Planar functions are functions defined over finite fields of odd characteristics such that their (nonzero) derivatives are permutations. They have connections with various areas of mathematics with applications in coding theory, cryptography, combinatorics, etc.
In this seminar, we will introduce planar functions, their connection with commutative semifields, and some equivalence notions used to classify them. We will recall a conjecture of 1968 by Dembowski and Ostrom stating that all planar functions are quadratic and we will see some recent results about this conjecture.