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1. Global-to-local-to-global interactions and climate change

   https://doi.org/10.1088/1748-9326/acc95c  

   Baldos, Uris_Lantz C; Chepeliev, Maksym; Cultice, Brian; Huber, Matthew; Meng, Sisi; Ruane, Alex C; Suttles, Shellye; van_der_Mensbrugghe, Dominique (April 2023, Environmental Research Letters)

   Abstract Climate change by its very nature epitomizes the necessity and usefulness of the global-to-local-to-global (GLG) paradigm. It is a global problem with the potential to affect local communities and ecosystems. Accumulation of local impacts and responses to climate change feeds back to regional and global systems creating feedback loops. Understanding these complex impacts and interactions is key to developing more resilient adaptation measures and designing more efficient mitigation policies. To this date, however, GLG interactions have not yet been an integrative part of the decision-support toolkit. The typical approach either traces the impacts of global action on the local level or estimates the implications of local policies at the global scale. The first approach misses cumulative feedback of local responses that can have regional, national or global impacts. In the second case, one undermines a global context of the local actions most likely misrepresenting the complexity of the local decision-making process. Potential interactions across scales are further complicated by the presence of cascading impacts, connected risks and tipping points. Capturing these dimensions is not always a straightforward task and often requires a departure from conventional modeling approaches. In this paper, we review the state-of-the-art approaches to modeling GLG interactions in the context of climate change. We further identify key limitations that drive the lack of GLG coupling cases and discuss what could be done to address these challenges. 

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2. A threefold violating a local-to-global principle for rationality

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   Frei, Sarah; Ji, Lena (June 2024, Research in Number Theory)
3. A local to global argument on low dimensional manifolds

   https://doi.org/10.1090/tran/7970  

   Nariman, Sam (February 2020, Transactions of the American Mathematical Society)
4. A non-loxodromic Morse element in a Morse local-to-global group

   Abbott, Carolyn; Zbinden, Stefanie (February 2025, arXiv.org)
5. Topology Applied to Machine Learning: From Global to Local

   https://doi.org/10.3389/frai.2021.668302  

   Adams, Henry; Moy, Michael (May 2021, Frontiers in Artificial Intelligence)

   null (Ed.)

   Through the use of examples, we explain one way in which applied topology has evolved since the birth of persistent homology in the early 2000s. The first applications of topology to data emphasized the global shape of a dataset, such as the three-circle model for 3 × 3 pixel patches from natural images, or the configuration space of the cyclo-octane molecule, which is a sphere with a Klein bottle attached via two circles of singularity. In these studies of global shape, short persistent homology bars are disregarded as sampling noise. More recently, however, persistent homology has been used to address questions about the local geometry of data. For instance, how can local geometry be vectorized for use in machine learning problems? Persistent homology and its vectorization methods, including persistence landscapes and persistence images, provide popular techniques for incorporating both local geometry and global topology into machine learning. Our meta-hypothesis is that the short bars are as important as the long bars for many machine learning tasks. In defense of this claim, we survey applications of persistent homology to shape recognition, agent-based modeling, materials science, archaeology, and biology. Additionally, we survey work connecting persistent homology to geometric features of spaces, including curvature and fractal dimension, and various methods that have been used to incorporate persistent homology into machine learning. 

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