This talk is about my joint work with Po Hu and Petr Somberg. "Brave new algebra", a.k.a. "derived algebraic geometry", refers (among other things) to the development of concepts of algebra and algebraic geometry where the ring of integers is replaced by "the stable sphere" S. I will discuss some new work in this area, namely Lie algebras and algebraic groups over S, their representations, and some aspects of Harish-Chandra theory. The specific concepts treated are guided by constructing a knot invariant called sl_k Khovanov homotopy type at a prime large relative to k. I will discuss the history of what led to this problem, and the current version of its solution. Speaker(s): Igor Kriz (University of Michigan)