David Reutter

Postdoctoral Fellow

Max Planck Institute for MathematicsVivatsgasse 7, 53111 Bonnemail: [lastname]@mpim-bonn.mpg.de


I am a postdoctoral fellow at the Max Planck Institute for Mathematics in Bonn. Previously, I was a DPhil student at the University of Oxford under the supervision of Jamie Vicary and the Strauch postdoctoral fellow at the Mathematical Sciences Research Institute (MSRI) in Berkeley for the spring 2020 programs Higher categories and categorification and Quantum symmetries. A brief version of my CV can be found here.

I am interested in applications of higher category theory to — and its interplay with — low-dimensional topology, homotopy theory and mathematical physics. I am currently working on various constructions of 4-dimensional topological field theories and related questions in quantum algebra. Another focus of my work is on applying ideas from low-dimensional category theory and homotopy theory to quantum information theory.

My research interests include:

    • higher category theory

    • topological field theory

    • quantum algebra (fusion categories, subfactors, quantum groups, . . .)

    • quantum information theory & its relations to topological field theory and noncommutative mathematics

On August 2019, I successfully defended my DPhil thesis on `Higher linear algebra in topology and quantum information theory', which can be found here.


11) A 3-categorical perspective on G-crossed braided categories

With Corey Jones and David Penneys

A braided monoidal category may be considered a 3-category with one object and one 1-morphism. In this paper, we show that, more generally, 3-categories with one object and 1-morphisms given by elements of a group G correspond to G-crossed braided categories, certain mathematical structures which have emerged as important invariants of low-dimensional quantum field theories. More precisely, we show that the 4-category of 3-categories C equipped with a 3-functor BG→C which is essentially surjective on objects and 1-morphisms is equivalent to the 2-category of G-crossed braided categories. This provides a uniform approach to various constructions of G-crossed braided categories.

arXiv 2020

10) A type theory for strictly unital ∞-categories

With Eric Finster and Jamie Vicary

We present a type theory for strictly unital ∞-categories, in which a term computes to its strictly unital normal form. Using this as a toy model, we argue that it illustrates important unresolved questions in the foundations of type theory, which we explore. Furthermore, our type theory leads to a new definition of strictly unital ∞-category, which we claim is stronger than any previously described in the literature.

arXiv 2020

9) Semisimple 4-dimensional topological field theories cannot detect exotic smooth structure

We prove that semisimple 4-dimensional oriented topological field theories lead to stable diffeomorphism invariants and can therefore not distinguish homeomorphic closed oriented smooth 4-manifolds and homotopy equivalent simply connected closed oriented smooth 4-manifolds. We show that all currently known 4-dimensional field theories are semisimple, including unitary field theories, and once-extended field theories which assign algebras or linear categories to 2-manifolds. As an application, we compute the value of a semisimple field theory on a simply connected closed oriented 4-manifold in terms of its Euler characteristic and signature. Moreover, we show that a semisimple 4-dimensional field theory is invariant under ℂP^2-stable diffeomorphisms if and only if the Gluck twist acts trivially. This may be interpreted as the absence of fermions amongst the `point particles' of the field theory. Such fermion-free field theories cannot distinguish homotopy equivalent 4-manifolds. Throughout, we illustrate our results with the Crane-Yetter-Kauffman field theory associated to a ribbon fusion category. As an algebraic corollary of our results applied to this field theory, we show that a ribbon fusion category contains a fermionic object if and only if its Gauss sums vanish.

arXiv 2020

8) Uniqueness of unitary structure for unitarizable fusion categories

A unitary fusion category is a fusion category over the complex numbers with a compatible positive dagger structure. Such unitary structures naturally arise in many applications and constructions of fusion categories, most notably in the context of operator algebras, subfactor theory, and mathematical physics. Not every fusion category admits a unitary structure, and a fusion category could, in principle, have more than one unitary structure. Especially in light of the recent powerful operator algebraic classification techniques of unitary fusion categories, the question of uniqueness of unitary structure has become a significant open problem. In this paper, we prove that every unitarizable fusion category, and more generally every semisimple C*-tensor category, admits a unique unitary structure. Our proof is based on a categorified polar decomposition theorem for monoidal equivalences between such categories.

arXiv 2019

7) High-level methods for homotopy constructions in associative n-categories

With Jamie Vicary.

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. We develop a general contraction principle in ANCs which forms the mathematical heart of homotopy.io, allowing string diagram manipulations in arbitrary dimensions.

arXiv 2019

Proceedings of the 34th Annual ACM/IEEE Symposium on Logic in Computer Science, 2019 (LICS 2019)

6) Fusion 2-categories and a state-sum invariant for 4-manifolds

With Christopher Douglas.

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 oriented singular piecewise-linear 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.

arXiv 2018

5) The Morita theory of quantum graph isomorphisms

With Benjamin Musto and Dominic Verdon.

Using the machinery developed in `A compositional approach to quantum functions', 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.

arXiv 2018

Communications in Mathematical Physics, 365(2):797-845, 2019

4) A compositional approach to quantum functions

With Benjamin Musto and Dominic Verdon.

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.

arXiv 2018

Journal of Mathematical Physics, 59(8):081706, 42, 2018

3) Shaded tangles for the design and verification of quantum programs

With Jamie Vicary.

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. We analyze 11 known quantum procedures in this way -- including entanglement manipulation, error correction and teleportation -- and in each case present a fully-topological formal verification, yielding generalized procedures in some cases.

arxiv 2018

Proceedings of the Royal Society A, 475(2224):20180338, 2019

2) A classical groupoid model for quantum networks

With Jamie Vicary.

We describe a certain combinatorial 2-category and use it to give a mathematical analysis of a new type of classical computer network architecture, intended as a model of a new technology that has recently been proposed in industry. Our approach is based on groubits, generalizations of classical bits based on groupoids. This network architecture allows the direct execution of a number of protocols that are usually associated with quantum networks, including teleportation, dense coding and secure key distribution.

arXiv 2017

Logical Methods in Computer Science, 15(1), 2019

1) Biunitary constructions in quantum information theory

With Jamie Vicary.

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. In this paper, 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.

arXiv 2016

Higher Structures, 3(1):109–154, 2019