From topological Hochschild homology to algebraic K-theory

Algebraic K-theory lies at the intersection of algebraic topology and algebraic geometry, and gives deep information about things as far apart as number theory and the geometry of manifolds. Unfortunately it is extremely difficult to compute in all but the simplest cases. One way forward goes through stable homotopy theory, a subfield of algebraic topology. Topological Hochschild homology serves as an approximation to algebraic K-theory, and is more computable. We propose to study this approximation in a number of cases to gain an increased understanding of algebraic K-theory.