Research Interest

Algebraic Topology, Homotopy Theory of classifying spaces of finite groups, Classical Homotopy Theory, Representation Theory of finite groups and interactions with homotopy theory

Current Research

p-Local Groups. These are algebraic objects modelled on the homotopy theory associated to p-completed classifying spaces of finite groups. They enable one to relate the p-local structure of a finite group to related aspects of the homotopy theory of the resulting p-completed classifying space. The concept also allows for the construction of exotic homotopy types, i.e., spaces which behave from a homotopy theoretic point of view like a p-completed classifying space of a finite group, but in fact are not such spaces. The theory extends to the concept of p-local compact groups which is modelled on the p-local homotopy theory of classifying spaces of compact Lie groups and p-compact groups.

Algebraic Models for Iterated Loop spaces. I am interested in creating "small" computable models for the classical algebraic invariants of iterated loop space. In particular work with Cohen established free simplicial group models for iterated loop spaces. In a separate project with Hess we constructed a model for the double loop space homology of a space. These models are constructed with a view to theoretical appplications, particularly the study of homology and homotopy exponent problems.

Collaborations

• p-Local Groups and the homotopy theory of classifying spaces: Carles Broto (Barcelona), Bob Oliver (Paris 13), Natalia Castellana (Barcelona), Jesper Grodal (Kopenhagen) Dietrich Notbohm (Leicester).
• Classical Homotopy Theory: Fred Cohen (Rochester), Kathryn Hess (EPFL)

Teaching Responsibilities

On leave until 1/1/2008

Web Links

• University of Aberdeen, Mathematical Sciences 
• Math Sci Net (subscription site) 

Publications

• Levi,R.; Broto,C.; Castellana,N.; Grodal,J.; Oliver,B.; Extensions of p-local finite groups , Original Article in Peer Reviewed Journal, in press, Transactions of the American Mathematical Society, ISSN/ISBN: 0002-9947
• Levi,R.; Castellana,N.; Notbohm,D.; Homology decompositions for p-compact groups , Original Article in Peer Reviewed Journal, in press, Advances in Mathematics, ISSN/ISBN: 0001-8708
• Oliver,B.; Broto,C.; Levi,R.; Discrete models for the p-local homotopy theory of compact Lie groups and p-compact groups , Original Article in Peer Reviewed Journal, 2007, GEOMETRY & TOPOLOGY, 11, pp315-427, ISSN/ISBN: 1364-0380
• Levi,R.; Broto,C.; Oliver,B.; A geometric construction of saturated fusion systems , Original Article in Peer Reviewed Journal, 2006, Contemporary Mathematics, 399, pp11-40, ISSN/ISBN: 0271-4132
• Levi,R.; Oliver,B.; Correction to: Construction of 2-local finite groups of a type studied by Solomon and Benson; [Geom. Topol. 6 (2002), 917-990 (electronic)] , Original Article in Peer Reviewed Journal, 2005, GEOMETRY & TOPOLOGY, 9, pp2395-2415, ISSN/ISBN: 1364-0380
• Levi,R.; Broto,C.; Castellana,N.; Grodal,J.; Oliver,B.; Subgroup families controlling p-local finite groups , Original Article in Peer Reviewed Journal, 2005, Proceedings of the London Mathematical Society, 91, 2, pp325-354, ISSN/ISBN: 0024-6115
• Arone,G.; Hubbuck,J. R.; Levi,R.; Weiss,M.; Categorical decomposition techniques in algebraic topology. Proceedings of the International Conference on Algebraic Topology, Isle of Skye, 2001 , Edited Book, 2004, Progress in Mathematics 215, Birkhauser Verlag, Basel, pp302, ISSN/ISBN: 3-7643-0400-6
• Broto,C.; Levi,R.; Oliver,B.; The theory of ρ-local groups: a survey. Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic K-theory , Original Article in Peer Reviewed Journal, 2004, Contemporary Mathematics, 346, pp51-84, ISSN/ISBN: 0271-4132
• Broto,C.; Levi,R.; Oliver,B.; Homotopy Equivalences of p-Completed Classifying Spaces of Finite Groups , Original Article in Peer Reviewed Journal, 2003, Inventiones Mathematicae, 151, pp611-664, ISSN/ISBN: 0020-9910
• Broto,C.; Levi,R.; Oliver,B.; The Homotopy Theory of Fusion Systems , Original Article in Peer Reviewed Journal, 2003, Journal of the American Mathematical Society, 16, 4, pp779-856, ISSN/ISBN: 0894-0347
• Levi,R.; Oliver,B.; Construction of 2-Local Finite Groups of a Type studied by Solomon and Benson , Original Article in Peer Reviewed Journal, 2002, GEOMETRY & TOPOLOGY, 6, pp917-990, ISSN/ISBN: 1364-0380
• Broto,C.; Levi,R.; On Spaces of Self-Homotopy Equivalences of ρ-Completed Classifying Spaces of Finite Groups and Homotopy Group Extension , Original Article in Peer Reviewed Journal, 2002, Topology, 41, pp229-255, ISSN/ISBN: 0040-9383
• Levi,R.; Priddy,S.; On certain homotopy actions of general linear groups on iterated products , Original Article in Peer Reviewed Journal, 2001, Annales de l'Institut Fourier, 51, 6, pp1719-1739, ISSN/ISBN: 0373-0956