We try to investigate a generalization of the Heisenberg commutation re- lation [p; q] = ô€€€i}, 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 elds over a closed even-dimensional spin manifold. Existence of a solution of the one-sided equation implies that the manifold decomposes into a dis- connected sum of spheres of unit volume which represent quanta of geometry. The two-sided equation, as the rened 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.

I shall give a detailed construction of the BPST instantons.

Continuing the previous talk, I will present some properties of (-1)- and 0-types; propositions and sets.

We introduce polynomially convex subsets of the n-dimensional Euclidean complex space and expose some of their key properties. We discuss the presence of an analytic structure in the polynomially convex hull of a compact set, Rossi's local maximum principle and Oka's characterization theorem. This talk is given in fulfillment of the requirements for Part II of the PhD comprehensive examination.

We introduce polynomially convex subsets of the n-dimensional Euclidean complex space and expose some of their key properties. We discuss the presence of an analytic structure in the polynomially convex hull of a compact set, Rossi's local maximum principle and Oka's characterization theorem.

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.

In this talk we will introduce the theory combinatorial games, a simple mathematical structure with incredibly rich algebraic properties. As well as containing all real numbers, the class of games contains all ordinals, a collection of infinitesimals and plenty in between. The study of combinatorial games can be applied directly to the analysis of actual strategy games, including Chess and Go. If time permits, we will use the techniques of the talk to analyse a curious chess endgame.

Given a d-dimensional convex polytope, its k-th face number is the number of (k 􀀀-1)-dimensional faces it has. The f-vector of a polytope is the sequence of its face numbers. Beside Euler's formula, these numbers satisfy further equalities and inequalities. Characterization of f-vector of d-dimensional convex polytope is already known for d ≤ 3 . For d ≥ 3, we do not have a complete answer, but g-theorem gives us a characterization for simplicial (and dually simple) case. Here we review g-theorem and its various proofs.