Abstract: Recent decades saw a strong symbiosis between Mathematics and Theoretical High Energy Physics.

Contributions to mathematics include the following very partial list: link invariants, Seiberg-Witten invariants for 4 manifolds, Gromov-Witten invariants, Mirror symmetry between pairs of Calabi-Yau 3-folds (CY3) and resolution of singularities such as the flop transition in CY3. Applications in string theory got as far as K theory, non-commutative geometry, gerbes, sheaves, and even p-adic numbers.

The talk will present the general trend of modern physics to become more and more geometrical and will present a "road map" for orientation in physics. The talk will concentrate on the geometry of candidate compact manifolds in extra dimensions of spacetime and their relevant mathematical properties. Some of the main concepts that will be discussed are: fiber bundles and gauge theories, differential geometry and General Relativity, black holes, Supersymmetry (or ``super-mathematics''), String Theory, manifolds with reduced holonomy, Homology, and moduli spaces for embedded objects.