# Research

## Interests

My research focuses on topology, category theory, and their applications, studying a variety of complicated systems by modeling networks of relationships within them to leverage their compositional structure. I'm generally interested in exploring new contexts in which to apply these concepts, principles, and techniques in the search of novel insight or creative solutions.

Additionally, I'm of the opinion that an essential part of a mathematician's job is to help other people understand mathematics. As such, I'm interested in how humans conceptualize, communicate, and learn mathematics, especially in regard to the use of metaphorical reasoning and visualization.

## Student opportunities

Research in mathematics can be extremely rewarding if you enjoy creating, discovering, learning, or communicating new ideas. Interested students are encouraged to contact me regarding research projects, independent/individual studies, summer classes, honors options, and topics courses that they might wish to pursue.

## Publications

"On the topological characterization of gestures in a convenient category of spaces," Journal of Mathematics and Music (2021)

Description. In mathematical music theory, a gesture is a formal way of mapping a directed graph into a topological space. The resulting network of paths in the space can then be used to model the physical motion of performers or listeners as well as the conceptual motion of musical relationships in a piece. This work revisits a topological (as opposed to combinatorial) characterization of gestures in an attempt to simplify the presentation of the underlying theory.

"Categorical equivalences in the theory of sharp transitivity," Quasigroups and Related Systems (2019)

Description. We exhibit well-known correspondences between non-associative algebraic structures and sharply transitive actions as a related assortment of categorical equivalences.

"Rational spaces in resolving classes," Topology and its Applications (2016)

Description. Using algebraic data to model rational topological spaces, we uncover a quotient of the Witt group (of rational quadratic forms) embedded in a complicated structure of topological spaces induced by resolving classes.

## Other work

"Resolving classes and resolvable spaces in rational homotopy theory," Doctoral Dissertation (2016)

Description. In homotopy theory, there are essentially two ways of constructing new spaces: (1) "gluing" known spaces together (via homotopy colimits), or (2) "carving" from blocks of known spaces (via homotopy limits). In this work, we examine the use of homotopy limits in the construction and understanding of topological spaces. Focusing on a class of computationally well-behaved spaces (rational spaces), we associate algebraic data (commutative differential graded algebras) to spaces and use these algebraic models to characterize when we can think of one space as carved from another. We then use this characterization to investigate spaces carved from well-known spaces such as spheres, Eilenberg-MacLane spaces, and projective spaces. Further, we study applications to the theory of rational quadratic forms, including an explicit cone decomposition for the spatial model assigned to a quadratic form.