Cherednik Algebras at Infinity

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Many people delight in the formal elegance the symmetry can bring to a design. Less well known is the 19th century mathematical discovery that symmetry forms the framework for Geometry. Later, 20th century physicists discovered that these same ideas lie at the heart of our understanding of the Universe. There is a special branch of Mathematics, called Representation Theory, which studies symmetries revealed (or sometimes concealed) by objects and phenomena in Mathematics and other fields of Science including Biology, Chemistry and Physics. That branch is the area of the project. The aim of the project is to study particular objects, called Cherednik algebras or double affine Hecke algebras. They appeared about 20 years ago as a result of blending ideas from the branch of modern Theoretical Physics called Conformal Field Theory, and from a more traditional branch of Mathematics, called the Theory of Special Functions. Since then the Cherednik algebras were found to play major roles in several other branches of Mathematics. We aim to advance the theory of these algebras, and to find more links between Mathematics and Theoretical Physics.