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1. Persistent Topological Laplacians—A Survey

   https://doi.org/10.3390/math13020208  

   Wei, Xiaoqi; Wei, Guo-Wei (January 2025, Mathematics)

   Persistent topological Laplacians constitute a new class of tools in topological data analysis (TDA). They are motivated by the necessity to address challenges encountered in persistent homology when handling complex data. These Laplacians combine multiscale analysis with topological techniques to characterize the topological and geometrical features of functions and data. Their kernels fully retrieve the topological invariants of corresponding persistent homology, while their non-harmonic spectra provide supplementary information. Persistent topological Laplacians have demonstrated superior performance over persistent homology in the analysis of large-scale protein engineering datasets. In this survey, we offer a pedagogical review of persistent topological Laplacians formulated in various mathematical settings, including simplicial complexes, path complexes, flag complexes, digraphs, hypergraphs, hyperdigraphs, cellular sheaves, and N-chain complexes. 

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2. Topological metasurfaces [Invited]

   https://doi.org/10.1364/OME.529092  

   Smirnova, Daria; Kiriushechkina, Svetlana; Vakulenko, Anton; Khanikaev, Alexander B (January 2024, Optical Materials Express)

   Topological photonics allows for the deterministic creation of electromagnetic modes of any dimensionality lesser than that of the system. In the context of two-dimensional systems such as metasurfaces, topological photonics enables trapping of light in 0D cavities defined by boundaries of higher-order topological insulators and topological defects, as well as guiding of optical fields along 1D boundaries between topologically distinct domains. More importantly, it allows engineering interactions of topological modes with radiative continuum, which opens new opportunities to control light-matter interactions, scattering, generation, and emission of light. This review article aims at highlighting recent work in the field focusing on the control of radiation and generation of light in topological metasurfaces. 

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3. The topological complexity of pure graph braid groups is stably maximal

   https://doi.org/10.1017/fms.2022.83  

   Knudsen, Ben (January 2022, Forum of Mathematics, Sigma)

   Abstract We prove Farber’s conjecture on the stable topological complexity of configuration spaces of graphs. The conjecture follows from a general lower bound derived from recent insights into the topological complexity of aspherical spaces. Our arguments apply equally to higher topological complexity. 

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4. Lattice materials with topological states optimized on demand

   https://doi.org/10.1073/pnas.2506787122  

   Azizi, Pegah; Kundu, Rahul Dev; Li, Weichen; Sun, Kai; Zhang, Xiaojia Shelly; Gonella, Stefano (August 2025, Proceedings of the National Academy of Sciences)

   Topological states of matter, first discovered in quantum systems, have opened new avenues for wave manipulation beyond the quantum realm. In elastic media, realizing these topological effects requires identifying lattices that support the corresponding topological bands. However, among the vast number of theoretically predicted topological states, only a small fraction has been physically realized. To close this gap, we present a strategy capable of systematically and efficiently discovering metamaterials with desired topological state. Our approach builds on topological quantum chemistry, which systematically classifies topological states by analyzing symmetry properties at selected wavevectors. Because this method condenses the topological character into mathematical information at a small set of wavevectors, it encodes a clear and computationally efficient objective for topology optimization algorithms. We demonstrate that, for certain lattice symmetries, this classification can be further reduced to intuitive morphological features of the phonon band structure. By incorporating these band morphology constraints into topology optimization algorithms and further fabricating obtained designs, we enable the automated discovery and physical realization of metamaterials with targeted topological properties. This methodology establishes a paradigm for engineering topological elastic lattices on demand, addressing the bottleneck in material realization and paving the way for a comprehensive database of topological metamaterial configurations. 

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5. Manipulation of the magnetic monopole injection for topological transition

   https://doi.org/10.1038/s41427-024-00529-9  

   Han, Hee-Sung; Montoya, Sergio A; Fullerton, Eric E; Chao, Weilun; Je, Soong-Geun; Lee, Ki‐Suk; Im, Mi-Young (December 2024, NPG Asia Materials)

   Abstract Manipulating the topological properties of spin textures in magnetic materials is of great interest due to the rich physics and promising technological applications of these materials in advanced electronic devices. A spin texture with desired topological properties can be created by magnetic monopole injection, resulting in topological transitions involving changes in the topological charge. Therefore, controlling magnetic monopole injection has paramount importance for obtaining the desired spin textures but has not yet been reported. Here, we report the use of reliably manipulated magnetic monopole injection in the topological transition from stripe domains to skyrmions in an Fe/Gd multilayer. An easily tunable in-plane magnetic field applied to an Fe/Gd multilayer plays a key role in the magnetic monopole injection by modulating the local exchange energy. Our findings facilitate the efficient management of topological transitions by providing an important method for controlling magnetic monopole injection. 

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