Some thoughts on algebraic $K$-theory of topological algebras

Algebraic $K$-theory of (primarily real) topological algebras has been the subject of numerous studies, especially its relationship with topological (aka operator) $K$-theory. I first give the motivation behind my thesis project (the modified Hodge conjecture) and then explain how the formalism of $K$-theory of analytic rings encourages a reexamination of this topic. Subsequently, I illustrate how fundamental tools in condensed mathematics give us a useful picture. Time permitting, I will discuss one outcome from my study, which confirms previously unverified claims of Rosenberg made in 1990.