Lectures on cohomology, T-duality, and generalized geometry

These are notes for lectures, originally entitled "Selected Mathematical Aspects of Modern Quantum Field Theory", presented at the Summer School "New Paths Towards Quantum Gravity", Holbæ k, Denmark, 10-16 May 2008. My aim for these lectures was to introduce a mixture of physics and mathematics postgraduate students into a selection of exciting new developments on the interface of mathematics and quantum field theory. This write-up covers three topics: (1) cohomology and differential characters, (2) T-duality, and (3) generalized geometry. The three chapters can be read, more or less, independent of each other, but there is a common central theme, namely the occurrence of a (local) 2-form gauge field in certain quantum fields theories, the so-called B-field, which plays a role analogous to the electromagnetic gauge field. The notes are suitable for beginning postgraduate students in mathematical physics with some background in differential geometry and algebraic topology, but some sections may need a slightly more sophisticated background. I hope these notes fill a gap between undergraduate coursework and current research at the cutting edge of the field. The notes certainly do not offer an exhaustive discussion of the topics mentioned above, but rather serve as an introduction after which the reader should feel comfortable to study research papers in these areas.