David Reutter
David Reutter
Room
204,
Wolfson Building,
Parks Road, Oxford OX1 3QD
United Kingdom
Interests
I am interested in the applications of higher category theory to lowdimensional topology, homotopy theory and mathematical physics.
Upcoming
 From September to December 2019 and from June 2020, I will be a postdoc at the MaxPlanck 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.
Research
1. Fusion 2categories and 4dimensional topological field theories
One of the early successes of quantum topology was the construction of a threedimensional topological field theory by TuraevViro, and its generalization by BarrettWestbury, starting from the data of a spherical fusion category. In this work, we introduce spherical fusion 2categories and construct a statesum invariant of 4manifolds, which categorifies the TuraevViroBarrettWestbury invariant. There are many examples of fusion 2categories, including the 2category of module categories of a braided fusion category, the 2category of 2representations of a finite 2group and twisted linearizations of finite 2groups.
 Fusion 2categories and a statesum invariant of 4manifolds (with Christopher Douglas).
arXiv:1812.11933
» We introduce semisimple 2categories, fusion 2categories, and spherical fusion 2categories. For each spherical fusion 2category, we construct a statesum invariant of oriented singular piecewiselinear 4manifolds.
2. Associative ncategories and homotopy.io
The theory of associative ncategories (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.
 Highlevel methods for homotopy construction in associative ncategories (with Jamie Vicary).
arXiv:1902.03831
» 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 pseudotelepathy
Pseudotelepathy is a phenomenon in quantum information, where two noncommunicating parties can use preshared entanglement to perform a task classically impossible without communication. In this work, we uncover a connection between pseudotelepathy and quantum symmetry, expressed in the language of fusion categories. We use this connection to classify instances of pseudotelepathy 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 2categorical framework for finite quantum set theory.

A compositional approach to quantum functions (with Benjamin Musto and Dominic Verdon).
arXiv:1711.07945
J. Math. Phys. 59, 081706 (2018). doi: 10.1063/1.5020566
» We describe a 2categorical approach to finite quantum set and quantum graph theory.

The Morita theory of quantum graph isomorphisms (with Benjamin Musto and Dominic Verdon).
arXiv:1801.09705
Commun. Math. Phys. (2018), to appear. doi: 10.1007/s0022001832256
» We classify instances of quantum pseudotelepathy 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 grouptheoretical 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 2categories to quantum and classical information theory.
 Biunitary constructions in quantum information (with Jamie Vicary).
arXiv:1609.07775
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).
arXiv:1805.01540
Submitted for publication (2017).
» We give a tanglebased 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).
arXiv:1707.00966
Log. Meth. Comput. Sci. (2017), to appear.
» We describe a new combinatorial 2category and use it to model a classical network architecture allowing the execution of several quantumlike tasks such as teleportation, dense coding and secure key distribution.
Talks
 October 2018. Invited talk "From quantum automorphisms to pseudotelepathy" 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 "Pseudotelepathy 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 3categories" 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 3categories" 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 2categorical 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.
Public Engagement
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 handheld electronic qubit simulators.
 June 2018. Two public workshops at the Hay festival (with Jamie Vicary).
 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 highachieving Year 12 students (with Dominic Verdon).
 January 2018. Two public workshops at the Radley college STEMFest for students from local schools (with Dominic Verdon).
 June 2017. Two public workshops at the Hay festival (with Jamie Vicary).
 June 2017. Two public workshops at the Cheltenham science festival (with Jamie Vicary).
Biography
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 AtiyahSinger 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.