Explore trace methods in algebraic K-theory in this 55-minute lecture by David Gepner, part of the "Cyclic Cohomology at 40: achievements and future prospects" series at the Fields Institute. Delve into the computational aspects of algebraic K-theory, focusing on the cyclotomic trace and its approximation of K-theory through cyclic homology. Examine the concept of topological cyclic homology (TC) and its significance when computed over the sphere. Learn about recent advances in understanding TC, attributed to Nikolaus and Scholze, and their impact on computational control. Investigate trace maps from algebraic K-theory and their applications in chromatic homotopy theory. Cover topics such as S-linearity, K-theory, stable infinity categories, cyclotomic spectra, and higher chromatic rings.