Martin Dunwoody
Encyclopedia
Martin John Dunwoody is an Emeritus Professor of mathematics
at the University of Southampton
, England.
He earned his Ph.D. in 1964 from the Australian National University
. He held positions at the University of Sussex
before becoming full Professor at the University of Southampton
in 1992. He has been Emeritus Professor since 2003.
Dunwoody works on geometric group theory
and low-dimensional topology
. He is a leading expert in splittings and accessibility of discrete groups
, Groups acting on graphs and trees, JSJ-decompositions
, the topology of 3-manifold
s and the structure of their fundamental group
s.
From 1971 on several mathematicians have been working on Wall's conjecture
, posed by Wall in a 1971 paper, which said that all finitely generated groups were accessible. Roughly, this means that every finitely generated group can be constructed from finite and one-ended
groups via a finite number of amalgamated free products
and HNN extension
s over finite subgroups. In view of the Stallings theorem about ends of groups
, one-ended groups are precisely those finitely generated infinite groups that cannot be decomposed nontrivially as amalgamated products or HNN-extensions over finite subgroups.
Dunwoody proved the Wall conjecture for finitely presented groups in 1985. In 1991 he finally disproved Wall's conjecture by finding a finitely generated group that is not accessible.
Dunwoody found a graph-theoretic proof of Stallings' theorem about ends of groups
in 1982, by constructing certain tree-like automorphism invariant graph decompositions. This work has been developed to an important theory in the book "Groups acting on graphs", Cambridge University Press, 1989, with Warren Dicks. In 2002 Dunwoody put forward a proposed proof of the Poincaré conjecture
. The proof generated considerable interest among mathematicians, but a mistake was quickly discovered and the proof was withdrawn. The conjecture was later proven by Grigori Perelman
, following the program of Richard Hamilton.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
at the University of Southampton
University of Southampton
The University of Southampton is a British public university located in the city of Southampton, England, a member of the Russell Group. The origins of the university can be dated back to the founding of the Hartley Institution in 1862 by Henry Robertson Hartley. In 1902, the Institution developed...
, England.
He earned his Ph.D. in 1964 from the Australian National University
Australian National University
The Australian National University is a teaching and research university located in the Australian capital, Canberra.As of 2009, the ANU employs 3,945 administrative staff who teach approximately 10,000 undergraduates, and 7,500 postgraduate students...
. He held positions at the University of Sussex
University of Sussex
The University of Sussex is an English public research university situated next to the East Sussex village of Falmer, within the city of Brighton and Hove. The University received its Royal Charter in August 1961....
before becoming full Professor at the University of Southampton
University of Southampton
The University of Southampton is a British public university located in the city of Southampton, England, a member of the Russell Group. The origins of the university can be dated back to the founding of the Hartley Institution in 1862 by Henry Robertson Hartley. In 1902, the Institution developed...
in 1992. He has been Emeritus Professor since 2003.
Dunwoody works on geometric group theory
Geometric group theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act .Another important...
and low-dimensional topology
Low-dimensional topology
In mathematics, low-dimensional topology is the branch of topology that studies manifolds of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. It can be regarded as a part of geometric topology.A number of...
. He is a leading expert in splittings and accessibility of discrete groups
Stallings theorem about ends of groups
In the mathematical subject of group theory, the Stallings theorem about ends of groups states that a finitely generated group G has more than one end if and only if the group G admits a nontrivial decomposition as an amalgamated free product or an HNN extension over a finite subgroup...
, Groups acting on graphs and trees, JSJ-decompositions
Bass–Serre theory
Bass–Serre theory is a part of the mathematical subject of group theory that deals with analyzing the algebraic structure of groups acting by automorphisms on simplicial trees...
, the topology of 3-manifold
3-manifold
In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions...
s and the structure of their fundamental group
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
s.
From 1971 on several mathematicians have been working on Wall's conjecture
C. T. C. Wall
Charles Terence Clegg Wall is a leading British mathematician, educated at Marlborough and Trinity College, Cambridge. He is an emeritus professor of the University of Liverpool, where he was first appointed Professor in 1965...
, posed by Wall in a 1971 paper, which said that all finitely generated groups were accessible. Roughly, this means that every finitely generated group can be constructed from finite and one-ended
Stallings theorem about ends of groups
In the mathematical subject of group theory, the Stallings theorem about ends of groups states that a finitely generated group G has more than one end if and only if the group G admits a nontrivial decomposition as an amalgamated free product or an HNN extension over a finite subgroup...
groups via a finite number of amalgamated free products
Free product
In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “most general” group having these properties...
and HNN extension
HNN extension
In mathematics, the HNN extension is a basic construction of combinatorial group theory.Introduced in a 1949 paper Embedding Theorems for Groups by Graham Higman, B. H...
s over finite subgroups. In view of the Stallings theorem about ends of groups
Stallings theorem about ends of groups
In the mathematical subject of group theory, the Stallings theorem about ends of groups states that a finitely generated group G has more than one end if and only if the group G admits a nontrivial decomposition as an amalgamated free product or an HNN extension over a finite subgroup...
, one-ended groups are precisely those finitely generated infinite groups that cannot be decomposed nontrivially as amalgamated products or HNN-extensions over finite subgroups.
Dunwoody proved the Wall conjecture for finitely presented groups in 1985. In 1991 he finally disproved Wall's conjecture by finding a finitely generated group that is not accessible.
Dunwoody found a graph-theoretic proof of Stallings' theorem about ends of groups
Stallings theorem about ends of groups
In the mathematical subject of group theory, the Stallings theorem about ends of groups states that a finitely generated group G has more than one end if and only if the group G admits a nontrivial decomposition as an amalgamated free product or an HNN extension over a finite subgroup...
in 1982, by constructing certain tree-like automorphism invariant graph decompositions. This work has been developed to an important theory in the book "Groups acting on graphs", Cambridge University Press, 1989, with Warren Dicks. In 2002 Dunwoody put forward a proposed proof of the Poincaré conjecture
Poincaré conjecture
In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere , which is the hypersphere that bounds the unit ball in four-dimensional space...
. The proof generated considerable interest among mathematicians, but a mistake was quickly discovered and the proof was withdrawn. The conjecture was later proven by Grigori Perelman
Grigori Perelman
Grigori Yakovlevich Perelman is a Russian mathematician who has made landmark contributions to Riemannian geometry and geometric topology.In 1992, Perelman proved the soul conjecture. In 2002, he proved Thurston's geometrization conjecture...
, following the program of Richard Hamilton.
External links
- home page of Martin Dunwoody.