My own area of research is fractal geometry. In fractal geometry, there are various notions of dimension (Hausdorff, box, Minkowski, Assouad, packing, complex etc.). Very roughly speaking, the dimension of a set in a metric space describes how the "volume" of a set scales when the set is scaled, where "volume" may be in terms of a measure (e.g. in the case of the Minkowski dimension), in terms of ball-counting functions (e.g. the box and Assouad dimensions), or in terms of some other generalized notion of volume. These analytic notions of dimension help to characterize "fractals", are related to certain embedding problems in the study of metric spaces, and appear in the study of dynamics.
I follow the dimension-theory tag, and am interested in it from the point of view of fractal geometry.
On the other hand, there seems to be some notion of algebraic dimension theory. This is an area where I am far from expert---I got through my graduate algebra class, but the material never really clicked for me (in the sense that I never found it all that engaging). As far as I can tell, algebraic notions of dimension seek to generalize the idea of a basis or generating set for a vector space. I don't really grok what these algebraic notions of dimension are all about, though I can recognize that questions in algebraic dimension theory are very different from the kinds of questions that are in my wheelhouse.
Finally, there are a lot of very elementary questions which show up in the dimension-theory tag which really don't belong. These tend to be of the type "What is the dimension of this subspace of a vector space?" These questions should probably be tagged linear-algebra (and nothing else, probably). I think that they end up getting tagged with dimension-theory because the askers start typing "dimen..." and see a tag which they think is appropriate, without actually reading the tag description.
###Proposal###
I would be more than happy to divide the dimension-theory tag. I might suggest the creation of two new tags:
- dimension-theory-analysis, for questions about topological and metric notions of dimension (e.g. Hausdorff and box-counting dimension), and
- dimension-theory-algebra, for questions about algebraic notions of dimension (e.g. dimension of a vector space, Krull dimension).
This split makes the roles of the tags more clear, and may help to prevent novice users from using inappropriate tags.