Representation Theory Across The Channel

Lead Research Organisation: University of Aberdeen
Department Name: Mathematical Sciences

Abstract

A central theme in mathematics is to study symmetries: An object or a structure is symmetrical if it looks the same after a specific type of change is applied to it. This very general concept occurs in a huge variety of situations, ranging from classical geometry through computer science and on tonatural phenomena in physics and chemistry, e.g., crystal structures. Perhaps the most familiar symmetrical objects are the five Platonic solids, or regular polyhedra. Mathematicians study such symmetries, and their higher-dimensional analogues, by investigating the corresponding symmetry group , that is, the possible transformations of the object which preserve the symmetry.Initially, representation theory was concerned with the study of properties of such symmetry groups via their realisations by linear transformations of vector spaces. This idea was soon extended to include other mathematical structures, such as associative algebras, and Lie and Hopf algebras. The original theory was mainly concerned with representations over the fields of real or complex numbers. New phenomena and difficulties, as well as new connections with other areas (like number theory), arose through the study of representations obtained by a process of reducing modulo aprime number. The introduction of geometric methods, for example the linearisation of group actions by means of cohomology theories, has revolutionised the field. It led to a flow of new ideas and results between several disciplines. The theoretical development has been driven by an abundance of challenging basic problems, like finding character formulae for irreducible representations, whichform the building blocks for all representations. Major open conjectures have lead to vast research programmes, the most prominent of which probably is the Langlands Programme. In this way, representation theory has become one of the most vibrant fields of mathematics today, deeply connected to a variety of areas such as group theory, ring theory, number theory, geometry, algebraic topology, computational algebra, integrable systems, category theory, combinatorics, and mathematical physics. Given this background, it is essential to provide researchers (of all ages!) with regular opportunities for interdisciplinary interaction, the chance to develop a broader vision of the fundamental underpinnings and future directions in representation theory, and to learn about neighbouring disciplines.The idea for launching this network arose from discussions between EPSRC and its Frenchcounterpart, the CNRS, mediated by the British Embassy in Paris. The plan is to ``twin'' this proposed network with two already existing networks on representation theory in France, for the mutual benefit of mathematicians in both countries, especially for supporting and encouraging mobility of early career researchers. Thus, the proposed network is much more ambitious than already existing frameworks for regionalor interregional cooperation; it is truly national in scale, interdisciplinary in nature, with an addedinternational component. It is expected to have a significant positive impact on the quality of research in the UK. The proposed activities include annual instructional conferences, 1-2 day meetings spread geographically and scientifically, invited lecture series, public interest lectures, and, last but not least, short visit schemes (both for visits within the UK and between France and the UK).

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