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1. On the degeneration of asymptotically conical Calabi–Yau metrics

   https://doi.org/10.1007/s00208-021-02229-z  

   Collins, T. (July 2021, Mathematische Annalen)

   null (Ed.)

   We study the degenerations of asymptotically conical Ricci-flat Kahler metrics as the Kahler class degenerates to a semi-positive class. We show that under appropriate assumptions, the Ricci-flat Kahler metrics converge to a incomplete smooth Ricci-flat Kahler metric away from a compact subvariety. As a consequence, we construct singular Calabi–Yau metrics with asymptotically conical behaviour at infinity on certain quasi-projective varieties and we show that the metric geometry of these singular metrics are homeomorphic to the topology of the singular variety. Finally, we will apply our results to study several classes of examples of geometric transitions between Calabi–Yau manifolds. 

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2. Enhanced sensitivity via non-Hermitian topology

   https://doi.org/10.1038/s41377-024-01667-z  

   Parto, Midya; Leefmans, Christian; Williams, James; Gray, Robert_M; Marandi, Alireza (January 2025, Light: Science & Applications)

   Abstract Sensors are indispensable tools of modern life that are ubiquitously used in diverse settings ranging from smartphones and autonomous vehicles to the healthcare industry and space technology. By interfacing multiple sensors that collectively interact with the signal to be measured, one can go beyond the signal-to-noise ratios (SNR) attainable by the individual constituting elements. Such techniques have also been implemented in the quantum regime, where a linear increase in the SNR has been achieved via using entangled states. Along similar lines, coupled non-Hermitian systems have provided yet additional degrees of freedom to obtain better sensors via higher-order exceptional points. Quite recently, a new class of non-Hermitian systems, known as non-Hermitian topological sensors (NTOS) has been theoretically proposed. Remarkably, the synergistic interplay between non-Hermiticity and topology is expected to bestow such sensors with an enhanced sensitivity that grows exponentially with the size of the sensor network. Here, we experimentally demonstrate NTOS using a network of photonic time-multiplexed resonators in the synthetic dimension represented by optical pulses. By judiciously programming the delay lines in such a network, we realize the archetypal Hatano-Nelson model for our non-Hermitian topological sensing scheme. Our experimentally measured sensitivities for different lattice sizes confirm the characteristic exponential enhancement of NTOS. We show that this peculiar response arises due to the combined synergy between non-Hermiticity and topology, something that is absent in Hermitian topological lattices. Our demonstration of NTOS paves the way for realizing sensors with unprecedented sensitivities. 

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3. Dynamics and topology of non-Hermitian elastic lattices with non-local feedback control interactions

   https://doi.org/10.1088/1367-2630/ab81b6  

   Rosa, Matheus I. N.; Ruzzene, Massimo (May 2020, New Journal of Physics)

   Abstract We investigate non-Hermitian elastic lattices characterized by non-local feedback interactions. In one-dimensional lattices, proportional feedback produces non-reciprocity associated with complex dispersion relations characterized by gain and loss in opposite propagation directions. For non-local controls, such non-reciprocity occurs over multiple frequency bands characterized by opposite non-reciprocal behavior. The dispersion topology is investigated with focus on winding numbers and non-Hermitian skin effect, which manifests itself through bulk modes localized at the boundaries of finite lattices. In two-dimensional lattices, non-reciprocity is associated with directional wave amplification. Moreover, the combination of skin effect in two directions produces modes that are localized at the corners of finite two-dimensional lattices. Our results describe fundamental properties of non-Hermitian elastic lattices, and suggest new possibilities for the design of meta materials with novel functionalities related to selective wave filtering, amplification and localization. The considered non-local lattices also provide a platform for the investigation of topological phases of non-Hermitian systems. 

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4. Ricci flow does not preserve positive sectional curvature in dimension four

   https://doi.org/10.1007/s00526-022-02335-z  

   Bettiol, Renato G.; Krishnan, Anusha M. (November 2022, Calculus of Variations and Partial Differential Equations)

   Abstract We find examples of cohomogeneity one metrics on$$S^4$$ $S^4$and$$\mathbb {C}P^2$$ $C{P}^2$with positive sectional curvature that lose this property when evolved via Ricci flow. These metrics are arbitrarily small perturbations of Grove–Ziller metrics with flat planes that become instantly negatively curved under Ricci flow. 

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5. Hodge theory on ALG ^∗ manifolds

   https://doi.org/10.1515/crelle-2023-0016  

   Chen, Gao; Viaclovsky, Jeff; Zhang, Ruobing (May 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We develop a Fredholm theory for the Hodge Laplacian in weighted spaces on ALG ∗ manifolds in dimension four.We then give several applications of this theory.First, we show the existence of harmonic functions with prescribed asymptotics at infinity.A corollary of this is a non-existence result for ALG ∗ manifolds with non-negative Ricci curvature having group Γ = { e } \Gamma=\{e\} at infinity.Next, we prove a Hodge decomposition for the first de Rham cohomology group of an ALG ∗ manifold.A corollary of this is vanishing of the first Betti number for any ALG ∗ manifold with non-negative Ricci curvature.Another application of our analysis is to determine the optimal order of ALG ∗ gravitational instantons. 

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