University of Wales, Bangor

Bangor, Gwynedd LL57 1UT

UK

http://www.bangor.ac.uk/ r.brown

r.brown@bangor.ac.uk

BORN: January 4, 1935, London, England NATIONALITY: English

UNIVERSITY EDUCATION:

· 1956 - B.A. - Oxford
University

· 1962 - D.Phil. - Oxford University

ACADEMIC POSITIONS:

· 1959-64 Assistant Lecturer, then Lecturer,
Liverpool University.

· 1964-70 Senior Lecturer, then Reader, Hull
University.

· 1970-1999 Professor of Pure Mathematics,
University of Wales, Bangor.

· 1983-84 Professeur associé pour
un mois, Université Louis Pasteur, Strasbourg.

· 1999-2001 Half time Research
Professorship.

· September 2001 Professor Emeritus, University
of Wales.

· 2002-2004 Leverhulme Emeritus Research
Fellowship for a project `Crossed complexes and homotopy groupoids'.

UNIVERSITY OF WALES SERVICE:

· 1970-1993 Headship of Pure Mathematics
or of a School of Mathematics in various forms for parts of this period.

· In the above Member of Council, Court,
Academic Board, and various Committees at various times.

· 1990-93 Chairman, University of Wales
Validation Board.

EDITING

· 1968-86 Editor, Chapman & Hall Mathematics
Series.

· 1975-1994 Editorial Advisory Board, London
Mathematical Society.

· 1995- One of the founding members and on
the Management Committee of Editorial Board, Electronic Journal:
Theory and Applications of
Categories.

· 1996-2007
Editorial Board: Applied Categorical
Structures (Kluwer).

· 1999- One of the founding members of
the Electronic Journal: Homology, Homotopy and Applications

. 2006 - Journal of Homotopy and
Related Structures

SELECTED OTHER PROJECTS

· 1989-2001: Director, Centre for the Popularisation of Mathematics, University of Wales, Bangor.

· 1995-2000: Coordinator, INTAS Project
`Algebraic K-theory, groups and categories', for Bangor, the University of
Bielefeld, Georgian Mathematical Institute, State Universities of Moscow
and of St. Petersburg, and the Steklov Institute, St. Petersburg.

· 2000: Grant to produce a CDRom as part
of an EC Project `Raising Public Awareness of Mathematics in WMY2000'.

· 2003-2005: EPSRC Grant: Higher Dimensional
algebra and Differential Geometry (Visiting Fellowship for J.F. Glazebrook,
Eastern Illinois).

· August, 2003: Opening lecture, `Global
actions and groupoid atlases', to the conference
`Directions in K-theory',
Poznan, in honour of the 60th birthday of A. Bak.

ADVISOR OF 23 successful Ph.D. students

The numbers refer to the full publication list.

[Book] *Elements of Modern Topology*, McGraw Hill, Maidenhead, (1968).

second edition: *Topology: a geometric account of general topology, homotopy
types, and the fundamental groupoid*, Ellis Horwood, Chichester (1988)
460 pp.
Third edition: Topology and Groupoids, Booksurge
LLC, (2006) xxv+525p

[Book2] *Nonabelian algebraic topology*, with P.J.
HIGGINS, R.SIVERA, (in preparation).

[5] ``The twisted Eilenberg-Zilber theorem'', * Celebrazioni Archimedi
de secolo xx, Syracusa, 1964, Simposi di topologia * (1967) 33-37.

[27] (with P.I. BOOTH), ``On the application of fibred mapping spaces to
exponential laws for bundles, ex-spaces and other categories of maps'',
*Gen. Top. Appl.* 8 (1978) 165-179.

[35] (with J. HUEBSCHMANN), ``Identities among relations'', in *Low
dimensional topology*, London Math. Soc. Lecture Note Series 48 (ed.
R. Brown and T.L. Thickstun, Cambridge University Press) (1982), pp. 153-202.

[47] (with S.P. HUMPHRIES), ``Orbits under symplectic transvections II: the
case K = **F**_{2}'', *Proc. London Math. Soc.* (3) 52
(1986) 532-556.

[48] (with P.J. HIGGINS), ``Tensor products and homotopies for
omega-groupoids and crossed complexes'', *J. Pure
Appl. Alg.* 47 (1987) 1-33.

[49] (with J.-L. LODAY), ``Homotopical excision, and Hurewicz theorems, for
n-cubes of spaces'', *Proc. London Math. Soc.* (3) 54 (1987) 176-192.

[50] ``From groups to groupoids: a brief survey'', *Bull. London Math.
Soc.* 19 (1987) 113-134.

[51] (with J.-L. LODAY), ``Van Kampen theorems for diagrams of spaces'',
*Topology* 26 (1987) 311-334.

[59] (with N.D. GILBERT), ``Algebraic models of 3-types and automorphism
structures for crossed modules'', *Proc. London Math. Soc.* (3) 59
(1989) 51-73.

