摘要: In 1957, in order to formalize his work on Riemann-Roch theorem, Grothendieck introduced the K-theory in algebraic geometry. In 1959, Atiyah and Hirzebruch constructed the topological K-theory as the real analogue. In 1968, using the topological K-theory, Atiyah and Singer gave a new proof of the famous index theorem as a part of the topological version of the Riemann-Roch theorem. In 1990, in order to extend the Arakelov theory to the higher dimension, Gillet and Soule extended the K-theory to the arithmetic K-theory in arithmetic algebraic geometry. In 2008, Gillet, Roessler and Soule proved the famous arithmetic Riemann-Roch theorem in Arakelov geometry. In this talk, we will discuss the real analogue of the arithmetic K-theory, called the differential K-theory. It is a new research field in 21 century which is motivated by the study of the superstring theory in theoretical physics. We will mainly focus on the comparison of these K-theories and study the corresponding Riemann-Roch theorem in differential K-theory. In particular, we will explain the Lefschetz formula in differential K-theory, which is a joint work with Xiaonan Ma recently.