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In these lectures, we examine how twisted generalized Calabi-Yau (GCY) manifolds arise in the construction of a general class of topological sigma models with non-trivial three-form flux. The topological sigma model defined on a twisted GCY can be regarded as a simultaneous generalization of the more familiar A-model and B-model. Emphasis will be given to the relation between topological observables of the sigma model and a Lie algebroid cohomology intrinsically associated with the twisted GCY. If time permits, we shall also discuss topological D-branes in this more general setting, and explain how the viewpoint from the Lie algebroid helps to elucidate certain subtleties even for the conventional A-branes and B-branes. The lectures will be physically motivated, although I will try to make the presentation self-contained for both mathematicians and physicists.

The strange paradoxes and puzzles of the quantum behaviour of black holes and the things that fall into them led to a spirited battle of ideas between Stephen Hawking, Leonard Susskind and other scientists. Resolving the debate may change our entire understanding of space, time, matter and information – is the entire world, for example, a quantum hologram? black hole, Leonard Susskind, space-time, Einstein, Hawking, complementarity, uncertainty principle, holographic principle, string theory,