This series of three lectures aims to introduce the audience to a bunch of related problems in hyperkahler (quaternionic, N=4 supersymmetric) geometry. In the first lecture I start with listing several famous and by now classical results from Kahler (complex, N=2 supersymmetric) geometry. I quickly go through the definition of Kahler manifolds, the Kahler quotient construction, Hodge theory and the Hard Lefschetz theorem and index theory on them. I finish with three exciting applications. The first is Stanley's solution of McMullen's conjecture about the classification of face vectors of simple convex polytopes (as an application of the Hard Lefschetz theorem). The other two are 2+1 dimensional Topological Quantum Field Theories. One leads to the Casson invariant of three manifolds (as an application of intersection theory), the other (as an application of an index theorem) leads to the Jones polynomial and its generalization the Reshetikhin-Turaev-Witten invariant. These invariants are cooked out from the cohomology of a particular Káhler manifold: the moduli space of stable bundles on a complex curve. This first talk of the lecture series serves to explain the Kahler package that I am intending to carry through in the hyperkahler setting.

Completely opposite to the first talk, this second one will be a thorough introduction to hyperkahler manifolds with lots of examples. After some general ideas and definitions about hyperkahler manifolds I will go through the simplest hyperkahler manifolds, namely 4-dimensional gravitational instantons. In particular I will explain the Gibbons-Hawking ansatz, which gives the multi-Eguchi-Hanson metrics and multi-Taub-NUT metrics, they are all 4-dimensional hyperkahler manifolds. The next topic of the talk will be a more general method of constructing hyperkahler manifolds: the hyperkahler quotient construction. As an example I will detail the toric case, that is the construction of toric hyperkahler varieties; the simplest of this family, the Calabi metric on the cotangent bundle to CP^n will be treated explicitely. Finally, if time permits, I may talk about the moduli space of Higgs bundles, which also has a natural hyperkahler metric.

This is the third of three talks.

This is the first of three talks.