ABHYANKAR 82nd BIRTHDAY CONFERENCE SPEAKER: Nick Inglis (Edinburgh) TITLE: Small polynomials for big groups ABSTRACT: The title is, of course, an homage to Abhyankar's paper "Nice polynomials for nice groups", which was near the start of a sequence of papers in which explicit polynomials were given in non-zero characteristic, whose Galois groups were (almost) simple groups. These polynomials were small in the sense that they had few non-zero terms. We revisit polynomials affording the classical (linear, symplectic, orthogonal and unitary) groups over finite fields and aim to show, using tools such as E. H. Moore's generalisation of Fermat's little theorem, and Abhyankar's mantra, how much of the structure of these big groups is naturally encoded in these small polynomials.