Professor Ran Levi

Professor Ran Levi The University of Aberdeen School of Natural & Computing Sciences Professor Ran Levi Chair in Mathematical Sciences work +44 (0)1224 272753 www.abdn.ac.uk/~mth139 pref Fraser Noble 138

Chair in Mathematical Sciences

BSc (Hebrew University), PhD (University of Rochester)

Professor Ran Levi

Personal Details

Telephone: +44 (0)1224 272753
Email: r.levi@abdn.ac.uk
Personal website: www.abdn.ac.uk/~mth139
Address: Fraser Noble 138
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Research Interests

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


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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.


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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), Assaf Libman (Aberdeen), Alex Gonzalez (Barcelona), Emannuel Farjoun (Hebrew University, Jerusalem)
  • Classical Homotopy Theory: Fred Cohen (Rochester), Kathryn Hess (EPFL)

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Teaching Responsibilities

2010/11: MX4546 Algebraic Topology, EG2510 Maths Engineering 3

2011/12: Topics, Maths Engineering 3, Algebraic Topology


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Admin Responsibilities

Institute of Mathematics Web Officer


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Publications

Contributions to Journals

Articles

  • Levi, R. & Seeliger, NIH. (in press). 'Loop space homology associated to mod-2 Dickson invariants'. Forum Mathematicum.
    [Online] DOI: 10.1515/FORM.2011.099
  • Levi, R. & Ragnarsson, K. (2011). 'p-local finite group cohomology'. Homology, Homotopy and Applications, vol 13, no. 1, pp. 223-257.
    [Online] DOI: doi:10.4310/HHA.2011.v13.n1.a8
  • Castellana, N., Levi, R. & Notbohm, D. (2007). 'Homology decompositions for p-compact groups'. Advances in Mathematics, vol 216, no. 2, pp. 491-534.
    [Online] DOI: 10.1016/J.AIM.2007.05.004
  • Broto, C., Levi, R. & Oliver, B. (2007). 'Discrete models for the p-local homotopy theory of compact Lie groups and p-compact groups'. Geometry & Topology, vol 11, pp. 315-427.
    [Online] DOI: 10.2140/GT.2007.11.315
  • Broto, C., Castellana, N., Grodal, J., Levi, R. & Oliver, B. (2007). 'Extensions of p-local finite groups'. Transactions of the American Mathematical Society, vol 359, no. 8, pp. 3791–3858.
  • Levi, R., Broto, C. & Oliver, B. (2006). 'A geometric construction of saturated fusion systems'. Contemporary Mathematics, vol 399, pp. 11-40.
  • Levi, R. & Oliver, B. (2005). 'Correction to: Construction of 2-local finite groups of a type studied by Solomon and Benson; [Geom. Topol. 6 (2002), 917-990 (electronic)]'. Geometry & Topology, vol 9, pp. 2395-2415.
    [Online] DOI: 10.2140/gt.2005.9.2395
  • Levi, R., Broto, C., Castellana, N., Grodal, J. & Oliver, B. (2005). 'Subgroup families controlling p-local finite groups'. Proceedings of the London Mathematical Society, vol 91, no. 2, pp. 325-354.
    [Online] DOI: 10.1112/S0024611505015327
  • Broto, C., Levi, R. & Oliver, B. (2004). 'The theory of p-local groups: a survey. Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic K-theory'. Contemporary Mathematics, vol 346, pp. 51-84.
  • Broto, C., Levi, R. & Oliver, B. (2003). 'Homotopy Equivalences of p-Completed Classifying Spaces of Finite Groups'. Inventiones Mathematicae, vol 151, pp. 611-664.
    [Online] DOI: 10.1007/S00222-002-0264-5
  • Broto, C., Levi, R. & Oliver, B. (2003). 'The Homotopy Theory of Fusion Systems'. Journal of the American Mathematical Society, vol 16, no. 4, pp. 779-856.
    [Online] DOI: 10.1090/S0894-0347-03-00434-X
  • Levi, R. & Oliver, B. (2002). 'Construction of 2-Local Finite Groups of a Type studied by Solomon and Benson'. Geometry & Topology, vol 6, pp. 917-990.
    [Online] DOI: 10.2140/gt.2002.6.917
  • Broto, C. & Levi, R. (2002). 'On Spaces of Self-Homotopy Equivalences of p-Completed Classifying Spaces of Finite Groups and Homotopy Group Extension'. Topology, vol 41, no. 2, pp. 229-255.
    [Online] DOI: 10.1016/S0040-9383(00)00031-8
  • Levi, R. & Priddy, S. (2001). 'On certain homotopy actions of general linear groups on iterated products'. Annales de l'Institut Fourier, vol 51, no. 6, pp. 1719-1739.
  • Levi, R. (1999). 'On p-completed classifying spaces of discrete groups and finite complexes'. Journal of the London Mathematical Society, vol 59, no. 3, pp. 1064-1080.
    [Online] DOI: 10.1112/S0024610799007279

Special Issues

  • Levi, R. (2008). 'The work of Fred Cohen: Groups, homotopy and configuration spaces'. Geometry & Topology Monographs, vol 13, pp. 529–546.
    [Online] DOI: 10.2140/gtm.2008.13.529

Books and Reports

Books

  • Arone, G., Hubbuck, JR., Levi, R. & Weiss, M. (2004). 'Categorical decomposition techniques in algebraic topology. Proceedings of the International Conference on Algebraic Topology, Isle of Skye, 2001'. Unknown Publisher, Progress in Mathematics 215, Birkhauser Verlag, Basel.

Contributions to Specialist Publications

Editorials

  • Levi, R., Iwase, N., Kono, T., Tamaki, D. & Wu, J. (eds) (2008). 'Groups, homotopy and configuration spaces: Proceedings of the conference in honor of the 60th birthday of Fred Cohen held at the University of Tokyo, Tokyo, July 5–11, 2005'. ', Geometry & Topology Monographs.

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