[104] (with A. RAZAK SALLEH), ``Free crossed resolutions of groups and
presentations of modules of identities among relations'', *LMS J. Comp.
and Math*. 2 (1999) 28-61.

[107] (with A. HEYWORTH), ``Using rewriting systems to compute left Kan
extensions and induced actions of categories'', *J. Symbolic Computation
*29 (2000) 5-31.

[113] (with I. IÇEN), ``Locally Lie subgroupoids and their Lie holonomy
and monodromy groupoids'', *Top. and its Appl.* 115 (2001) 125-138.

[114] (with M. GOLASINSKI, T.PORTER and A.P.TONKS), ``On function spaces of
equivariant maps and the equivariant homotopy theory of crossed complexes
II: the general topological group case'', * K-Theory* 23 (2001)129-155.

[116] (with A. AL-AGL and R. STEINER), ``Multiple categories: the equivalence
between a globular and cubical approach'', * Advances in Mathematics*,
170 (2002) 71-118.

[123] (with I. IÇEN), ``Towards a 2-dimensional notion of holonomy'',
*Advances in Mathematics*, 178 (2003) 141-175.

[124] (with C.D.WENSLEY), ``Computation and homotopical applications of induced
crossed modules'', *J. Symbolic Computation* 35 (2003) 59-72.

[132] ``Crossed complexes and homotopy groupoids as non commutative tools for higher dimensional local-to-global problems'', Proceedings of the Fields Institute Workshop on Categorical Structures for Descent and Galois Theory, Hopf Algebras and Semiabelian Categories, September 23-28, Fields Institute Communications 43 (2004) 101-130. math.AT/0212274

[149] (with Bak, A., Minian, G., and Porter, T.), `Global actions, groupoid atlases and applications', J. Homotopy and Related Structures, 1 (2006) 101-167.

[159] (with I. C. Baianu and J.F. Glazebrook), CATEGORICAL ONTOLOGY OF COMPLEX SYSTEMS, META--SYSTEMS AND LEVELS: The Emergence of Life, Human Consciousness and Society'. In: ``Theory and Applications of Ontology." vol.1, R. Poli, et al., eds. 2008, Springer:Berlin (in press).

**Personal statement**

The publications listed above are chosen to represent a range of work. For full list, click here, which contains some links to pdf files.

The paper [5] became suprisingly (to me) influential, since it contained
the first version of what is now known as the * homological perturbation
lemma*. The resulting *homological perturbation theory* (click
here for
a link to a brief survey) has proved an important theoretical and computational
tool in algebraic topology and in the computation of resolutions.

I have an interest in the general topology of function spaces, dating back
to my first papers in 1963-4, which introduced the notion of an `adequate
and convenient category of topological spaces for homotopy theory', stimulating
a wide range of work on *convenient categories*. The collaboration with
Peter Booth [27] develops these notions in a wider context.

The collaboration resulting in [47] helped me to learn some aspects of linear groups, and also of mapping class groups.

Writing and revising the topology text [Book] has been a great influence on my research, since it led me to the concept of groupoid, [50]. A major theme of the book is that all of 1-dimensional homotopy theory is better expressed in terms of groupoids rather than groups. This raised the question of applications of groupoids in higher homotopy theory, and so to a long march to higher order Van Kampen Theorems, which give new higher dimensional, nonabelian, local-to-global methods, with relations to homology and K-theory.

[35] on *identities among relations* has been useful to many as a basic
source. This work is continued in [104] which introduces algorithmic procedures
for computations of these identities, using techniques of crossed complexes
worked out since 1974 mostly with P.J. Higgins, and which is surveyed in
[132]. Paper [48], one of 14 papers with Higgins, represents one of the harder
technical aspects of this work, which is vital also for [59] and [114].

Interest in algorithmic procedures and specific computations is shown in
[107] and [124]. Such computations also occur in [51], which introduced a
*non abelian tensor product of groups which act on each other*, and
for which the bibliography is now up to 90 papers.

The term `higher dimensional algebra' was introduced in the 1987 survey paper [50], following from the earlier `higher dimensional group theory, 1982, [36], and this area has been successful in applications in mathematics, physics, and computer science, as a web search shows.

The paper [116] is the culmination of work since 1966 on the development of cubical methods in higher category theory.

Paper [123] combines methods of double groupoids with differential ideas on holonomy, to make a start on developing higher order notions of `flows', analogous to evolving systems in concurrency theory.

Current work, in collaboration with Glazebrook and Porter, is on developing smooth analogues of techniques outlined in [132], with potential applications to gerbes and stacks.

The Leverhulme Emeritus Fellowship gave support for the republication of [Book] and the preparation of [Book2], which is intended to be a connected and full account, accessible to postgraduate students, of work since the 1970s, in collaboration with Higgins and others, on crossed complexes and the related higher homotopy groupoids.

Back to home page

August 22, 2007