I am primarily interested in different kinds of higher categorical structures and how they relate to each other, and am currently focused on how techniques from universal algebra can be used to describe different sorts of higher categories. When I say "higher categories" I mean any structure involving collections of cells in various shapes and operations for composing certain kinds of diagrams of those cells into a single cell. For instance, n-categories are algebraic structures on n-globular sets, and n-tuple categories are algebraic structures on diagrams of cubical cells which keep track of dimensions.
In infinite dimensions, these algebraic structures can be modeled as a homotopy theory for diagrams over various more complicated categories. My work focuses on classifying these models and establishing analogues of results from ordinary category theory for higher category theories with more general cell shapes.
Algebraic K-theory describes how various types of objects are built up out of smaller pieces, where "smaller" is typically tracked by either monomorphisms in a category or reversed epimorphisms. I am interested in further developing an axiomatic framework for K-theory defined by Campbell and Zakharevich, which uses double categories to model objects with two different ways of mapping a smaller object to a bigger one. This approach helps to generalize classical theorems of K-theory to non-abelian settings and relates to broader relationships between n-tuple categories and factorization systems on ordinary categories.
In addition to using algebraic approaches in categorical topology, I am also interested in using techniques from homotopy theory to describe computational properties of algebraic structures. For instance, in the theory of monoids, a formal expression like 1+2+3 "partially evaluates" to either 3+3 or 1+5 before reaching its "total evaluation" 6. These partial evaluations can be treated like directed paths in a space of such formal expressions in a monoid. Properties of both the theory of monoids and a particular monoid are encoded in properties of these computation spaces.
As an undergrad I did research projects in polyhedral decompositions of manifolds, hyperbolic knot theory, formal languages for concurrent processes, and computational astrophysics of plasma jets and supernova neutrinos.
- Partial Evaluations and the Compositional Structure of the Bar Construction. Joint with Carmen Constantin, Tobias Fritz, and Paolo Perrone. [PDF] [arXiv]
- Compositional Structure of Partial Evaluations. MIT Category Seminar, September 2020. [Animated Slides] [Less Animated Slides]
- Cubical omega-Categories and Cubical Theta. Cubical Sets Seminar, MSRI, May 2020. [Slides]
- Constructing Cubes from Semicubes. Cubical Sets Seminar, MSRI, May 2020. [Slides]
- Shape Independent Category Theory and Enrichment. Category Theory OctoberFest, Johns Hopkins University, October 2019. [Slides]
- Densities of Hyperbolic Cusp Invariants. Proceedings of the American Mathematical Society. Volume 146, Number 9, September 2018, Pages 4073–4089. [PDF] [arXiv]
- specgen: A Tool for Modeling Statecharts in CSP. Nasa Formal Methods 282, 2017.
- Nonstandard Neutrino Interactions In Supernovae. Physical Review D 94, 093007, 2016. [PDF] [arXiv]
- Density Constructions for Hyperbolic Link Invariants. Olivetti Club, Cornell University, October 2017 and Graduate Student Seminar, Brandeis University, February 2017. [Slides]
- The Geometry of Knots (Cusp Densities of Links). MathFest, August 2016. [Slides]
- Cusp Density: Dense or Knot?. Unknot III Conference, August 2016. [Slides]
- Non-Standard Interactions and Neutrino Oscillations in Core-Collapse Supernovae. Astrophysics Group, Brandeis University, October 2014. [Slides]
- Neutrinos and the Unknown. North Carolina Museum of Natural Sciences, July 2014. [Slides]
- Cusp Invariants: Dense or Knot?. Summer Research Symposium, Williams College, August 2016. [Poster]
- Non-Standard Interactions and Neutrino Oscillations in Core-Collapse Supernovae II. NCSU Undergraduate Research Symposium, July 2014. [Poster]
- Non-Standard Interactions and Neutrino Oscillations in Core-Collapse Supernovae I. UNC/NCSU CAP/URCA Joint Physics Poster Session, July 2014. [Poster]