Professor Ran Levi
BSc (Hebrew University), PhD (University of Rochester)
Chair in Mathematical Sciences
- BSc Mathematics1988 - Hebrew University, Jerusalem, Israel
- PhD Mathematics1993 - University of Rochester
Mathematics Post Graduate Coordinator
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
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.
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.
- 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)
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Existence and uniqueness of classifying spaces for fusion systems over discrete p-toral groupsJournal of the London Mathematical Society, vol. 91, no. 1, pp. 47-70Contributions to Journals: Articles
An Algebraic Model for Finite Loop SpacesAlgebraic & Geometric Topology, vol. 14, no. 5, pp. 2915-2982Contributions to Journals: Articles
The rational cohomology of a p-local compact groupProceedings of the American Mathematical Society, vol. 142, no. 3, pp. 1035-1043Contributions to Journals: Articles
Loop space homology associated with the mod 2 Dickson invariantsForum Mathematicum, vol. 24, no. 6, pp. 1161–1182Contributions to Journals: Articles
Unstable Adams operations on p-local compact groupsAlgebraic & Geometric Topology, vol. 12, no. 1, pp. 49-74Contributions to Journals: Articles
p-local finite group cohomologyHomology, Homotopy and Applications, vol. 13, no. 1, pp. 223-257Contributions to Journals: Articles
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, vol. 13Contributions to Specialist Publications: Editorials
The work of Fred Cohen: Groups, homotopy and configuration spacesGeometry & Topology Monographs, vol. 13, pp. 529–546Contributions to Journals: Special Issues
Homology decompositions for p-compact groupsAdvances in Mathematics, vol. 216, no. 2, pp. 491-534Contributions to Journals: Articles
Extensions of p-local finite groupsTransactions of the American Mathematical Society, vol. 359, no. 8, pp. 3791–3858Contributions to Journals: Articles