"Calabi–Yau manifolds ... The geometry of a space is a local feature, Yau and Nadis explain, which turns out to be related to its topology, a global feature. ... A general manifold is complicated — indeed a full classification of their topology is impossible according to Go?del’s incompleteness theorem. ... Ricci-flat, source-free solutions. ... The geometry and topology of any extra dimensions figure in the nature and interactions of low-energy particles, the statistical mechanics of black holes, and the rate of expansion of the Universe. ... If one assumes that supersymmetry, a hypothetical spacetime symmetry relating bosons to fermions, exists at energy scales below the inverse size of string theory’s extra dimensions, this restricts attention to ten-dimensional backgrounds, with the six extra dimensions curled up in the form of a Calabi–Yau or similar space. ...  This possibility is to be tested by the Large Hadron Collider at CERN in Geneva. ... mirror symmetry, a physical equivalence in string theory between Calabi–Yau manifolds of different topology.... This also leads to topology- changing processes in which doughnuts can lose holes and Calabi–Yau manifolds of different topology can connect smoothly. Mirror symmetry makes calculations that are difficult in one Calabi–Yau space simple in terms of the equivalent, mirror manifold. ... Black holes form, merge and evaporate according to simple equations of the same form as the laws of thermodynamics, with the area of the event horizon in Planck units behaving like entropy. This invites a statistical interpretation, an enumeration of microstates of the black hole to be compared to its area. Using tractable (although unrealistic) black-hole solutions, string theorists have reproduced precisely this result. ... the inverse problem of trying to reconstruct a string background from observations is clearly difficult if not impossible. ... cosmological inflation requires knowledge of certain quantum-gravity corrections to the Lagrangian, with some string-theoretic models producing distinctive signatures. The late Universe has recently been determined to contain dark energy, the simplest model for which is Einstein’s original cosmological constant leading to late-time de Sitter spacetime. Its tiny magnitude remains mysterious, if not a selection effect (observers like us would not form in a generic, rapidly inflating Universe). Various string compactifications — including Calabi–Yau manifolds with additional sources — seem to produce a finely spaced set of possible values for the cosmological constant, which fits with this observation. ... " by Eva Silverstein who is in the Department of Physics and SLAC, Stanford University,