Nonultralocality and new mathematical structures in quantum integrability

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The crucial natural question to ask of a problem in mathematical physics is whether we can solve it. Since most such problems are couched in the language of differential equations, the question is therefore of whether we can integrate the equations, of whether the problem is 'integrable'. The generalized mathematics of integrability is that of the structures which make some problems solvable in a way that others are not. Perhaps surprisingly, it turns out that fundamental physics makes much use of integrable models. The mathematical analysis of the central issue in modern fundamental physics, the relationship between the gauge theories of particle physics and the much more speculative string theory (the 'gauge/string correspondence'), has proved to be full of integrable models in recent years - but whereas conventional integrability refers to time-evolution, the crucial integrability in gauge theory is with the evolution of the energy scale.Some integrable models are 'ultralocal' - that is, very well-behaved in terms of the localization of their interactions. Most of the mathematical techniques of quantum integrability apply principally to such models. Less well-behaved, 'nonultralocal' (non-UL) models are harder to handle, but keep re-appearing and increasing in importance. Sometimes their problems can be accommodated in a rather ad-hoc way, but a systematic understanding of them is lacking. This project attacks the problem from a number of directions. Some involve attempting to understand better the algebraic structures underlying known non-UL models, with or without a boundary, especially those of, or generalized from, the gauge/string correspondence. However, a new departure is to take existing ideas for handling non-UL models in the abstract and attempt 'reverse-engineer' the associated non-UL integrable models.