I am interested in the applications of higher category theory to low-dimensional topology, homotopy theory and mathematical physics.
- From September to December 2019 and from June 2020, I will be a postdoc at the Max-Planck Institute for Mathematics in Bonn, continuing my work on topological field theories and other applications of higher category theory.
- From January to May 2020, I will be a postdoc at the Mathematical Sciences Research Institute (MSRI) in Berkeley during the programs Higher Categories and Categorification and Quantum Symmetries.
1. Fusion 2-categories and 4-dimensional topological field theories
One of the early successes of quantum topology was the construction of a three-dimensional topological field theory by Turaev-Viro, and its generalization by Barrett-Westbury, starting from the data of a spherical fusion category. In this work, we introduce spherical fusion 2-categories and construct a state-sum invariant of 4-manifolds, which categorifies the Turaev-Viro-Barrett-Westbury invariant. There are many examples of fusion 2-categories, including the 2-category of module categories of a braided fusion category, the 2-category of 2-representations of a finite 2-group and twisted linearizations of finite 2-groups.
- Fusion 2-categories and a state-sum invariant of 4-manifolds (with Christopher Douglas).
» We introduce semisimple 2-categories, fusion 2-categories, and spherical fusion 2-categories. For each spherical fusion 2-category, we construct a state-sum invariant of oriented singular piecewise-linear 4-manifolds.
2. Associative n-categories and homotopy.io
The theory of associative n-categories (ANC) is a model for semistrict higher categories, recently developed by Christoph Dorn, Christopher Douglas, and Jamie Vicary, and based on a combinatorial representation of higher string diagrams. In this work, we continue the development of ANCs and introduce the proof assistant homotopy.io.
- High-level methods for homotopy construction in associative n-categories (with Jamie Vicary).
» We develop a general contraction principle in ANCs which forms the mathematical heart of the proof assistant homotopy.io, allowing string diagram manipulations in arbitrary dimensions.
3. Quantum symmetry and pseudo-telepathy
Pseudo-telepathy is a phenomenon in quantum information, where two non-communicating parties can use pre-shared entanglement to perform a task classically impossible without communication. In this work, we uncover a connection between pseudo-telepathy and quantum symmetry, expressed in the language of fusion categories. We use this connection to classify instances of pseudo-telepathy in the `graph isomorphism game'. In this case, the relevant categories are (co)representation categories of `quantum automorphisms groups' of graphs, studied in compact quantum group theory. Along the way, we develop a 2-categorical framework for finite quantum set theory.
A compositional approach to quantum functions (with Benjamin Musto and Dominic Verdon).
J. Math. Phys. 59, 081706 (2018). doi: 10.1063/1.5020566
» We describe a 2-categorical approach to finite quantum set and quantum graph theory.
The Morita theory of quantum graph isomorphisms (with Benjamin Musto and Dominic Verdon).
Commun. Math. Phys. (2018), to appear. doi: 10.1007/s00220-018-3225-6
» We classify instances of quantum pseudo-telepathy in the graph isomorphism game by showing that graphs quantum isomorphic to a graph G correspond to certain Frobenius algebras in a monoidal category of quantum graph automorphisms of G. In certain cases, this classification can be expressed in group-theoretical terms.
4. Biunitaries in quantum information theory and beyond
Biunitary connections are central tools in the study and classification of subfactors. At the same time, several important quantum informatic quantities such as Hadamard matrices, unitary error bases and quantum Latin squares can be expressed as special cases of biunitaries. This leads to a range of applications of biunitaries in various 2-categories to quantum and classical information theory.
- Biunitary constructions in quantum information (with Jamie Vicary).
Higher Structures (2019), to appear.
» We use the interpretation of Hadamard matrices, unitary error bases and quantum Latin squares as biunitaries to obtain many new construction methods for these quantities. An extended abstract can be found here.
- Shaded tangles for the design and verification of quantum programs (with Jamie Vicary).
Submitted for publication (2017).
» We give a tangle-based graphical language for the description of quantum circuits such that isotopic tangles yield equivalent programs. This leads to several new protocols and topological insight into known phenomena such as error correcting codes and local inversion of cluster states.
- A classical groupoid model for quantum networks (with Jamie Vicary).
Log. Meth. Comput. Sci. (2017), to appear.
» We describe a new combinatorial 2-category and use it to model a classical network architecture allowing the execution of several quantum-like tasks such as teleportation, dense coding and secure key distribution.
- October 2018. Invited talk "From quantum automorphisms to pseudo-telepathy" at the workshop Cohomology of quantum groups and quantum automorphism groups of finite graphs, at the University of Saarbrücken.
- September 2018. Contributed talk "A compositional approach to quantum functions" at the First Symposium on Compositional Structures (SYCO 1) at the University of Birmingham.
- May 2018. Invited talk "Pseudo-telepathy via graphs and fusion categories" at the IOP's nonlinear and complex physics group's spring meeting on graph theory and physics at Imperial College, London.
- April 2018. Invited talk "Hopf algebras and 3-categories" at the University of Cambridge's junior geometry seminar.
- March 2018. Invited talk "Quantum functions and the Morita theory of quantum graph isomorphisms" at the workshop Combining viewpoints in quantum theory at the ICMS Edinburgh. A video of the talk can be found here.
- March 2018. Invited talk "Higher algebra in quantum information theory" at Stanford University's quantum information seminar.
- August 2017. Invited talk "Frobenius algebras, Hopf algebras and 3-categories" at the conference Hopf algebras in Kitaev's quantum double models at the Perimeter Institute, Canada. A video of my talk can be found here.
- July 2017. Contributed talks "Biunitary constructions in quantum information" and "Shaded tangles for the design and verification of quantum programs" at QPL 2017. Videos of my talks can be found here and here.
- June 2017. Contributed talks "A 2-categorical approach to composing quantum structures" and "A classical groupoid model for quantum networks" at CALCO 2017. Slides can be found here.
- January 2017. Contributed talk "Biunitary constructions in quantum information" at QIP 2017, the top international quantum information conference. Slides and video of the talk can be found here and here.
With Jamie Vicary, I have developed a public engagement workshop, called "Build your own quantum computer!", where participants can explore quantum concepts—including superposition, entanglement and teleportation—using hand-held electronic qubit simulators.
- April 2018. Three public workshops at the Department of Physics, Oxford, as part of the event Marie Curious — Girls Exploring Science (with Fatimah Ahmadi).
- February 2018. Two public workshops at University College, Oxford as part of a mathematical sciences study day for high-achieving Year 12 students (with Dominic Verdon).
- January 2018. Two public workshops at the Radley college STEMFest for students from local schools (with Dominic Verdon).
From 2010 to 2014, I did a Bachelor's and Master's degree in physics at ETH Zürich, mainly focussing on mathematical physics and geometry. In Zürich, I wrote a semester thesis on mutual information quantities associated to a generalized quantum entropy and a master's thesis on geometric constructions of conformal blocks. I then went on to do Part III in mathematics at the University of Cambridge where I wrote an essay on the Atiyah-Singer index theorem and got interested in category theory and its applications to physics. Since 2015, I have been a DPhil student in the quantum group.