Analysis of the Finite State Ergodic Master Equation

Joint work with Asaf Cohen.

What is this? Here, partial differential equations, stochastic control theory, and game theory are used to model large, interacting systems. A school of fish is a good toy example of this. A more serious example might be a bank run model.

Abstract: Mean field games model equilibria in games with a continuum of players as limiting systems of symmetric n-player games with weak interaction between the players. We consider the finite-state, infinite-horizon problem with two cost criteria: discounted and ergodic. We characterize the stationary ergodic mean field game equilibrium by a mean field game system of two coupled equations: one for the value and the other for the stationary measure. This system is linked with the ergodic master equation. Several discounted mean field game systems are utilized in order to set up the relevant discounted master equations. We show that the discounted master equations are smooth, uniformly in the discount factor. Taking the discount factor to zero, we achieve the smoothness of the ergodic master equation.

Limiting Speed of a Second Class Particle in ASEP

Assisted Promit Ghosal and Axel Saenz-Rodriguez.

What is this? There is a deep connection between asymptotics of certain particle systems and particular PDEs (see, e.g. KPZ Universality). My job in this project was to model these systems in Python and analyze the numerics. I also generated GIFs like the one to the right that represent particle movement.

Short Abstract: We study the asymptotic speed of a second class particle in two-species asymmetric simple exclusion process (ASEP) on the integer lattice with each particle belonging to either the first class or the second class.

Credit: Wikipedia.

Some Tables of Right Set Properties of Affine Weyl Groups of Type A

Joint with Leonard L. Scott.

What is this? Abstract algebra related to symmetries in different dimensions. My job was to model certain objects in Sage (a programming language) and compute certain elements to produce the paper's tables.

Abstract: The tables of this title are a first attempt to understand empirically the sizes of certain distinguished sets, introduced by Hankyung Ko, of elements in affine Weyl groups. The sizes are relevant to the computational efficiency of direct approaches to computing characters of modular representations of algebraic groups from characters of corresponding irreducible representations of quantum groups.