We present a strong connection between quantum information and the theory of quantum permutation groups. Specifically, we define a notion of quantum isomorphisms of graphs based on quantum automorphisms from the theory of quantum groups, and then show that this is equivalent to the previously defined notion of quantum isomorphism corresponding to perfect quantum strategies to the isomorphism game. Moreover, we show that two connected graphs X and Y are quantum isomorphic if and only if there exists x ∈ V (X) and y ∈ V (Y ) that are in the same orbit of the quantum automorphism group of the disjoint union of X and Y . This connection links quantum groups to the more concrete notion of nonlocal games and physically observable quantum behaviours. In this work, we exploit this by using ideas and results from quantum information in order to prove new results about quantum automorphism groups of graphs, and about quantum permutation groups more generally. In particular, we show that asymptotically almost surely all graphs have trivial quantum automorphism group. Furthermore, we use examples of quantum isomorphic graphs from previous work to construct an infinite family of graphs which are quantum vertex transitive but fail to be vertex transitive, answering a question from the quantum permutation group literature.Our main tool for proving these results is the introduction of orbits and orbitals (orbits on ordered pairs) of quantum permutation groups. We show that the orbitals of a quantum permutation group form a coherent configuration/algebra, a notion from the field of algebraic graph theory. We then prove that the elements of this quantum orbital algebra are exactly the matrices that commute with the magic unitary defining the quantum group. We furthermore show that quantum isomorphic graphs admit an isomorphism of their quantum orbital algebras which maps the adjacency matrix of one graph to that of the other.We hope that this work will encourage new collaborations among the communities of quantum information, quantum groups, and algebraic graph theory.

PrefaceAnalogy is one of the most effective techniques of human reasoning: When we face new problems we compare them with simpler and already known ones, in the attempt to use what we know about the latter ones to solve the former ones. This strategy is particularly common in Mathematics, which offers several examples of abstract and seemingly intractable objects: Subsets of the plane can be enormously complicated but, as soon as they can be approximated by rectangles, then they can be measured; Uniformly finite metric spaces can be difficult to describe and understand but, as soon as they can be approximated by Hilbert spaces, then they can be proved to satisfy the coarse Novikov's and Baum-Connes's conjectures.These notes deal with two particular instances of such a strategy: Sofic and hyperlinear groups are in fact the countable discrete groups that can be approximated in a suitable sense by finite symmetric groups and groups of unitary matrices. These notions, introduced by Gromov and Rȃdulescu, respectively, at the end of the 1990s, turned out to be very deep and fruitful, and stimulated in the last 15 years an impressive amount of research touching several seemingly distant areas of mathematics including geometric group theory, operator algebras, dynamical systems, graph theory, and more recently even quantum information theory. Several long-standing conjectures that are still open for arbitrary groups were settled in the case of sofic or hyperlinear groups. These achievements aroused the interest of an increasing number of researchers into some fundamental questions about the nature of these approximation properties. Many of such problems are to this day still open such as, outstandingly: Is there any countable discrete group that is not sofic or hyperlinear? A similar pattern can be found in the study of II 1 factors. In this case the famous conjecture due to Connes (commonly known as Connes' embedding conjecture) that any II 1 factor can be approximated in a suitable sense by matrix algebras inspired several breakthroughs in the understanding of II 1 factors, and stands out today as one of the major open problems in the field.The aim of this monograph is to present in a uniform and accessible way some cornerstone results in the study of sofic and hyperlinear groups and Connes' embedding conjecture. These notions, as well as the proofs of many results, are here presented in the framework of model theory for metric structures. We believe that this point of view, even though rarely explicitly adopted in the literature, can contribute to a better understanding iii iv of the ideas therein, as well as provide additional tools to attack many remaining open problems. The presentation is nonetheless self-contained and accessible to any student or researcher with a graduate-level mathematical background. In particular no specific knowledge of logic or model theory is required.Chapter 1 presents the conjectures and open problems that will serve as common thread and motivation for the rest of the survey: Connes' em...

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AcknowledgementsThe collaboration between the authors first began when they participated in an American Institute of Mathematics (AIM) Structured Quartet Research Ensemble (or SQuaRE) program together with Renling Jin, Steven Leth, and Karl Mahlburg. We thus want to thank AIM for all of their support during our three year participation in the SQuaRE program as well as their encouragement to organize a larger workshop on the subject. A preliminary version of this manuscript was distributed during that workshop and we want to thank the participants for their valuable comments. In particular, Steven Leth and Terence Tao gave us a tremendous amount of feedback and for that we want to give them an extra expression of gratitude. i ii INTRODUCTION iii and measurable sets in a dynamical system), with the extra feature that the dynamical system obtained perfectly reflects all the combinatorial properties of the set that one started with. The achievements of the nonstandard approach in this area include the work of Leth on arithmetic progressions in sparse sets, Jin's theorem on sumsets, as well as Jin's Freiman-type results on inverse problems for sumsets. More recently, these methods have also been used by Jin, Leth, Mahlburg, and the present authors to tackle a conjecture of Erdős concerning sums of infinite sets (the so-called B +C conjecture), leading to its eventual solution by Moreira, Richter, and Robertson.Nonstandard methods are also tightly connected with ultrafilter methods. This has been made precise and successfully applied in recent work of Di Nasso, where he observed that there is a perfect correspondence between ultrafilters and elements of the nonstandard universe up to a natural notion of equivalence. On the one hand, this allows one to manipulate ultrafilters as nonstandard points, and to use ultrafilter methods to prove the existence of certain combinatorial configurations in the nonstandard universe. One the other hand, this gives an intuitive and direct way to infer, from the existence of certain ultrafilter configurations, the existence of corresponding standard combinatorial configurations via the fundamental principle of transfer in the nonstandard method. This perspective has successfully been applied by Di Nasso, Luperi Baglini, and co-authors to the study of partition regularity problems for Diophantine equations over the integers, providing in particular a far-reaching generalization of the classical theorem of Rado on partition regularity of systems of linear equations. Unlike Rado's theorem, this recent generalization also includes equations that are not linear.Finally, it is worth mentioning that many other results in combinatorics can be seen, directly or indirectly, as applications of the nonstandard method. For instance, the groundbreaking work of Hrushovski and Breuillard-Green-Tao on approximate groups, although not originally presented in this way, admit a natural nonstandard treatment. The same applies to the work of Bergelson and Tao on recurrence in quasirandom groups.The goal of t...

Abstract. We realize the Jiang-Su algebra, all UHF algebras, and the hyperfinite II 1 factor as Fraïssé limits of suitable classes of structures. Moreover by means of Fraïssé theory we provide new examples of AF algebras with strong homogeneity properties. As a consequence of our analysis we deduce Ramsey-theoretic results about the class of full-matrix algebras.