This project is a review of the paper Mean Field Games in a Stackelberg problem with an informed major player. The paper analyzes the effect that an imbalance of information has on a Stackelberg problem with a large number of small followers. In particular, the leader receives a private signal about the world and must optimize their cost, taking into account the decisions of the small players. The small players learn this signal through the major player’s action. This report consists of a review of the current literature, the mathematical background required to understand the analysis, and a detailed summary of the paper with some ideas for possible future extensions.

In this project, my partner and I present findings related to the implementation of the deep reinforcement learning algorithm DDPG to geological carbon storage. Pressure buildup is a major concern in large scale carbon storage operations. The DDPG agent is able to find a schedule of injection and extraction rates that optimizes storage with this constraint.

Hamiltonian Monte Carlo (HMC), a method to sample from high dimensional distributions, is introduced. Standard sampling methods run into problems in high dimensional space and with distributions that have sticking points. Applications of HMC are described, in particular that of sampling from various empirical distributions.

This project attempts to answer whether all convergences in math can be described topologically. This turns out not to be the case. Convergence almost everywhere, which is fundamental in probability theory, is not topological. I present these findings, the equivalency of the various formulations of compactness in metric spaces, and more.