Algebraic K-theory is a very general cohomology theory for rings, schemes (and more) which single-handedly captures information about numerous other invariants: algebraic cycles, class groups, étale cohomology, special values of zeta functions, etc. The main modern method through which it is studied is to approximate it via cyclic and topological cyclic homology; these latter theories are closer to de Rham, crystalline, and prismatic cohomology. In this talk I will provide an introduction to algebraic K-theory, including some of the main historical examples, before overviewing more recent advances.
