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Tuesday, April 8, 2014

**Abstract:** n the early 80's, Yau posed the problem of establishing the rigidity of the classical singularity theorems of Hawking and Penrose. This lead to the introduction of the Busemann function and its horospheres (level sets) in the Lorentzian setting. While a Lorentzian splitting theorem was successfully established by the end of the decade, the original rigidity question remained, and was formulated concretely in a splitting conjecture by Bartnik in '88. We will discuss a new approach to (generalized) Lorentzian horospheres, and a generalized splitting theorem, with applications to Bartnik's conjecture. This is joint work with G. Galloway.