Introduction to Mochizuki's works on Inter-universal Teichmuller theory

Abstract. Inter-universal Teichmuller theory, as developed by Shinichi Mochizuki in the past decade, is an analogue for number fields of the classical Teichmuller theory, and also of the p-adic Teichmuller theory of Mochizuki. In this theory, the ring structure of a number field is subject to non-scheme theoretic deformation. Absolute anabelian geometry, a refinement of anabelian geometry, plays a crucial role in inter-universal Teichmuller theory. In this talk, we will try to give an introduction to these ideas, using analogy from the theory of hyperbolic manifolds. Although F_1-geometry is not used in Mochizuki's works, some of the ideas involved could be regarded as a manifestation of some kind of F_1-geometry, for example the crucial use of monoid structures.