Ronald BROWN: BIOGRAPHICAL SKETCH

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

[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 = F₂'', 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.

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August 22, 2007