We consider the problem of reconstructing energies, momenta, and masses in
collider events with missing energy, along with the complications introduced by
combinatorial ambiguities and measurement errors. Typically, one reconstructs
more than one value and we show how the wrong values may be correlated with the
right ones. The problem has a natural formulation in terms of the theory of
Riemann surfaces. We discuss examples including top quark decays in the
Standard Model (relevant for top quark mass measurements and tests of spin
correlation), cascade decays in models of new physics containing dark matter
candidates, decays of third-generation leptoquarks in composite models of
electroweak symmetry breaking, and Higgs boson decay into two tau leptons.Comment: 28 pages, 6 figures; version accepted for publication, with
discussion of Higgs to tau tau deca