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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?

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3 Answers 3

Given the context, I guess that by 'graduate level' you mean 'currently taught at the graduate level,' which of course raises the question 'where?' As you invariably know, different universities (esp. in different countries) take very different approaches in defining "grad-level stuff." Also, do you mean 'Math sophomore?' I guess you do.

Those objections aside, I'd say anything from various applied topics in diff. eqs./dyn. syst. theory. For example, huge chunks of the Lin & Segel book 'Mathematics Applied to Deterministic Problems in the Natural Sciences' incl. (several) multiscale notions, heat conduction & Sturm-Liouville theory (which used to be there but got pushed out as it got older). I also tend to think that much of the theory in classic ODE books (Hale, Coddington & Levinson) would be accessible to such a student, but then again sophomore classes still focus on explicit solutions for special cases (not without reason, but still).

As for Calculus for the Life Sciences, which you mention, no worries: there's a strong drive towards powering up the math curriculum in the Life Sciences (& vice versa!). It might take one generation to fully appreciate the difference, but it'll happen.

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Complex analysis without excess concern for analytical niceties like interchange of integrals and sums and such, notoriously has a high ratio of interesting results to set-up costs. Applications to evaluation of integrals by residues, summing series by residues, as well as more serious things (zeta function, elliptic functions, modular forms) whose introductory aspects require little preparation.

A version of "theory of equations" in which Lagrange resolvents are emphasized and cyclotomic polynomials used as significant example does not need much preparation after an awareness of complex numbers.

As @JoeJohnson126 notes, homology with coefficients in $\mathbb Z/2$ avoids the issue of motivating or explaining the signs in boundary maps and orientations and such. Alexandroff wrote a very slim book on algebraic topology from this viewpoint.

The non-Euclidean geometries of spheres and hyperbolic spaces can be studied with only a little linear algebra and ideas of calculus, rather than imbedding the topics in general Riemannian geometry and Lie theory.

(Yes, it does seem to be the case that the undergrad math curriculum all-too-often acquires a dreary, tedious air, since the good stuff has been pushed later, on grounds of "logical ordering"... the problem being that this tends to push all the motivating examples years later, which is ridiculous.)

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Geometries and Groups by Nikulin & Shafarevich (a delightful book) even studies spherical and hyperbolic geometry with just high school background. –  Marius Kempe Mar 26 '14 at 4:51

You can do a bit of homology with $\mathbb{Z}/2$ coefficients, as in the book "A Combinatorial Introduction to Topology".

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Perhaps you might elaborate a bit. –  Erik G. Jan 21 '13 at 0:32

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