His early work was in
cobordism theory in
algebraic topology; this includes his 1959 Cambridge PhD thesis entitled "Algebraic aspects of cobordism", written under the direction of
Frank Adams and
Christopher Zeeman. His research was then mainly in the area of
manifolds, particularly
geometric topology and related
abstract algebra included in
surgery theory, of which he was one of the founders. In 1964 he introduced the
Brauer–Wall group of a field. His 1970 research monograph "Surgery on Compact Manifolds" is a major reference work in geometric topology. In 1971 he
conjectured that every
finitely generated group is
accessible. The conjecture motivated much progress in the understanding of splittings of
groups. In 1985
Martin Dunwoody proved the conjecture for the class of
finitely presented groups. The resolution of the full conjecture took until 1991 when, surprising to most mathematicians at the time, Dunwoody found a finitely generated group that is not accessible and hence the conjecture turned out to be not correct in its general formulation. Wall's work since the mid-1970s has mostly been in
singularity theory as developed by
R. Thom,
J. Milnor and
V. Arnold, and especially concerns the classification of
isolated singularities of
differentiable maps and of
algebraic varieties. He has written two research monographs on singularity theory, "The Geometry of Topological Stability" (1995) (containing a great deal of original work) with Andrew du Plessis, and "Singular Points of Plane Curves" (2004). His notable students include
Michael Boardman,
Bill Bruce,
Andrew Casson,
Francis E. A. Johnson,
David Mond, Andrew du Plessis, and
David Trotman. == Awards ==