# Colleen Delaney

### Research

Broadly speaking my research interests lie in algebraic and computational aspects of quantum field theories.

I work with fusion categories that model particle-like objects that arise in 2+1 dimensional topological phases of matter and their application for topological quantum information processing. These include modular tensor categories and $$G$$-crossed braided fusion categories. More generally I enjoy pure quantum algebra, and I like computing examples of fusion categories and writing code to investigate their properties and relationships to one another.

• Symmetry defects and their applications to topological quantum computing. C. Delaney, Zhenghan Wang. (To appear in AMS Contemporary Math Series.)
• A systematic search of knot and link invariants beyond modular data. C. Delaney, A. Tran. arXiv:1806.02843
• On invariants of modular categories beyond modular data. Parsa Bonderson, C. Delaney, Cesar Galindo, Eric C. Rowell, Alan Tran, Zhenghan Wang. arXiv:1805.05736 Click here to see the W-matrix data.
• Local unitary representations of the braid group and their application to quantum computing. C. Delaney, Eric C. Rowell, Zhenghan Wang. arXiv:1604.06429

While I'm partial to the topological approach, I am interested in all aspects of quantum computing from quantum hardware design to quantum algorithms. In summer 2017 I was an intern in the Computational Physics department at HRL Laboratories, where I worked on randomized benchmarking for semiconductor quantum dots.

I've also studied various Hopf algebras of diagrams that appear in quantum field theories, where they describe the algebraic theory of renormalization. These combinatorial Hopf algebras have interesting connections with noncommutative geometry, and the machinery that has been developed to study them can be applied to new settings, like classical information theory.

• Dyson-Schwinger equations and the theory of computation. C. Delaney, Matilde Marcolli."Feynman Amplitudes, Periods and Motives", Clay Math Institute and AMS. arXiv:1302.5040
• Generalizing the Connes-Moscovici Hopf algebra to contain all rooted trees. Susama Agarwala, C. Delaney. Journal of Mathematical Physics, Vol. 56, No. 4, (2015). arXiv:1302.4004

Lately I've spent some time thinking about Wilson loop diagrams in $$\mathcal{N}=4$$ supersymmetric Yang-Mills theory and their relationship to the Amplituhedron of Arkani-Hamed/Trnka.