Quantum groups in action

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Symmetry is first encountered in geometry, for example the rotational and reflectional symmetry of regular polygons, or the three dimensional symmetries of polyhedra. In higher mathematics, symmetry is a core guiding principle, from symmetries of fields in Galois theory through to the use of symmetry in counting arguments in Combinatorics. This project has its roots in both algebra, through the notion of a group, an abstraction of the notion of a symmetry, and in analysis, through the study of continuity of symmetries. This can be seen by thinking about the rotational symmetries of a circle, for example, where it makes sense to speak of two rotations being "close together" or not.Quantum groups arose almost simultaneously from quantum physics (in the study of the quantum inverse scattering method) and from abstract harmonic analysis, as part of a quest to generalise Pontraygin duality to non-abelian groups. In modern mathematics, one often does not study a mathematical object directly, but instead studies maps between mathematical objects. This leads to the study of algebras, in our case, the group algebra. When the group is commutative (the order of symmetry does not matter, which is rare in examples from geometry) the two main group algebras are linked through the Fourier Transform. When the group is the circle group, this is the idea underpinning signal processing, a ubiquitous tool in digital communication, which allows one to transform a continuous signal into discrete data, reflected in the Mathematical fact that the "dual group" to the circle is the integers.For non-commutative groups, one has to give up on the "dual group" actually existing. Instead, it is represented by a different group algebra. Through the study of Kac algebras, Compact Quantum Groups, and motivated by various examples motivated from Quantum Physics, we have now arrived at the definition of a Locally Compact Quantum Group, which encompasses all known examples. While this study has its roots in simply understanding better actual groups, there are now many examples of genuinely "quantum groups": objects which are not real groups, but clearly have something to do with symmetry.This project explores a wide gamut of questions from quantum group theory. We will study structural properties of locally compact quantum groups: their "quantum subgroups". Symmetries in geometry are the study of group "actions" where a group "acts" on a geometric object. We will study actions of quantum groups, in particular, a class of quantum groups (the complex semisimple quantum groups) which arise directly from quantization, and classical groups, than other examples. Our aim is to use this study to work on the Baum-Connes conjecture for such quantum groups. An important mathematical use of groups is in providing examples of algebraic objects. Quantum groups give rise to a wide variety of Operator Algebras, and we will study properties of these algebras, in particular various "approximation properties". Finally, we will study some questions from Quantum Information Theory. As is common in mathematics, relations between diverse areas arise in unexpected ways. While quantum groups originally arose in part from Quantum Physics, in recent years quite different aspects of quantum groups (for example, the "combinatorial quantum groups" which at first glance have little to do with physics) have appeared in the study of Quantum Information Theory (itself related to the foundations of Quantum Computation). It is these links, and links with our other work, which we shall explore.