New examples of quantum projective spaces

I will discuss how to construct a large collection of “quantum projective spaces”, in the form of Koszul, Artin-Schelter regular quadratic algebras with the Hilbert series of a polynomial ring. I will do so by starting with the toric ones (the $q$-polynomial algebras), and then deforming their relations using a diagrammatic calculus, proving unobstructedness of such deformations under suitable nondegeneracy conditions. Time permitting, I will show that these algebras coincide with the canonical quantizations of corresponding families of quadratic Poisson structures. This provides new evidence to Kontsevich's conjecture about convergence of his deformation quantization formula. This is joint work with Brent Pym and Travis Schedler.