This project has three main areas: unbalanced allocations, subgraphs of random graphs, and modeling waves. Balanced allocations are a well-studied problem in computer science. A simple example is shoppers selecting cashiers at a grocery store; in a balanced allocation, most lines would be of the same (short) length. In this setting, the goal is to balance loads across all of the possible sites. For an unbalanced allocation, the opposite is the case. A simple example is selecting films at a multiplex; most filmgoers are wary of short ticket lines and choose the longer lines, assuming the popular films are better. This forms an unbalanced allocation, with many empty loads and a few very large ones. This type of unbalanced allocation arises naturally in physical processes such as crystal formation; it also minimizes costs in many settings. Through this project's work, we now understand the behavior of a broad class of unbalanced allocations: how heavy and light loads tend to be, and how the loads change over time. A random graph is a network that arises randomly; a simple example is how webpage links are built up over time, or how friendships are formed. This project studied substructures of random graphs and looked for patterns. We found that a specific class of substructures of random graphs are in fact also random; there are very few patterns that can appear. We did this by inventing a simple game that shows us exactly when the patterns are and are not random. Mathematical physicists have found differential equations that, when true, generate waves (like on the surface of the ocean). However, it can be difficult to find specific functions that make these differential equations true. This project built up a big family of functions that not only make waves but are also easy to describe and study.
