Topology and geometry are of fundamental importance in quantum physics. This proposal concerns applications of recent developments in topology and geometry to outstanding problems in condensed matter physics and quantum information theory. The first aim is to lay the foundations for the understanding of the influence of complex topology, which gives rise to generalized anyon statistics, on many-particle transport properties on networks, on entanglement of many-particle network states, and on topological quantum computing. The second aim is application of recent results on symplectic geometry and equivariant cohomology to the description and classification of quantum entanglement, the central concept in quantum information theory. The proposed research is multidisciplinary. On the mathematics side, it involves combinatorial graph theory, algebraic topology, cohomology of quotients and homogenous spaces. On the physics side, it concerns fundamental open problems in quantum many-body physics and quantum information theory. It is also very timely. The recent results on anyons on graphs may lead to new phenomena in quantum transport on networks, for example, generalizations of the integer and fractional quantum Hall effects. They can also provide new developments in topological quantum computing, especially in the topological quantum error correcting codes such as Kitaev's toric code. The distinguishing feature of the proposal is the unusual combination of methods, techniques and people involved. The research will be conducted in two world leading scientific institutions: Department of Physics, Massachusetts Institute of Technology and School of Mathematics of University of Bristol.