Brian H. Bowditch
   HOME

TheInfoList



OR:

Brian Hayward Bowditch (born 1961
Bowditch's personal information page at the
University of Warwick The University of Warwick ( ; abbreviated as ''Warw.'' in post-nominal letters) is a public research university on the outskirts of Coventry between the West Midlands and Warwickshire, England. The university was founded in 1965 as part of ...
) is a British mathematician known for his contributions to
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
and
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, particularly in the areas of
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 group ...
and
low-dimensional topology In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the theory of 3-manifolds and 4-manifolds, knot theory, ...
. He is also known for solving the
angel problem The angel problem is a question in combinatorial game theory proposed by John Horton Conway. The game is commonly referred to as the angels and devils game.John H. Conway, The angel problem', in: Richard Nowakowski (editor) ''Games of No Chance'' ...
. Bowditch holds a chaired Professor appointment in Mathematics at the
University of Warwick The University of Warwick ( ; abbreviated as ''Warw.'' in post-nominal letters) is a public research university on the outskirts of Coventry between the West Midlands and Warwickshire, England. The university was founded in 1965 as part of ...
.


Biography

Brian Bowditch was born in 1961 in
Neath Neath (; ) is a market town and Community (Wales), community situated in the Neath Port Talbot, Neath Port Talbot County Borough, Wales. The town had a population of 50,658 in 2011. The community of the parish of Neath had a population of 19,2 ...
, Wales. He obtained a B.A. degree from
Cambridge University The University of Cambridge is a Public university, public collegiate university, collegiate research university in Cambridge, England. Founded in 1209, the University of Cambridge is the List of oldest universities in continuous operation, wo ...
in 1983. He subsequently pursued doctoral studies in Mathematics at the
University of Warwick The University of Warwick ( ; abbreviated as ''Warw.'' in post-nominal letters) is a public research university on the outskirts of Coventry between the West Midlands and Warwickshire, England. The university was founded in 1965 as part of ...
under the supervision of David Epstein where he received a PhD in 1988. Bowditch then had postdoctoral and visiting positions at the
Institute for Advanced Study The Institute for Advanced Study (IAS) is an independent center for theoretical research and intellectual inquiry located in Princeton, New Jersey. It has served as the academic home of internationally preeminent scholars, including Albert Ein ...
in
Princeton Princeton University is a private Ivy League research university in Princeton, New Jersey, United States. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the Unit ...
,
New Jersey New Jersey is a U.S. state, state located in both the Mid-Atlantic States, Mid-Atlantic and Northeastern United States, Northeastern regions of the United States. Located at the geographic hub of the urban area, heavily urbanized Northeas ...
, the University of Warwick,
Institut des Hautes Études Scientifiques The Institut des hautes études scientifiques (IHÉS; English: Institute of Advanced Scientific Studies) is a French research institute supporting advanced research in mathematics and theoretical physics (also with a small theoretical biology g ...
at
Bures-sur-Yvette Bures-sur-Yvette (, "Bures-on- Yvette") is a commune in the Essonne department in the Île-de-France region in Northern France. It is a southern Parisian outer suburb in the Vallée de Chevreuse, with a population of 9,254 as of 2021. Geograp ...
, the
University of Melbourne The University of Melbourne (colloquially known as Melbourne University) is a public university, public research university located in Melbourne, Australia. Founded in 1853, it is Australia's second oldest university and the oldest in the state ...
, and the
University of Aberdeen The University of Aberdeen (abbreviated ''Aberd.'' in List of post-nominal letters (United Kingdom), post-nominals; ) is a public university, public research university in Aberdeen, Scotland. It was founded in 1495 when William Elphinstone, Bis ...
. In 1992 he received an appointment at the
University of Southampton The University of Southampton (abbreviated as ''Soton'' in post-nominal letters) is a public university, public research university in Southampton, England. Southampton is a founding member of the Russell Group of research-intensive universit ...
where he stayed until 2007. In 2007 Bowditch moved to the University of Warwick, where he received a chaired Professor appointment in Mathematics. Bowditch was awarded a
Whitehead Prize The Whitehead Prize is awarded yearly by the London Mathematical Society to multiple mathematicians working in the United Kingdom who are at an early stage of their career. The prize is named in memory of homotopy theory pioneer J. H. C. Whitehe ...
by the
London Mathematical Society The London Mathematical Society (LMS) is one of the United Kingdom's Learned society, learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh ...
in 1997 for his work in
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 group ...
and
geometric topology In mathematics, geometric topology is the study of manifolds and Map (mathematics)#Maps as functions, maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topo ...
. He gave an Invited address at the 2004
European Congress of Mathematics The European Congress of Mathematics (ECM) is the second largest international conference of the mathematics community, after the International Congresses of Mathematicians (ICM). The ECM are held every four years and are timed precisely betwee ...
in Stockholm. Bowditch is a former member of the Editorial Board for the journal ''
Annales de la Faculté des Sciences de Toulouse Annals are a concise form of historical writing which record events chronologically, year by year. The equivalent word in Latin and French is ''annales'', which is used untranslated in English in various contexts. List of works with titles contai ...
'' and a former Editorial Adviser for the
London Mathematical Society The London Mathematical Society (LMS) is one of the United Kingdom's Learned society, learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh ...
.


