Chair in Mathematical Sciences
Algebraic Topology, Homotopy Theory of classifying spaces of finite groups, Applications of topology and geometry to neuroscience, Classical Homotopy Theory, Representation Theory of finite groups and interactions with homotopy theory
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.
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.
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.
- 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)
- Classical Homotopy Theory: Fred Cohen (Rochester), Kathryn Hess (EPFL)
- Further Info
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