Low-dimensional topology and topological methods in condensed matter physics

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The proposed research concerns low-dimensional topology, specifically geometric classification of topological 4-manifolds and certain aspects of quantum topology. Krushkal proposes an approach to solving the topological 4-dimensional surgery conjecture in the context of the homotopy A-B slice problem. A convenient framework for this approach is provided by topological arbiters, a notion recently introduced in collaboration with M. Freedman. The project includes a number of problems in quantum topology, specific questions concern spin networks, and as a special case a categorified relation between the chromatic polynomial of planar graphs and the Temperley-Lieb algebra. The project also aims to use topological methods to address classification problems for free and certain classes of interacting fermions, an important class of systems in condensed matter physics.Topological invariants have played an important role in recent work on classification of gapped phases of non-interacting fermions by Kitaev and others. K-theory and Bott periodicity turned out to be crucial ingredients in classifying systems with various symmetries. Krushkal plans to work on a number of related problems, including invariants of families of free fermions and extensions to certain classes of interacting systems.One goal of the project concerns classification of possible large scale shapes of objects that locally look like the Euclidean space. The classification of three- and four-dimensional shapes is a particularly important and challenging problem. This project is aimed at a better understanding of 4-dimensional objects with large fundamental groups, which contain many loops that cannot be contracted. Another part of the project concerns quantum topology, a subject influenced by ideas from both physics and mathematics and connected with statistical mechanics, representation theory, topology, and combinatorics. In particular, the goal is to gain a new conceptual perspective on spin networks, a notion whose origins are in quantum physics, in the recently developed mathematical framework of categorification. The third part of the project explores interdisciplinary connections between Topology and Condensed Matter Physics, focusing on topological insulators. This research, in collabor ation with physicists, will build on recent advances in classification of this remarkable class of physical systems.