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Quantum many-body systems all of whose excitations are gapped fall into distinct equivalence classes - quantum phases of matter. Apart from the familiar cases described within Landau paradigm of symmetry breaking, there are also more interesting phases, for example those exhibiting Integer and Fractional Quantum Hall effects. Such exotic phases are usually called topological phases of matter, since their low-energy properties can often be described by Topological Quantum Field Theory (TQFT). However, this terminology may be becoming obsolete, since the recently discovered fracton phases cannot be described by TQFT. One direction of my work is to understand invariants which distinguish different topological phases. Another one is to classify a particular class of topological phases: invertible phases (including Symmetry Protected Topological phases). I use methods from quantum statistical mechanics, quantum information theory, differential geometry, and homotopy theory.

The Berry connection was discovered by Michael Berry while studying quantum systems with parameters. It refines the work of von Neumann and Wigner who studied energy level crossings in such systems and showed that in the absence of symmetries they occur in codimension 3. In my work, I study generalizations of the Berry connection for infinite-volume lattice systems. From the field theory viewpoint, these are known as Wess-Zumino-Witten terms. Equivalently, I am studying the topology of the space of gapped quantum lattice systems. The problem of the classification of gapped quantum phases is a special case, since it amounts to finding the set of connected components of the space of gapped systems.

Transport coefficients such as conductivity, heat conductivity, and thermoelectric coefficients, are often defined using Kubo formulas. There are many subtleties associated with them, the most basic one being why the limits occurring in these formulas exist. Consistency with the laws of thermodynamics is also far from obvious. I am also interested in nonlinear transport.

Hydrodynamics in a generalized sense describes homogeneous macroscopic systems which are in local equilibrium. Hydrodynamic equations of motion are strongly constrained by the local version of the Kubo-Martin-Schwinger condition. I am interested in classifying varieties of hydrodynamic behavior and finding examples of exotic hydrodynamics in nature.