Mathematical contributions

Early notable results of Bowditch include clarifying the classic notion of
geometric finiteness In geometry, a group of isometries of hyperbolic space is called geometrically finite if it has a well-behaved fundamental domain. A hyperbolic manifold is called geometrically finite if it can be described in terms of geometrically finite groups. ...
for higher-dimensional
Kleinian group In mathematics, a Kleinian group is a discrete subgroup of the group (mathematics), group of orientation-preserving Isometry, isometries of hyperbolic 3-space . The latter, identifiable with PSL(2,C), , is the quotient group of the 2 by 2 complex ...
s in constant and variable negative curvature. In a 1993 paper Bowditch proved that five standard characterisations of geometric finiteness for discrete groups of isometries of
hyperbolic 3-space In mathematics, hyperbolic space of dimension ''n'' is the unique simply connected, ''n''-dimensional Riemannian manifold of constant sectional curvature equal to âˆ’1. It is homogeneous, and satisfies the stronger property of being a symme ...
and
hyperbolic plane In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P' ...
, (including the definition in terms of having a finitely-sided fundamental polyhedron) remain equivalent for groups of isometries of hyperbolic ''n''-space where ''n'' â‰¥ 4. He showed, however, that in dimensions ''n'' â‰¥ 4 the condition of having a finitely-sided
Dirichlet domain In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. It can be classified also as a tessellation. In the simplest case, these objects are just finitely many points in the plane (called ...
is no longer equivalent to the standard notions of geometric finiteness. In a subsequent paper Bowditch considered a similar problem for discrete groups of isometries of
Hadamard manifold In mathematics, a Hadamard manifold, named after Jacques Hadamard — more often called a Cartan–Hadamard manifold, after Élie Cartan — is a Riemannian manifold (M, g) that is complete and simply connected and has everywhere non-positive sectio ...
of pinched (but not necessarily constant) negative curvature and of arbitrary dimension ''n'' â‰¥ 2. He proved that four out of five equivalent definitions of geometric finiteness considered in his previous paper remain equivalent in this general set-up, but the condition of having a finitely-sided fundamental polyhedron is no longer equivalent to them. Much of Bowditch's work in the 1990s concerned studying boundaries at infinity of
word-hyperbolic group In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a ''word hyperbolic group'' or ''Gromov hyperbolic group'', is a finitely generated group equipped with a word metric satisfying certain properties abstra ...
s. He proved the ''cut-point conjecture'' which says that the boundary of a one-ended word-hyperbolic group does not have any global
cut-point In topology, a cut-point is a point of a connected space such that its removal causes the resulting space to be disconnected. If removal of a point doesn't result in disconnected spaces, this point is called a non-cut point. For example, every po ...
s. Bowditch first proved this conjecture in the main cases of a one-ended hyperbolic group that does not split over a two-ended
subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation  ...
(that is, a subgroup containing
infinite cyclic subgroup In abstract algebra, a cyclic group or monogenous group is a group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of -adic numbers), that is generated by a single element. That is, it is a set of invertib ...
of finite
index Index (: indexes or indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on the Halo Array in the ...
) and also for one-ended hyperbolic groups that are "strongly accessible". The general case of the conjecture was finished shortly thereafter by G. Ananda Swarup who characterised Bowditch's work as follows: "The most significant advances in this direction were carried out by Brian Bowditch in a brilliant series of papers ( . We draw heavily from his work". Soon after Swarup's paper Bowditch supplied an alternative proof of the cut-point conjecture in the general case. Bowditch's work relied on extracting various discrete tree-like structures from the
action Action may refer to: * Action (philosophy), something which is done by a person * Action principles the heart of fundamental physics * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video gam ...
