Past Projects

This was one of my first major projects in graduate school with interesting results. In 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 feasability of a virtual higher dimension quantum computer, which may be impossible to engineer. 

This was my second major project that I spent quite a while thinking about in the world of quantum codes. I felt that a lot of the examples and models in topological quantum computing, where almost defined to be topological because they were based off of something topological. Topological quantum field theories (TQFTs) for instance are morally tied to topological quantum computing. I was very curious about this and decided that I should try to reverse engineer these codes, so that they have all the structure that one would want, and then we could see if that if we go from the quantum code out we still get these types of structures. This led me to a complicated path of defining a very specific subset of topological codes with an explicit definition which is in complete agreement, and has interesting higher dimensional cases. As this work ended up much larger than expected it should be split into atleast two seperate papers, included is just the chapter from my dissertation, which should be understood as a set of complete notes on this question as opposed to a formal and directed article.

This is the first of two shorter papers on unfair games. In this one I give the definition of unfairness 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 fair 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 game 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 similair to the topological idea of blowing up a manifold at a point by pasting in an entire manifold in that point in some cannonical 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 higher number of players. As with anything in game theory, when adding players the complexity increases massively. In a rigourous 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 unfair playable game we find a game that is almost as unfair as possible. (in that as the number of players increase the game approaches the maximum possible unfairness). The majority of the paper is in proving that this game is in fact strongly unfair. 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 are able to 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.