Associated to any four-dimensional symplectic toric manifold is a Delzant polygon, which is a very special type of polygon that encodes all of the information of the manifold, and a smooth fan, which is a special finite set of integer vectors related to that polygon. By studying the Delzant polygon and fan of the manifold, it is possible to use two dimensional combinatorial techniques to learn about the four dimensional object, making this a good starting point to study four-dimensional geometry without having to actually work with anything four-dimensional.
Delzant polygons and smooth fans are well understood, but can only be used to study symplectic toric manifolds, which makes their use somewhat restrictive. A more complicated, and more interesting, situation arises in the study of what are called symplectic semitoric manifolds, where the Delzant polygon is replaced with an infinite family called a semitoric polygon and the fan is replaced with an infinite set of vectors called a semitoric helix. This project is concerned with the study of these two-dimensional objects, and producing examples of them making use of what are called the semitoric minimal models and certain operations called blowups and blowdowns.
Linear algebra and abstract algebra