of a word-hyperbolic group on its boundary. Bowditch also proved that (modulo a few exceptions) the boundary of a one-ended word-hyperbolic group ''G'' has local cut-points if and only if ''G'' admits an essential splitting, as an amalgamated free product or an
HNN extension In mathematics, the HNN extension is an important construction of combinatorial group theory. Introduced in a 1949 paper ''Embedding Theorems for Groups'' by Graham Higman, Bernhard Neumann, and Hanna Neumann, it embeds a given group ''G'' into ...
, over a virtually infinite cyclic group. This allowed Bowditch to produce a theory of
JSJ decomposition In mathematics, the JSJ decomposition, also known as the toral decomposition, is a topological construct given by the following theorem: : Irreducible orientable closed (i.e., compact and without boundary) 3-manifolds have a unique (up to isoto ...
for word-hyperbolic groups that was more canonical and more general (particularly because it covered groups with nontrivial torsion) than the original JSJ decomposition theory of
Zlil Sela Zlil Sela () is an Israeli mathematician working in the area of geometric group theory. He is a Professor of Mathematics at the Hebrew University of Jerusalem. Sela is known for the solution of the isomorphism problem for torsion-free word-hype ...
. One of the consequences of Bowditch's work is that for one-ended word-hyperbolic groups (with a few exceptions) having a nontrivial essential splitting over a virtually cyclic subgroup is a
quasi-isometry In mathematics, a quasi-isometry is a function between two metric spaces that respects large-scale geometry of these spaces and ignores their small-scale details. Two metric spaces are quasi-isometric if there exists a quasi-isometry between them. ...
invariant. Bowditch also gave a topological characterisation of word-hyperbolic groups, thus solving a conjecture proposed by Mikhail Gromov. Namely, Bowditch proved that a group ''G'' is word-hyperbolic if and only if ''G'' admits an
action Action may refer to: * Action (philosophy), something which is done by a person * Action principles the heart of fundamental physics * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video gam ...
by
homeomorphism In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...
s on a perfect metrisable compactum ''M'' as a "uniform convergence group", that is such that the diagonal action of ''G'' on the set of distinct triples from ''M'' is properly discontinuous and co-compact; moreover, in that case ''M'' is ''G''-equivariantly homeomorphic to the boundary ∂''G'' of ''G''. Later, building up on this work, Bowditch's PhD student Yaman gave a topological characterisation of relatively hyperbolic groups. Much of Bowditch's work in 2000s concerns the study of the curve complex, with various applications to
3-manifold In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane (geometry), plane (a tangent ...
s,
mapping class group In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space. Mo ...
s and
Kleinian group In mathematics, a Kleinian group is a discrete subgroup of the group (mathematics), group of orientation-preserving Isometry, isometries of hyperbolic 3-space . The latter, identifiable with PSL(2,C), , is the quotient group of the 2 by 2 complex ...
s. The curve complex ''C''(''S'') of a finite type surface ''S'', introduced by Harvey in the late 1970s, has the set of free homotopy classes of essential simple closed curves on ''S'' as the set of vertices, where several distinct vertices span a simplex if the corresponding curves can be realised disjointly. The curve complex turned out to be a fundamental tool in the study of the geometry of the
Teichmüller space In mathematics, the Teichmüller space T(S) of a (real) topological (or differential) surface S is a space that parametrizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmülle ...
, of
mapping class group In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space. Mo ...
s and of
Kleinian group In mathematics, a Kleinian group is a discrete subgroup of the group (mathematics), group of orientation-preserving Isometry, isometries of hyperbolic 3-space . The latter, identifiable with PSL(2,C), , is the quotient group of the 2 by 2 complex ...
s. In a 1999 paper
Howard Masur Howard Alan Masur is an American mathematician who works on topology, geometry, and combinatorial group theory. Biography Masur was an invited speaker at the 1994 International Congress of Mathematicians in Zürich. and is a fellow of the Amer ...
and
Yair Minsky Yair Nathan Minsky (born in 1962) is an Israeli- American mathematician whose research concerns three-dimensional topology, differential geometry, group theory and holomorphic dynamics. He is a professor at Yale University. He is known for havin ...
