Professor Ran Levi

Professor Ran Levi
Professor Ran Levi

Professor Ran Levi

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

Chair in Mathematical Sciences

About

Qualifications

  • BSc Mathematics 
    1988 - Hebrew University, Jerusalem, Israel 
  • PhD Mathematics 
    1993 - University of Rochester 

Internal Memberships

Mathematics Post Graduate Coordinator

Research

Research Overview

Algebraic Topology, Homotopy Theory of classifying spaces of finite groups, Applications of topology and geometry to science and particularly neuroscience, Classical Homotopy Theory, Representation Theory of finite groups and interactions with homotopy theory

Current Research

Combinatorial Topology. Combinatorial constructions have played a major role in algebraic since the beginning of the subject. With the growing interest in real world applications of algebraic topology, there is an ever growing interest in the interaction between combinatorics and topology. I am spefically interested in constructions such as polyhedral products, configuration spaces, complexes of injective words and stochastic topology.

Neuoro-topology: The geometry and topology of neural systems. Mathematics has played a central role in neuroscience since its inception. More recently, particularly with the  emergence of extremely powerful computational models of the brain (for instance the EPFL's Blue Brain Project),  new mathematical approaches to neuroscience are emerging, which are analogous to the highly productive feedback loop between topology and physics. The methods of algebraic and geometric topology are perfectly suited for modelling, analysing, and predicting structures that arise in neuroscience, which in turn inspire new directions for research within topology. Indeed, topology is already both providing significant insight in current research and contributing to shaping future research in neuroscience.  Moreover it is probable that major questions of neuroscience could inspire the creation of new and exciting topological concepts that could then provide powerful new tools for neuroscience. 

In a collaboration with the Blue Brain Project, we address the following challenges:

  • Describe possible approaches for applying topology to neuroscience, both conceptually and computationally. 
  • Explore new ways of applying topological and computational techniques to the identification, analysis and development of modelled neural structures.
  • Build a theoretical framework for neuroscience that is based on the topology, geometry and category theory that naturally emerge when studying neural systems.
  • Explain mechanisms by which major questions of neuroscience could inspire the creation of new and exciting topological concepts that, in turn, could provide powerful tools for neuroscience.

Aberdeen Neuro-Topology Research Group

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.

 

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)
  • Neurotopology: Kathryn Hess (EPFL), Henry Markram (Blue Brain Project, EPFL), Dejan Govc (RF, Aberdeen), Janis Lazovskis (RF, Aberdeen), Henri Riihimaki (RF, Aberdeen), Jason Smith (RF, Aberdeen) 
  • Classical Homotopy Theory: Fred Cohen (Rochester), Kathryn Hess (EPFL)

Funding and Grants

EPSRC, Topological Analysis of Neural Systems EP/P025072/1

Ecole Polytechnique Federale de Lausanne (Collaboration agreement Blue Brain Project)

Publications

Page 3 of 4 Results 21 to 30 of 32

  • Discrete models for the p-local homotopy theory of compact Lie groups and p-compact groups

    Broto, C., Levi, R., Oliver, B.
    Geometry & Topology, vol. 11, pp. 315-427
    Contributions to Journals: Articles
  • A geometric construction of saturated fusion systems

    Levi, R., Broto, C., Oliver, B.
    Contemporary Mathematics, vol. 399, pp. 11-40
    Contributions to Journals: Articles
  • Correction to: Construction of 2-local finite groups of a type studied by Solomon and Benson; [Geom. Topol. 6 (2002), 917-990 (electronic)]

    Levi, R., Oliver, B.
    Geometry & Topology, vol. 9, pp. 2395-2415
    Contributions to Journals: Articles
  • Subgroup families controlling p-local finite groups

    Levi, R., Broto, C., Castellana, N., Grodal, J., Oliver, B.
    Proceedings of the London Mathematical Society, vol. 91, no. 2, pp. 325-354
    Contributions to Journals: Articles
  • The theory of p-local groups: a survey

    Broto, C., Levi, R., Oliver, B.
    Homotopy Theory. Goerss, P., Priddy, S. (eds.). Providence, RI, USA: American Mathematical Society pp. 51-84, 34 pages.
    Chapters in Books, Reports and Conference Proceedings: Conference Proceedings
  • Categorical decomposition techniques in algebraic topology. Proceedings of the International Conference on Algebraic Topology, Isle of Skye, 2001

    Arone, G., Hubbuck, J. R., Levi, R., Weiss, M.
    Unknown Publisher, Progress in Mathematics 215, Birkhauser Verlag, Basel. 302 pages
    Books and Reports: Books
  • The Homotopy Theory of Fusion Systems

    Broto, C., Levi, R., Oliver, B.
    Journal of the American Mathematical Society, vol. 16, no. 4, pp. 779-856
    Contributions to Journals: Articles
  • Homotopy Equivalences of p-Completed Classifying Spaces of Finite Groups

    Broto, C., Levi, R., Oliver, B.
    Inventiones Mathematicae, vol. 151, pp. 611-664
    Contributions to Journals: Articles
  • On Spaces of Self-Homotopy Equivalences of p-Completed Classifying Spaces of Finite Groups and Homotopy Group Extension

    Broto, C., Levi, R.
    Topology, vol. 41, no. 2, pp. 229-255
    Contributions to Journals: Articles
  • Construction of 2-Local Finite Groups of a Type studied by Solomon and Benson

    Levi, R., Oliver, B.
    Geometry & Topology, vol. 6, pp. 917-990
    Contributions to Journals: Articles

Refine

Books and Reports

Chapters in Books, Reports and Conference Proceedings

Contributions to Journals

Contributions to Specialist Publications

Working Papers and Discussion Papers