I was browsing through a book entitled: Field Theory and Its Classical Problems by Charles Hadlock, one of the Carus Mathematical Monographs titles. In his preface he says the following:
In particular, this book presents an exposition of those portions of classical field theory which are encountered in the famous geometric construction problems of antiquity and the problem of solving polynomial equations by radicals. Some time ago much of this material was covered in undergraduate courses in the 'theory of equations'. Paradoxically, as the theory matured and became more elegant, it moved higher into the curriculum, so that nowadays it is not uncommon for it first to be encountered at the graduate level. It seems to me that this is most unfortunate, for this important and beautiful area of mathematics deserves to be studied early. It can then lend perspective and motivation to the later abstract study of mathematical structures.
This is an arresting paragraph, because it suggests that there is a distinction between a logical ordering of the material in the undergraduate curriculum and its ordering as a field that is driven by interesting questions and problems, the view of which is highly obscured if seen at all in the typical undergraduate curriculum. This will, of course, vary with the school and instructor, but in general, the standard undergraduate curriculum is boring, and to the uninitiated, there is something deadening in the way the material is presented.
So, the question is: What areas of graduate level mathematics can be made accessible to, say, a sophomore, after a rigorous calculus sequence?
You can do a bit of homology with $\mathbb{Z}/2$ coefficients, as in the book "A Combinatorial Introduction to Topology".