Green function are imprtotant tools part of every physicists arsenal. They appear every where fron electrodynamics to optics and fluid dynamics andto quantum field theory. Here is an extract briefly describing his times. He entered Cambridge as graduate student at the age of 40.
The Miller of Nottingham: The theory developed in this section first appeared in an 1828 memoir by George Green (1793-1841) entitled An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism. The Essay is remarkable, not least because the young Green received only one year of schooling before he was apprenticed to his father’s flour mill. Only at the age of 30 did Green join the Nottingham Subscription Library and discover higher mathematics through the works of Laplace, Lagrange, Legendre, and Poisson. Greatly stimulated, Green worked out his ideas and published his 72–page Essay (privately) for a subscription list of local notables. It received little attention and the discouraged Green returned to milling. A few years later, he followed the advice of friends and entered Cambridge as a 40-year-old undergraduate. Green graduated in 1837 and quickly published six papers on hydrodynamics, acoustics, and optics before returning abruptly to Nottingham in 1841. He died there at the age of 47, almost entirely unknown. William Thomson (later Lord Kelvin) found a copy of the Essay just after his own graduation from Cambridge in 1845. He arranged for its publication (in Germany) in 1850. Riemann coined the term “Green function” soon thereafter. The first widely seen edition in English was published in 1871.
Andrew Zangwill, "Modern ELectrodynamics" Cambridge University Press (2013)