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Spatial symmetries can enrich the topological classification of interacting quantum matter and endow additional ``weak” topological indices upon systems with non-trivial strong topological invariants (protected by internal symmetries). In this talk, I will discuss the boundary physics of charge conserving systems with a non-zero shift invariant, which is protected by either a continuous U(1) or discrete CN rotation symmetry. In particular, I will discuss an interface between two systems with the same Chern number but distinct shift invariants and show that the interface hosts protected gapless edge modes. For general Abelian topological orders in 2D, I will prove sufficient conditions for gapless edge states protected by continuous rotation symmetry. For the case of discrete rotation symmetries, I will show that the Chern-Simons field theory for systems with gappable edges predicts fractional corner charges. These can also be computed when the system is placed on the two-dimensional surface of a Platonic solid, which relates to the fractional charge bound at disclination defects. Time permitting, I will discuss recent results regarding the relation between the shift and many-body real-space invariants for 2D systems with crystalline symmetry.

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The implementation of optical astronomical interferometry presents several technical challenges, such as establishing a shared reference frame and conducting nonlocal measurements. This presentation aims to establish a connection between the theoretical framework developed in quantum information science and the schemes employed in astronomical interferometry. Our discussion will focus on the trade-offs between required resource and operational performance. We will categorize these methods into three types: nonlocal schemes, local schemes with a reference frame, and local schemes operating without a reference frame. By using this interdisciplinary connection, we also explore a generalized intensity interferometer. This approach achieves performance levels comparable to traditional intensity interferometers but introduces interesting fundamental distinctions.

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