Brent Pym, University of Oxford
"Holomorphic Poisson brackets and noncommutative geometry''
One of the basic problems in ring theory is to classify noncommutative algebras that behave like polynomial rings, or like rings of functions on smooth manifolds. A possible strategy is to start with a commutative algebra, and ask how to deform the product in order to make it noncommutative. To first order, such a deformation is given by a geometric structure known as a Poisson bracket. The "deformation quantization" theorem of Kontsevich shows that, in principle, one can recover the deformed algebra from knowledge of the Poisson bracket alone. In practice, applying this technique requires a thorough understanding of the geometry of the Poisson bracket, and elliptic curves often feature prominently. I will give an overview of this approach, and describe several recent results concerning the classification of holomorphic Poisson brackets on projective space and their associated noncommutative algebras. For the projective space of dimension three, we obtain the solution of a problem that has been open since the late 1980s: the classification of Artin--Schelter regular deformations of the polynomial ring in four variables.