Professor Ran Levi

Internal Memberships

Mathematics Post Graduate Coordinator

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)