Im starting to learn about modeling of moving interfaces and am feeling daft about the basic definition itself:
Given an $n$ dimensional space $\Omega$ an interface $\Gamma$ is a co-dimension 1 object that splits the domain into two distinct regions.
Ok so if $\Omega$ is a 2d domain, $\Gamma$ is a 1 dimensional object, i.e. a line that splits the domain.
The problem I have is with the motivating example, a circle in a 2d domain. Isn't a circle a 2d object, so it shouldn't count as an interface since its not codim 1? Or is it because we can parameterize the circle with just 1 degree of freedom that makes it 'count'?
It's not clear where you get the idea that a circle is a 2D object (or what exactly you mean by that). There are various notions of dimension, but usually if one can parametrize something by a function $\mathbb{R}^m\to\mathbb{R}^n$, one would see it as an $m$-dimensional surface in $\mathbb{R}^n$; alternatively, if a locus is defined by a set of $m$ equations in $n$ real variables, it can be seen as $n-m$ dimensional. A circle in $\mathbb{R}^2$ is of course defined by one equation as in $\{(x,y)\in\mathbb{R}^2:x^2+y^2=b\}$. There are other more sophisticated notions of dimension.