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The goal of my research is to understand in mathematical terms the novel materials known as topological insulators, and in particular, explore their hallmark property, the bulk-edge correspondence, under the physically relevant regime of strong disorder. This correspondence refers to the fact that certain topological invariants (e.g. the quantized Hall conductivity) are the same whether we compute them for an infinite system (bulk) or for the related half-infinite system (edge). In addition, I also seek to understand the topological classification of such materials under strong disorder. This regime is important for many reasons, for example, without it one couldn't explain the plateaus of the integer quantum Hall effect.
To insure topological invariants are well-defined, the system must be in a localized phase. Thus it is also important to know under what conditions localization happens, a study via the theory of random Schrödinger operators.
The tools to tackle such problems are algebraic topology of operators (the Hamiltonians) which is not covered by Fredholm theory and K-Theory as now operators have essential (but localized) spectrum near the Fermi energy. The study of localization (and delocalization) proceeds via functional analysis in a random setting as well as dynamical systems.