FOM: What is "Core Mathematics"?

[Joe Shipman]

Professor Insall's description of "Core Mathematics" as being primarily
related to the mathematics used in physics may be adequate to describe
undergraduate curricula, but the "core mathematics" that professional
mathematicians (in my opinion) have in mind is significantly different.
They would characterize "core mathematics" so as to include four major
areas (Analysis, Algebra, Topology, Number Theory) together with all
those parts of mathematics which are (logically) strongly connected to
them.  This includes mathematical physics of various kinds (classical
mechanics, operator theory, real and complex analysis, Riemannian
geometry, Lie groups, ODE's and PDE's), but also Number Theory (e.g.
Wiles's work), Algebraic and Differential Topology (e.g. proof of the
Poincare conjecture in dim > 3), Abstract Algebra (e.g. classification
of finite simple groups), and several "bridge" fields (measure theory,
homological algebra, convex and discrete geometry) that would be in the
"convex hull" of the four major areas.

The criterion for being included in "core mathematics" is an internal
one, based on strength of connections with other branches of "core
mathematics".  Fields like numerical analysis, set theory, logic,
combinatorics, and theory of algorithms are seen as impinging upon core
mathematics in peripheral rather than essential ways.  This perception
may be mistaken (in the case of model theory in particular the
connections to core mathematics are being taken more seriously
nowadays), and the implicit identification of "closeness to the core" as
a standard of value is certainly wrong, but I think the notion of an
INTRINSIC core is a reasonable one, even if it is dependent on what
mathematics has actually been done (connections we don't yet understand
may eventually be discovered that will make previously peripheral areas
appear central).

-- Joe Shipman