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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Polyhedral products over finite posetsKyoto Journal of Mathematics, vol. 62, no. 3, pp. 615-654Contributions to Journals: Articles
An application of neighbourhoods in digraphs to the classification of binary dynamicsNetwork Neuroscience, vol. 6, no. 2, pp. 528-551Contributions to Journals: Articles
Topology of synaptic connectivity constrains neuronal stimulus representation, predicting two complementary coding strategiesPloS ONE, vol. 17, no. 1, e0261702Contributions to Journals: Articles
Complexes of Tournaments, Directionality Filtrations and Persistent HomologyJournal of applied and computational topology, vol. 5, pp. 313-337Contributions to Journals: Articles
Loop space homology of a small categoryWorking Papers: Working Papers
Characterisation of polyhedral products with finite generalised Postnikov decompositionForum Mathematicum, vol. 32, no. 5, pp. 1253–1269Contributions to Journals: Articles
Computing Persistent Homology of Directed Flag ComplexesAlgorithms, vol. 13, no. 1, 19Contributions to Journals: Articles
On the topology of complexes of injective wordsJournal of applied and computational topology, vol. 4, pp. 29-44Contributions to Journals: Articles
A Topological Representation of Branching Neuronal MorphologiesNeuroinformatics, vol. 16, no. 1, pp. 3-13Contributions to Journals: Articles
Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and FunctionFrontiers in Computational Neuroscience, vol. 11, 48Contributions to Journals: Articles
Groups of unstable Adams operations on p-local compact groupsAlgebraic & Geometric Topology, vol. 17, no. 1, pp. 355-418Contributions to Journals: Articles
Automorphisms of p-local compact groupsJournal of Algebra, vol. 467, pp. 202-236Contributions to Journals: Articles
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
Discrete models for the p-local homotopy theory of compact Lie groups and p-compact groupsGeometry & Topology, vol. 11, pp. 315-427Contributions to Journals: Articles
A geometric construction of saturated fusion systemsContemporary Mathematics, vol. 399, pp. 11-40Contributions 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)]Geometry & Topology, vol. 9, pp. 2395-2415Contributions to Journals: Articles
Subgroup families controlling p-local finite groupsProceedings of the London Mathematical Society, vol. 91, no. 2, pp. 325-354Contributions to Journals: Articles
The theory of p-local groups: a surveyChapters 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, 2001Unknown Publisher, Progress in Mathematics 215, Birkhauser Verlag, Basel. 302 pagesBooks and Reports: Books
The Homotopy Theory of Fusion SystemsJournal of the American Mathematical Society, vol. 16, no. 4, pp. 779-856Contributions to Journals: Articles
Homotopy Equivalences of p-Completed Classifying Spaces of Finite GroupsInventiones Mathematicae, vol. 151, pp. 611-664Contributions to Journals: Articles
On Spaces of Self-Homotopy Equivalences of p-Completed Classifying Spaces of Finite Groups and Homotopy Group ExtensionTopology, vol. 41, no. 2, pp. 229-255Contributions to Journals: Articles
Construction of 2-Local Finite Groups of a Type studied by Solomon and BensonGeometry & Topology, vol. 6, pp. 917-990Contributions to Journals: Articles
On certain homotopy actions of general linear groups on iterated productsAnnales de l'Institut Fourier, vol. 51, no. 6, pp. 1719-1739Contributions to Journals: Articles
On p-completed classifying spaces of discrete groups and finite complexesJournal of the London Mathematical Society, vol. 59, no. 3, pp. 1064-1080Contributions to Journals: Articles