Magnetic topological quantum chemistry

For over 100 years, the group-theoretic characterization of crystalline solids has provided the foundational language for diverse problems in physics and chemistry. However, the group theory of crystals with commensurate magnetic order has remained incomplete for the past 70 years, due to the complicated symmetries of magnetic crystals. In this work, we complete the 100-year-old problem of crystalline group theory by deriving the small corepresentations, momentum stars, compatibility relations, and magnetic elementary band corepresentations of the 1,421 magnetic space groups (MSGs), which we have made freely accessible through tools on the Bilbao Crystallographic Server. We extend Topological Quantum Chemistry to the MSGs to form a complete, real-space theory of band topology in magnetic and nonmagnetic crystalline solids – Magnetic Topological Quantum Chemistry (MTQC). Using MTQC, we derive the complete set of symmetry-based indicators of electronic band topology, for which we identify symmetry-respecting bulk and anomalous surface and hinge states.

Integral models of local Shimura varieties

This chapter explains an application of the theory developed in these lectures towards the problem of understanding integral models of local Shimura varieties. As a specific example, it resolves conjectures of Kudla-Rapoport-Zink and Rapoport-Zink, that two Rapoport-Zink spaces associated with very different PEL data are isomorphic. The basic reason is that the corresponding group-theoretic data are related by an exceptional isomorphism of groups, so such results follow once one has a group-theoretic characterization of Rapoport-Zink spaces. The interest in these conjectures comes from the observation of Kudla-Rapoport-Zink that one can obtain a moduli-theoretic proof of Čerednik's p-adic uniformization for Shimura curves using these exceptional isomorphisms. The chapter defines integral models of local Shimura varieties as v-sheaves.