Joakim Arnlind: Noncommutative algebras related to Poisson structures on the intersection of hypersurfaces

Noncommutative versions of manifolds can be defined in many different ways, depending on both aim and original motivation. In this talk, I will define a Poisson structure on the intersection of d-2 hypersurfaces, where each hypersurface is described as the zero set of a real polynomial in d variables. A noncommutative algebra A is then defined via a set of relations, which are obtained from the structure of the Poisson algebra on the intersection, and A may be regarded as a "noncommutative coordinate ring". For specific families of hypersurfaces, it turns out that the (hermitian) representation theory of A respects certain geometric properties of the intersection. Apart from some remarks on the general low-degree cases, I will present a family of hypersurfaces (of varying geometry) that provides an example where one can explicitly study the relationship between geometry and representation theory.