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Abstract

The present thesis consists of 2 parts. Chapter 1 is about applications of twisted equivariant K-theory to condensed matter. We consider non-interacting electrons on a half-crystal (a crystal with a boundary), with a gapped bulk condition, under quasi-adiabatic evolution. In A. Adem, O. Antolin, G. Semenoff and D. Sheinbaum JHEP, 2016 we found that Fermi surfaces for these systems under quasi-adiabatic evolution are classified by the K⁻¹-group of the surface Brillouin zone Td⁻¹. Systems with time-reversal and particle-hole symmetry were also considered and we obtained different KR-groups for the different cases. In Chapter 1 I rewrite A. Adem, O. Antolin, G. Semenoff and D. Sheinbaum JHEP, 2016 in a more function-analytic language and further solve technical issues to extend it to include crystallographic symmetries on the directions parallel to the boundary. In Chapter 2 I reproduce the relevant parts of my joint work with C. Okay (C. Okay and D. Sheinbaum arXiv:1905.07723). There we explored a connection between twisted equivariant K-theory to contextuality in quantum mechanics. We also reformulated the sheaf-theoretic framework of S. Abramsky and A. Brandenburger New Journal of Physics, 2011 for contextuality and connect it to another one employing a group cohomology approach of C. Okay, S. Roberts, S.D Bartlett, and R. Raussendorf Quantum Information and Computation, 2017. This leads to the construction of a classifying space for contextuality, from which Wigner functions are classes in its twisted K-theory.