He is a professor at Complutense University of Madrid (UCM)[1] and a member of the Spanish Royal Academy of Sciences.

In 1998, he co-founded the journal Revista Matemática de la UCM and served on its editorial board from 1988 to 1995.

[3] He founded the Department of Applied Mathematics at the Facultad de Matemáticas of UCM in the early 1980s and led it for several years.

He has also worked in areas closer to pure mathematics, such as nonlinear analysis, focusing on accretive operators, rearrangement and gradient estimates.

His popular science works include 43 articles, two memoirs, two contributions to published proceedings and 17 chapters in books.

[5] Díaz was instrumental in the organization of the Spanish Congress of Differential Equations and Applications, an annual national meeting held from 1978 to 1987.

[11] From September 2003 to June 2014 he was the only Spanish person on the editorial board of the Journal of the European Mathematical Society.

[2] The meeting took place on June, 14-17, 2011, in the Palacio de Lorenzana, (UCLM) in Toledo, Spain (Ildefonso's birthplace).