Quantum field theory (QFT) is an outstandingly successful description of matter at short distance or high energy scales. However, its mathematical status is problematic: while it is possible to write down a short list of properties (or axioms) that seem reasonable assumptions on any QFT, no one has succeeded in giving a mathematically precise construction of a QFT on four-dimensional Minkowski spacetime that obeys all the assumptions and has nontrivial interactions. This may indicate that we do not fully understand what QFT really is and that our current understanding is tied too closely to classical field theory. Algebraic QFT (AQFT) focusses attention on the algebras of local observables with very few additional assumptions. It has been very fruitful in both understanding the general structures of QFT and in developing new mathematics in operator algebra theory. Furthermore, it provides a viewpoint that is ideal for studying QFT in curved spacetimes. Research in York covers AQFT in both flat and curved spacetimes, including new constructions of QFTs in low dimensions that have recently come out of the AQFT programme, and new mathematical frameworks for perturbative constructions of interacting theories.
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