On the \(K\)-theory of pullbacks and pushouts

A 2.5-hour expository talk given for zygotop in Spring 2023, a learning seminar for young graduate students interested in homotopy theory and related areas. I gave an overview of the Land-Tamme paper “On the \(K\)-theory of pushouts” shortly after it was uploaded to the arXiv. I stated and proved their main results, namely that we can measure the failure of excision for localising invariants (think \(K\)-theory) via the \(\odot\)-construction, and that this \(\odot\)-construction can be expressed as a pushout in \(\text{Alg}(k)\) in good circumstances. This makes it computable in many situations of interest. The talk finished with a tour of some applications.