The second part of the project studies supersymmetry breaking in string theory

compactifications with fluxes, with the purpose of finding a fully satisfactory de Sitter solution to string theory.

Among the topics are elusive worldsheet instantons in heterotic string

compactifications, the Witten equation and the geometry of the Landau-Ginzburg model, algebraic topological string theory, the fibrancy of symplectic homology in cotangent bundles, and the theory of higher rank stable pairs and virtual localization.

From subsequent investigations concept of grills has shown to be a powerful supporting and useful tool like nets and filters, further we get a deeper insight into studying some topological notions such as proximity spaces, closure spaces and the theory of

compactifications and extension problems of different kinds.

The concept of grills has shown to be a powerful supporting and useful tool like nets and filters, for getting a deeper insight into further studying some topological notions such as proximity spaces, closure spaces and the theory of

compactifications and extension problems of different kinds ([2], [3], [9]).

The first seven chapters cover the usual topics of point-set or general topology, including topological spaces, new spaces from old ones, connectedness, the separation and countability axioms, and metrizability and paracompactness, as well as special topics such as contraction mapping in metric spaces, normed linear spaces, the Frechet derivative, manifolds, fractals,

compactifications, the Alexander subbase, and the Tychonoff theorems.

Let (Y, f) and (Z, g) be two

compactifications of a topological space X.

In order to describe the physics at infinity we will recur to Penrose's ideas [12] of conformal

compactifications of Minkowski spacetime by attaching the light-cones at conformal infinity.

Ultrafilters, extensions of continuous functions, rings of continuous functions, uniformities,

compactifications, and measure theoretic techniques have all been profitably employed to provide methods for defining realcompactness.

Also for the investigations of many topological notions like

compactifications, proximity spaces, theory of grill topology was used.

Then they explore some applications of the algebra to the part of combinatorics known as Ramsey Theory and demonstrate its connections with topological dynamics, ergodic theory, and the general theory of semigroup

compactifications.

of Michigan) and their contributors cover a wide range of subjects, including Gromov-Witten theory, mirror symmetry in complex and symplectic geometry, and important ramifications in enumerative geometry, with specific papers addressing strong coupling, brane-induced gravity, heterotic geometry influxes, pure Yang-Mills theory, black hole microstates, gravitational singularity, recent developments in heterotic

compactifications, instantons and torsion curves.

In order to describe the physics at infinity we will recur to Penrose's ideas [10] of conformal

compactifications of Minkowski spacetime by attaching the light-cones at conformal infinity.