PhD Supervision

I. Operator methods in non-local game theory

In a non-local game, players Alice and Bob aim to convince a third party, the Verifier, that they jointly possess a certain piece of information. The Verifier tests the players by asking them questions; upon receiving their answers, the Verifier checks whether they satisfy the rules of the game. In order to win (if at all possible), the players synchronise their responses using a pre-agreed strategy. It is known that some games cannot be won if the players use only classical resources; however, if they have access to quantum entanglement, they can do much better or even sometimes win perfectly the game. In the past several years is has become clear that non-local games can be studied using operator theory. The underlying idea is to associate with the game a certain operator theoretic object, for example a C*-algebra, that encodes in itself the properties of the game, and to reduce (at least a part of) its study to the study of the associated object. While this approach has proven very fruitful, a number of intriguing questions remain unexplored.

II. Non-commutative graphs and zero-error information theory

In zero-error information transmission, Alice aims at transmitting a message to Bob in a way that, despite the noise that affects the information channel she uses, he recovers the message without error probability of error. The first questions of this type were studied by the founder of information theory, Claude Shannon, in the 1950's. Shannon showed to each classical information channel one can associate a graph (called the confusability graph of the channel) whose graph theoretic properties describe completely the zero-error transmission properties of the channel. It was discovered in 2013 by Duan, Severini, and Winter that non-commutative confusability graphs of quantum (as opposed to classical) channels can still be defined and still capture to perfection the zero-error quantum transmission properties of the quantum channels, but that in this case, they are operator systems - self-adjoint spaces of matrices containing the identity matrix. Every classical graph gives rise to a canonical non-commutative graph, and the latter remembers the former up to a graph isomorphism. These observations set the start of the area of non-commutative combinatorics, which bridges operator system theory in functional analysis with quantum information theory and graph theory. The field is in the early stages of its development and many questions are still open. They can be divided in two classes: on one hand, one is interested to what extent classical combinatorics can be lifted to the quantum realm; on the other, one aims at describing the properties of zero-error capacities of quantum channels via their non-commutative confusability graphs. It is still outstanding, for example, to make a precise link between the problems and methods used in zero-error quantum information theory and the ones in non-local game theory despite the evidence for such relation arising from non-commutative graphs.


III. Interactions between operator algebras and harmonic analysis

The work of P. Eymard, W. Arveson and Th. Varopoulos in the 1960's and the 1970's uncovered the first substantive links between harmonic analysis and operator algebra theory. During the decades to follow, it became clear that many concepts in harmonic analysis, for example, spectral synthesis, uniqueness and multiplicity, have operator counterparts, and that there exists a precise and rigorous way to pass from the classical harmonic analysis version to the operator one. This phenomenon of operator transference has attracted considerable attention, and while well-understood in some cases, offers some interesting outstanding questions. In particular, many properties of the fundamental Herz-Schur multipliers and their operator counterpart, Schur multipliers, are still unexplored when these maps are considered in the more general setting of dynamical systems. Further, their study is at its start in the more general context of quantum groups (as opposed to classical non-commutative locally compact groups). In another direction, subsets of groups give rise to natural  group operator systems in a similar fashion groups yield group C*-algebras. Operator systems from discrete groups have been shown to be related to questions in quantum information theory. Operator systems of non-discrete groups have, on the other hand, not been explored at all. 

Present PhD students:

Georgios Baziotis,  "Measurable no-signalling correlations", 2022-present
Alexandros Chatzinikolaou, "Locality and contextuality via operator systems", 2022-present
Gage Hoefer, "Strategy transport for non-local games", 2021-present

Past PhD students:

Joseph L. Habgood, "Convergence properties of masa-bimodules", 2007
Martin McGarvey, "Normalisers of operator algebras", 2009
Savvas Papapanayides, "Properties of subspace lattices related to reflexivity", 2011
Naomi Steen, "Unbounded generalisations of operator multipliers", 2013
Linda Mawhinney, "Inductive limits of operator systems", 2016
Andrew McKee, "Multipliers of dynamical systems", 2017
Weijao He, "Compact multipliers and their properties", 2019
Gareth Boreland, "Non-commutative graphs in quantum information theory", 2020