In algebraic topology, we study geometric objects using algebraic invariants. One important invariant is the homotopy groups of spheres, which gives information about how to build geometric objects out of spheres. In chromatic homotopy theory, we study the homotopy groups of spheres by analyzing periodic behavior that appears. My research is on the interaction of this periodic behavior and an invariant of algebraic objects, called algebraic K-theory. A priori, one may not expect a connection, but the proof of an important conjecture related to number theory, called the Lichtenbaum-Quillen conjecture, gives the first evidence for such a relationship. I am also interested in how there might be a similar connection when we also take into account added symmetries.