Past Projects

This was one of my first major projects in graduate school, with interesting results. Here, I describe a new method for encoding arbitrary homologies and cohomologies over any finitely generated group as the ground state of some quantum computer. I then analyze some interesting new properties available in this construction that are uniquely higher-dimensional. Finally, I use this construction as a base to build off of to construct an encoding of obstruction classes as necessary errors of a code. There are some ideas in here that I think have a lot of potential, but really depend on the feasibility of a virtual higher dimension quantum computer, which may be impossible to engineer. 

This is the first of two shorter papers on imbalanced games. In this one, I give the definition of balance and playability to certain types of games. I then provide some examples and context for which these ideas could be directly applied, notably as a base theory to understand certain highly sensitive ecologies, and as a basis to understand an optimal metagame in certain trading card games. I then use these definitions to find the least balanced but playable form of Rock-Paper-Scissors with an odd number of objects and two players. In doing so, I prove that there are no playable Rock-Paper-Scissors games with an even number of objects, and I find a novel "topological" construction in games of the game "blow-up". This blow-up in certain types of 2-person games acts very similarly to the topological idea of blowing up a manifold at a point by pasting in an entire manifold at that point in some canonical way. 

This paper is the direct extension of the previous paper to multiplayer Rock-Paper-Scissors games. Here, we allude to the previous constructions in order to fully generalize them to a higher number of players. As with anything in game theory, when adding players, the complexity increases massively. In a rigorous sense, one can consider an n-player game as a study of a function from an n-dimensional manifold to real n-dimensional space. For this reason, instead of finding the most imbalanced playable game, we find a game that is almost as imbalanced as possible. (In that, as the number of players increases, the game approaches the maximum possible imbalance). The majority of the paper is devoted to proving that this game is, in fact, strongly imbalanced. This turns out to be very hard, involving solving inequalities of very complicated functions. Instead, by fixing the number of players and the number of players doing a specific strategy, we can restrict these functions to be polynomials, which, when put into a computer algebra system, prove that this game is strongly playable for up to 100 players. Generalizing these constructions to higher numbers of players while keeping them canonical required several assumptions; however, I was very glad that even in this, we expanded to give definitions that can be applied to a wide swath of different types of games.