We shall discuss a recent joint work with Makoto Kawashima and Noriko Hirata-Kohno. For any set of algebraic numbers in a fixed number field K satisfying standard metric conditions in the theory (close enough to zero), we prove that the values of polylogarithms are linearly independent over K. This is done via a construction of an explicit system of Padé approximation. We shall also discuss extensions to lerch functions

Introductory reading:

http://www.fields.utoronto.ca/sites/default/files/uploads/transcend.pdf

http://www.fields.utoronto.ca/sites/default/files/uploads/HP-ankara.pdf

http://www.fields.utoronto.ca/sites/default/files/uploads/CMI122009.pdf

http://www.fields.utoronto.ca/sites/default/files/uploads/ASENS_1899_3_1...