The first IMUB colloquim of 2024 will take place at IMUB on the 21st of February, 2024.

Títol: E-functions
Abstract: The numbers e and pi are not roots of non-zero polynomials with rational coefficients. This is a special case of one of the most beautiful results of transcendence theory: the Hermite-Lindemann-Weierstrass theorem. Its proof at the end of the 19th century led to the question of how to obtain transcendence results for special functions other than the exponential. A case of interest was the Bessel function, whose zeros express the vibration modes of a circular membrane. With this goal in mind, Siegel introduced the notion of E-function in a 1929 paper which was a turning point for number theory. E-functions are power series which solve a differential equation and whose coefficients satisfy certain growth conditions of arithmetic nature. I will explain their history through many examples and then move to some of the most recent developments, including my joint solution with Peter Jossen to one of the problems in Siegel’s original paper.

Brief Bio: Javier Fresán is currently a professor at Sorbonne Université. He obtained his PhD thesis in 2013 from Paris 13 and was a postdoctoral fellow at MPIM Bonn and ETH Zürich, before joining École polytechnique as a Monge Professor in 2017. His contributions to number theory and algebraic geometry, namely to the study of periods, exponential motives and E-functions, have been recognised by the Vicent Caselles Award in 2015 and the Bronze Medal of the CNRS and a Frontiers of Science Award in 2023.