Auckland

Automatic structures are algebraic structures, such as graphs, groups
and partial orders, that can be presented by automata. By varying the 
classes of automata (e.g. finite automata, tree automata, omega-automata) 
one varies the classes of automatic structures. The class of all automatic 
structures is robust in the sense that it is closed under many natural
algebraic and model-theoretic operations.  
In this talk, we give formal definitions to 
automatic structures, motivate the study, present many examples, and
explain several fundamental theorems.  Some results in the area
are deeply connected  with algebra, additive combinatorics, set theory, 
and complexity theory. 
We then motivate and pose several important  unresolved questions in the
area.

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 $n$?

We consider a bounded connected open set

$\Omega \subset {\rm R}^d$ whose boundary $\Gamma$ has a finite

$(d-1)$-dimensional Hausdorff measure. Then we define the

Dirichlet-to-Neumann operator $D_0$ on $L_2(\Gamma)$ by form

methods. The operator $-D_0$ is self-adjoint and generates a

contractive $C_0$-semigroup $S = (S_t)_{t > 0}$ on

$L_2(\Gamma)$. We show that the asymptotic behaviour of

$S_t$ as $t \to \infty$ is related to properties of the

trace of functions in $H^1(\Omega)$ which $\Omega$ may or

may not have. We also show that they are related to the

essential spectrum of the Dirichlet-to-Neumann operator.

The talk is based on a joint work with W. Arendt (Ulm).