proved that for a finite type orientable surface ''S'' the curve complex ''C''(''S'') is Gromov-hyperbolic. This result was a key component in the subsequent proof of Thurston's Ending lamination conjecture, a solution which was based on the combined work of Yair Minsky, Howard Masur, Jeffrey Brock, and
Richard Canary Richard Douglas Canary (born in 1962) is an American mathematician working mainly on low-dimensional topology. He is a professor at the University of Michigan. Canary obtained his Ph.D. from Princeton University in 1989 under the supervision of ...
. In 2006 Bowditch gave another proof of hyperbolicity of the curve complex. Bowditch's proof is more combinatorial and rather different from the Masur-Minsky original argument. Bowditch's result also provides an estimate on the hyperbolicity constant of the curve complex which is logarithmic in complexity of the surface and also gives a description of geodesics in the curve complex in terms of the intersection numbers. A subsequent 2008 paper of Bowditch pushed these ideas further and obtained new quantitative finiteness results regarding the so-called "tight geodesics" in the curve complex, a notion introduced by Masur and Minsky to combat the fact that the curve complex is not locally finite. As an application, Bowditch proved that, with a few exceptions of surfaces of small complexity, the action of the
mapping class group In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space. Mo ...
Mod(''S'') on ''C''(''S'') is "acylindrical" and that the asymptotic translation lengths of pseudo-Anosov elements of Mod(''S'') on ''C''(''S'') are rational numbers with bounded denominators. A 2007 paper of Bowditch produces a positive solution of the
angel problem The angel problem is a question in combinatorial game theory proposed by John Horton Conway. The game is commonly referred to as the angels and devils game.John H. Conway, The angel problem', in: Richard Nowakowski (editor) ''Games of No Chance'' ...
of
John Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician. He was active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many br ...
: Bowditch provedB. H. Bowditch
"The angel game in the plane"
''
Combinatorics, Probability and Computing ''Combinatorics, Probability and Computing'' is a peer-reviewed scientific journal in mathematics published by Cambridge University Press. Its editor-in-chief is Béla Bollobás ( DPMMS and University of Memphis). History The journal was estab ...
'', vol. 16 (2007), no. 3, pp. 345–362
that a 4-angel has a winning strategy and can evade the devil in the "angel game". Independent solutions of the angel problem were produced at about the same time by András Máthé and Oddvar Kloster.Oddvar Kloster
"A solution to the angel problem"
''
Theoretical Computer Science Theoretical computer science is a subfield of computer science and mathematics that focuses on the Abstraction, abstract and mathematical foundations of computation. It is difficult to circumscribe the theoretical areas precisely. The Associati ...
'', vol. 389 (2007), no. 1-2, pp. 152–161


Selected publications

* * * * * *


See also

*
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 group ...
*
Geometric topology In mathematics, geometric topology is the study of manifolds and Map (mathematics)#Maps as functions, maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topo ...
*
3-manifold In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane (geometry), plane (a tangent ...
s *
Kleinian group In mathematics, a Kleinian group is a discrete subgroup of the group (mathematics), group of orientation-preserving Isometry, isometries of hyperbolic 3-space . The latter, identifiable with PSL(2,C), , is the quotient group of the 2 by 2 complex ...
s


References


External links


Brian H. Bowditch's HomePage
at the
University of Warwick The University of Warwick ( ; abbreviated as ''Warw.'' in post-nominal letters) is a public research university on the outskirts of Coventry between the West Midlands and Warwickshire, England. The university was founded in 1965 as part of ...
{{DEFAULTSORT:Bowditch, Brian 1961 births 20th-century Welsh mathematicians 21st-century Welsh mathematicians Academics of the University of Southampton Academics of the University of Warwick Alumni of the University of Warwick Combinatorial game theorists Differential geometers Group theorists People from Neath Living people British topologists Whitehead Prize winners