A polynomial f over a finite field k is said to be a permutation polynomial if it induces a bijection on k. A polynomial is then said to be exceptional if this permutation property holds for infinitely many extensions of k. A useful criterion then states that we can identify when a polynomial is exceptional by looking at absolutely irreducible factors of f(x)-f(y) in k[x,y].
We will then aim to generalise this situation from polynomials over finite fields to morphisms of schemes, and then later to morphisms of difference schemes. This will involve defining a more sophisticated notion of exceptionality, then formulating and proving a criterion in our new context.
We will have tea and biscuits in the common room from 10am, the talk will start at 10:30. Please bring a mug if you can.