts University. I plan to finish during the summer of 2025.

I am fully supported by the National Defense Science & Engineering Graduate (NDSEG) fellowship for a three-year period.

Interests: Gradient flows in the space of probability measures, conservation laws, infinite-dimensional variational methods, nonlinear stability, Markov processes, mean-field theory, McKean-Vlasov stochastical differential equations.

My research is in the interdisciplinary field of econophysics. My work centers around investigating a class of evolution equations that model time-dependent distributions of wealth. In particular, these equations are Fokker-Planck equations of the McKean-Vlasov type and they arise from a kinetic mean-field theory approximation to the evolution of the distribution of wealth as identical economic agents interact via stochastic binary transactions. The resulting evolution equations are second-order integro-differential equations, which are nonlinear and nonlocal in the dependent variable.

Such equations model the time evolution of a large population of identical agents interacting in a prescribed (but stochastic) way given some initial distribution of wealth.

It is known that without exogenous regularization a certain class of these models monotonically increases the Gini coefficient of economic inequality, which is a quadratic functional of the distribution of wealth. In this way, the Gini coefficient is a Lyapunov functional.

In order to prevent the condensation of all wealth into an infinitesimal population, it is necessary to introduce some regularization scheme. A simple one is a wealth tax from each agent that is then uniformly redistributed amongst the population. Such a tax is realized as a quadratic field that induces a restoring force towards the system’s mean wealth. However, there is no known Lyapunov functional for the resulting Fokker-Planck equation. This stands in contrast to more standard Fokker-Planck equations that can be represented as gradient flows in 2-Wasserstein space of a functional that is one part a convex, continuous potential field and one part Boltzmann entropy.

My research is divided into three primary area:

• identify a meaningful Lyapunov functional for the regularized model,
• show that the unregularized models may be viewed as a gradient flow of the Gini functional in the same way that the heat equation is the 2-Wasserstein gradient flow of the Boltzmann entropy, and
• investigate the rates of decay of certain functionals along the solutions to the evolution equations in the sense of entropy methods for diffusive PDE.

I am drawing on mathematical techniques from functional analysis in Banach spaces, the modeling of complex fluids, notions of differential geometry in infinite-dimensional spaces, and studies of simpler but still nonlinear Fokker-Planck equations.