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The International Centre for Mathematical Sciences (ICMS)

The origin of quantum representation theory dates back to Drinfeld and Jimbo’s discovery of quantum groups in 1985. From the outset, the mathematical theory was inextricably linked with various aspects of mathematical physics, including, prominently, the inverse scattering method and quantum integrability. In the 1990s, quantum representation theory was revolutionized by the introduction of powerful geometric methods, pioneered by Nakajima. At the same time, an interplay of algebraic and geometric approaches gave rise to the influential theory of q,t-characters of quantum affine algebras. Since 2000, the field has branched off in multiple directions, with seemingly endless applications.

The aim of the workshop is to bring together the world’s leading experts to discuss new developments in quantum representation theory, with focus on the following thematic areas: (1) quantum symmetric pairs, (2) quantum affine and toroidal algebras, (3) categorification and geometric approaches.

The International Centre for Mathematical Sciences (ICMS)

The interplay between geometry and physics has been a driving force for new developments in both mathematics and physics for a very long time. In the last two decades, higher category theory has become a fine contributor in this dialogue, providing many invaluable ideas and very useful heuristics. Organised in partnership with the Clay Mathematics Institute, this workshop aims to present the latest results in the application of (higher) geometry to theoretical physics and to foster new collaborations between mathematicians and theoretical physicists.

Isaac Newton Institute for Mathematical Sciences

The International Centre for Mathematical Sciences (ICMS)

The UK has always been at the forefront of, and has a reputation in, integrable systems research. It is an area that glues together topics in mathematical physics, analysis, algebra, geometry, applied mathematics and probability. Research in non-commutative integrable hierarchies has had a recent resurgence – they have many applications, including (eg.) in non-commutative nonlinear optics, and have strong links to string theory. The main themes outlined align with, and will be led, by recent research by the invitees. We propose a different than usual workshop format, along the lines of those held at Oberwolfach and the American Institute of Mathematics. Nominated experts give feeder presentations in their areas on successive mornings, outlining the current frontier of research and open problems. In the afternoons, these are followed up by discussion sessions and working groups who report back to the workshop as a whole at the end of the day. The workshop structure develops dynamically throughout the week. The organisers and the Scientific Advisory Group meet daily to set the agenda ahead, using feedback from the end-of-day reports and participants.

(i) Regularity and solutions; (ii) Non-commutativity, both in terms of non-Abelian (eg.\/ matrix) versions of classical integrable systems and in terms of non-commutative independent variables; and (iii) The connections of such systems to random matrix theory.

The International Centre for Mathematical Sciences (ICMS)

The String Math conference has been running annually since 2011, bringing together mathematicians and physicists who work at the interface between string theory, quantum field theory, and mathematics.