Abstract: There is a deep and fascinating connection between the ring of integer numbers and the ring of univariate polynomials over a finite field.

In this talk I will discuss the classical theory, and I will present a new approach based on Galois theory and Field Arithmetic.

I will demonstrate the method by solving a function field version of the classic problem on primes in short intervals.

Abstract: Understanding the rich structure of the absolute Galois group of the field $Q$ of rational numbers is a central goal in number theory.

Following Serre's question, the Sylow subgroups of the absolute Galois group of the fields $Q_p$ of $p$-adic numbers were studied and completely understood by Labute. However, the structure of the $p$-Sylow subgroups of the absolute Galois group of $Q$ is much more subtle and mysterious.

We shall discuss the first steps towards its determination via a surprisingly simple decomposition.

Abstract: Last week, I outlined an approach due to Mathai and Quillen which uses Clifford algebras to give an explicit computation of the Chern character of a superconnection on the level of differential forms, in which we see the emergence of a Gaussian-shaped Thom form. This week, I will explain how one can simplify the construction using equivariant differential forms, and obtain a 'universal' Thom form as the result. Time permitting, I'll explain how this construction, which is also due to Mathai and Quillen, can easily be extended to define the equivariant characteristic classes that appear in the cohomological equivariant index.

Abstract: The Euclidean division is a basic tool when dealing with the ordinary integers. It does not extend to rings of integers of algebraic number fields in general. It is natural to ask how to measure the "deviation" from the Euclidean property, and this leads to the notion of Euclidean minimum. The case of totally real number fields is of special interest, in particular because of a conjectured upper bound (conjecture attributed to Minkowski). The talk will present some recent results concerning abelian fields of prime power conductor. We will also define Euclidean minima for function fields and give some bounds for this invariant. We furthermore show that the results are analogous to those obtained in the number field case.