Peter Booth

Professor Emeritus

PhD Hull, 1972
M.Sc. Hull, 1967
B.Sc. Hull, 1962
Dean of Science Distinguished Scholar Medal, 1996

Office: HH-2001
Phone: (709) 864-8782
Fax: (709) 864-3010

Biographical Information

Dr. Booth completed his bachelor's degree in Pure and Applied Mathematics from the University of Hull in 1962. His graduate work was in Algebraic Topology -- in particular in Homotopy Theory -- from the same university. He received his masters degree in 1967 and his Ph.D. “by published work” in 1972.

He held the position of Leverhulme Fellow at the University of Liverpool for the year 1965/66 and joined the faculty at MUN in the fall of 1966.

Dr. Booth has taught a wide range of courses at MUN, mostly in Pure Mathematics. In recent years he has also been teaching Discrete Mathematics for the Computer and Electrical Engineering Programmes.

He has been actively involved with the NLTA High School Mathematics League since its inception in 1987. He is an editor/author, along with Dr. John Grant McLoughlin and Dr. Bruce Shawyer, of two books of problems that have been used in the League.

Dr. Booth was winner of the Dean of Science Distinguished Scholar Medal for 1996/97, and has been a member of the Board of Directors of the Canadian Mathematical Society since 1999.

Research Interests

His research activities involve some Point Set Topology, but are primarily in Algebraic Topology. He has twenty-four papers published in refereed research publications and five in refereed conference proceedings. His current research interests are as follows:

  1. Fibred Mapping Spaces and their Applications,
  2. The Theory of Fibrations,
  3. Postnikov Factorization, and
  4. Equivariant Homotopy Theory.

If we are given two real numbers, then there are various ways in which we might combine them together to produce a new real number. We might add them together, multiply them together, raise one to the power of the other, and so on.

In Topology there is a similar phenomenon. If we are given two topological spaces, then there are several ways in which we might combine them together to produce a new topological space. We could form their Cartesian Product, their tological sum, the mapping space of maps from one to the other, their join, and so on.

The procedures with numbers are basic to Arithmetic; the topological constructions are fundamental to Algebraic Topology.

Most Dr Booth's research makes use of a family non-standard topological constructions of this type. He developed the basic theory of fibrewise mapping spaces in his Ph.D. work. He soon discovered that these gadgets have some surprisingly powerful applications. For example, he was able to retrieve results concerning the cohomology of fibrations that normally require spectral sequence methods. Since then his work has extended to include pairwise mapping spaces, and other types of fibred mapping spaces. He has published applications of this work in six of the eight areas into which the AMS Subject Classification Scheme divides Algebraic Topology. Quite a few other Algebraic Topologists have utilized these techniques, although Dr Booth remains convinced that this work is still in its early stages, and the potential for future developments is wide open. He looks forward to the day when these methods are part of the the standard toolkit used by most algebraic topologists.

In 1999 he completed a series of five papers, on the foundation of the Theory of Fibrations and their Classifying Spaces, that were published in Cahiers de Topologie et Géométrie Différentielle Catégoriques. He would claim that he has achieved an optimal version of this theory, in terms of criteria such as simplicity of concepts and results, generality and enhanced applicability.

A procedure known as Postnikov Factorization allows us to construct homotopy types of topological spaces out of basic building blocks. He is at present working on the problem of taking this semi-effective procedure, and attempting to develop a computationaly successful method of describing exactly what is obtained when these blocks are fitted together.

Sample Publications

  • The Exponential Law of Maps I. Proceedings of the London Mathematical Society (3) 20 (1970), 179-192.
  • A Unified Treatment of some Basic Problems in Homotopy Theory. Bulletin of the American Mathematical Society, 79 (1973), 331-336.
  • Monoidal Closed, Cartesian Closed and Convenient Categories of Topological Spaces. Pacific Journal of Mathematics, 88 (1980), 35-53, with Tillotson, J.
  • H-Spaces of Self-Equivalences of Fibrations and Bundles. Proceedings of the London Mathematical Society (3), 49 (1984), 111-127, with Heath, P.R., Morgan, C., and Piccinini, R.
  • Cancellation and Non-Cancellation amongst Products of Spherical Fibrations, Math. Scand., 66 (1990), 33-43.
  • Local to Global Properties in the Theory of Fibrations, Cahiers de Topologie et Géométrie Différentielle Catégorique XXIV-2 (1993), 127-151.
  • Fibrations and Classifying spaces: Overview and the Classical Examples, Cahiers de Topologie et Géométrie Différentielle Catégorique XLI-3 (2000), 162-206.
  • Equivariant Fibrations, to appear in Bulletin de la Société Mathématique de Belgique.