'Algebraic independence of the values of Mahler functions'

The theory of Mahler functions is one of the most successful subjects in transcendental number theory. For proving the algebraic independence of the values of Mahler functions, it is necessary to prove the algebraic independence of the Mahler functions themselves. In many cases the algebraic independence of the functions implies that of their values by using some criteria. K. Nishioka developed the study of Mahler functions of one variable by using p-adic numbers and Nesterenko's theory. On the other hand, we have many open problems in the case of several variables. To solve these problems, I will construct new criteria for the algebraic independence of Mahler functions by applying Nesterenko's theory, and I intend to prove the transcendence and the algebraic independence of new numbers.