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Tuesday, October 20, 2015

**Abstract:** Algebraic K-theory is a spectral invariant of module categories with applications to number theory and manifold geometry. Recently, various people have used the technology of $\infty$-categories to establish universal characterizations for K-theory. Many of the basic structural results about K-theory have been elevated to apply in the $\infty$-categorical context. I will describe Waldhausen's sphere theorem, a new analogous result for the algebraic K-theory of stable $\infty$-categories, and some applications of the new theorem.