Current research in mathematics involves a wide variety of interlocking ideas, old and new. For example, results about the curves and surfaces defined by polynomial equations, as in algebraic geometry, appear in the study of solitary waves and also in the gauge theories in physics. Centuries-old problems in number theory have been solved, while others have been revealed as insoluble. The classification of all finite simple groups is nearly achieved (and the full treatment will be voluminous); the representation of groups aids in their application to the study of symmetry. These developments, and many others, attest to the vitality of mathematics.