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Tuesday, October 29, 2019

**Abstract:** Homotopy groups and stable homotopy groups of spaces are main invariants in algebraic topology. Homotopy groups are very powerful but difficult to compute in practice. Stable homotopy groups, on the other hand, are easier to work with, at the expense of losing unstable information. Structured ring spectra are spectra with certain algebraic structure encoded by the action of an operad O. For such O-algebras, the analog of stable homotopy groups is played by Topological Quillen (TQ) homology groups. In this talk, we will discuss the following question: What information can be seen by TQ-homology? In particular, we will discuss TQ-localization of O-algebras and show the TQ-Whitehead theorem for homotopy pro-nilpotent O-algebras.