This is joint work with Angus Macintyre. I will discuss new developments in
our work on the model theory of adeles concerning model theoretic
properties of adeles and related issues on adelic geometry and number theory.
One of the most classical questions of modern algebra is whether the group algebra of a torsion-free group can be embedded into a skew field. I will give a short survey about embeddability of group algebras into skew fields, matrix rings and, in general, continuous rings.
Thoughts on the Burnside problem
Let $\Gamma$ be a group and let $r_n(\Gamma)$ denote the
number of isomorphism classes of irreducible $n$-dimensional complex
characters of $\Gamma$. Representation growth is the study of the
behaviour of the numbers $r_n(\Gamma)$. I will give a brief overview of
representation growth.
We say $\Gamma$ has polynomial representation growth if $r_n(\Gamma)$ is
bounded by a polynomial in $n$. I will discuss a question posed by
Brent Everitt: can a group with polynomial representation growth have
the alternating group $A_n$ as a quotient for infinitely many $n$?
In the 1930's E. Artin conjectured that a form over a p-adic field of degree d has a non-trivial zero whenever n>d^2. In this talk we will discuss this relatively old conjecture, focusing on recent developments concerning quartic and quintic forms.