Hamiltonian reduction arose as a mechanism for reducing complexity of systems in mechanics, but it also provides a tool for constructing complicated but interesting objects from simpler ones. I will illustrate how this works in representation theory and algebraic geometry via examples. I will describe a new structure theory, motivated by Hamiltonian reduction (and in particular the Morse theory that results), for some categories (of D-modules) of interest to representation theorists. I will then explain how this implies a modified form of "hyperkahler Kirwan surjectivity" for the cohomology of certain Hamiltonian reductions. The talk will not assume that members of the audience know the meaning of any of the above-mentioned terms. The talk is based on joint work with K. McGerty.
