Abstract: We try to investigate a generalization of the Heisenberg commutation relation ${[p,q]=-i \hslash}$, introduced by Chamseddine, Connes and Mukhanov as ``the one-sided and the two-sided quantization equations'', which captures the geometry. The momentum variable $p$ is encoded by the Dirac operator and the analogue of the position variable $q$ is the Feynman slash of real scalar fields over a closed even-dimensional spin manifold. Existence of a solution of the one-sided equation implies that the manifold decomposes into a disconnected sum of spheres of unit volume which represent quanta of geometry. The two-sided equation, as the refined version of the one-sided equation by involving the real structure on a spin manifold, implies the quantization of the volume of the spin manifold.

Abstract: The purpose of this talk will be to illustrate how various abstract techniques from algebraic geometry (i.e. cohomology, Riemann Roch) can be used to study algebraic surfaces. Algebraic surfaces are smooth projective varieties over \mathbb{C} of dimension 2. After reviewing the basics of linear systems and divisors on surfaces, we will study morphisms determined by linear systems on the Hirzebruch surfaces, a particularly nice class of algebraic surfaces. Depending on time, other applications and results will be described, such as the relation of Hirzebruch surfaces to the Enriques-Kodaira classification or the classication of degree n-1 nondegenerate surfaces in P^{n}.