Abstract: Let R be a finitely generated algebra over a field. Two natural questions are (i) how many generators are needed to express R as an algebra over that field (ii) given two sets of N generators of the algebra R over the field, does there exist a change of variables in the polynomial algebra in N variables (that is, an automorphism of the polynomial algebra) which expresses one generating set in terms of the other one?

These questions can be naturally interpreted in the language of algebraic geometry, as questions about affine varieties, and their embeddings in affine space. This lecture will give an introduction to this topic, touching on results of Abhyankar, Nori, and others, including myself.

Abstract: The first row $(a_1, \ldots, a_n)$ of an $n \times n$  invertible matrix has by Laplace expansion the property that  there exist $b_1, \ldots, b_n$ with $a_1b_1 + a_2b_2 + \ldots + a_nb_n = 1$. We shall discuss whether a row $(a_1, \ldots, a_n)$ which has the property that there is a row $(b_1, \ldots, b_n)$ with $a_1b_1 + a_2b_2 + \ldots + a_nb_n = 1$ can be completed to an invertible matrix.

Abstract: I will speak on my results on  Fermat-Wilson congruences and connections with arithmetic derivatives and zeta values, as this can be explained within framework of  Abhyankar's "high school math" preference!