Eclectic problems in topology, geometry and dynamics

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This project aims to resolve a number of problems across several broad areas of pure mathematics. The problems all have a geometric or topological flavour, and some deal with dynamics in the qualitative sense. The problems share two common themes: they have group theoretic aspects and homological aspects. Specifically, the problems lie in the following areas: 1. finite dimensional Lie algebras and their cohomology, 2. low dimensional combinatorial geometry: graph drawings on surfaces, 3. topological dynamics of group actions, 4. differentiable group actions and foliation theory. The most significant aims are to resolve two well known conjectures: Halperin's toral rank conjecture and Conway's thrackle conjecture.