50 Years of translation structures

Abstract

In this paper the term translation structure will denote any geometric object canonically constructed from an elementary abelian group. Hence, translation weak affine spaces, translation planes, linear sets, translation ovoids of polar spaces, translation generalized quadrangles and linear MRD-codes are examples of translation structures. I will present my personal excursus in the theory of finite translation structures, from the pioneering times of B. Segre, A. Barlotti and G. Tallini to these days. My aim is not to give a complete survey of the known results but to describe the influence of translation structures in proofs or disproofs of some conjectures in different areas of Finite Geometries and how valuable tool they were giving significant contributions in different areas of Discrete Mathematics.