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1. Boundary behavior of compact manifolds with scalar curvature lower bounds and static quasi-local mass of tori

   https://doi.org/10.1090/proc/17223  

   Alaee, Aghil; Hung, Pei-Ken; Khuri, Marcus (March 2025, Proceedings of the American Mathematical Society)

   A classic result of Shi and Tam states that a 2-sphere of positive Gauss and mean curvature bounding a compact 3-manifold with nonnegative scalar curvature must have total mean curvature not greater than that of the isometric embedding into Euclidean 3-space, with equality only for domains in this reference manifold. We generalize this result to 2-tori of Gauss curvature greater than $-<\#comment/>1$, which bound a compact 3-manifold having scalar curvature not less than $-<\#comment/>6$and at least one other boundary component satisfying a ‘trapping condition’. The conclusion is that the total weighted mean curvature is not greater than that of an isometric embedding into the Kottler manifold, with equality only for domains in this space. Examples are given to show that the assumption of a secondary boundary component cannot be removed. The result gives a positive mass theorem for the static Brown-York mass of tori, in analogy to the Shi-Tam positivity of the standard Brown-York mass, and represents the first such quasi-local mass positivity result for nonspherical surfaces. Furthermore, we prove a Penrose-type inequality in this setting. 

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2. Characterizing Designs Via Isometric Embeddings: Applications to Airfoil Inverse Design

   https://doi.org/10.1115/1.4063363  

   Chen, Qiuyi; Fuge, Mark (May 2024, Journal of Mechanical Design)

   Many design problems involve reasoning about points in high-dimensional space. A common strategy is to first embed these high-dimensional points into a low-dimensional latent space. We propose that a good embedding should be isometric—i.e., preserving the geodesic distance between points on the data manifold in the latent space. However, enforcing isometry is non-trivial for common neural embedding models such as autoencoders. Moreover, while theoretically appealing, it is unclear to what extent is enforcing isometry necessary for a given design analysis. This paper answers these questions by constructing an isometric embedding via an isometric autoencoder, which we employ to analyze an inverse airfoil design problem. Specifically, the paper describes how to train an isometric autoencoder and demonstrates its usefulness compared to non-isometric autoencoders on the UIUC airfoil dataset. Our ablation study illustrates that enforcing isometry is necessary for accurately discovering clusters through the latent space. We also show how isometric autoencoders can uncover pathologies in typical gradient-based shape optimization solvers through an analysis on the SU2-optimized airfoil dataset, wherein we find an over-reliance of the gradient solver on the angle of attack. Overall, this paper motivates the use of isometry constraints in neural embedding models, particularly in cases where researchers or designers intend to use distance-based analysis measures to analyze designs within the latent space. While this work focuses on airfoil design as an illustrative example, it applies to any domain where analyzing isometric design or data embeddings would be useful. 

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3. Characterizing Designs via Isometric Embeddings: Applications to Airfoil Inverse Design

   https://doi.org/10.1115/DETC2023-116743  

   Chen, Qiuyi; Fuge, Mark D (August 2023, American Society of Mechanical Engineers)

   Many data analysis and design problems involve reasoning about points in high-dimensional space. A common strategy is to embed points from this high-dimensional space into a low-dimensional one. As we will show in this paper, a critical property of good embeddings is that they preserve isometry — i.e., preserving the geodesic distance between points on the original data manifold within their embedded locations in the latent space. However, enforcing isometry is non-trivial for common Neural embedding models, such as autoencoders and generative models. Moreover, while theoretically appealing, it is not clear to what extent enforcing isometry is really necessary for a given design or analysis task. This paper answers these questions by constructing an isometric embedding via an isometric autoencoder, which we employ to analyze an inverse airfoil design problem. Specifically, the paper describes how to train an isometric autoencoder and demonstrates its usefulness compared to non-isometric autoencoders on both simple pedagogical examples and for airfoil embeddings using the UIUC airfoil dataset. Our ablation study illustrates that enforcing isometry is necessary to accurately discover latent space clusters — a common analysis method researchers typically perform on low-dimensional embeddings. We also show how isometric autoencoders can uncover pathologies in typical gradient-based Shape Optimization solvers through an analysis on the SU2-optimized airfoil dataset, wherein we find an over-reliance of the gradient solver on angle of attack. Overall, this paper motivates the use of isometry constraints in Neural embedding models, particularly in cases where researchers or designer intend to use distance-based analysis measures (such as clustering, k-Nearest Neighbors methods, etc.) to analyze designs within the latent space. While this work focuses on airfoil design as an illustrative example, it applies to any domain where analyzing isometric design or data embeddings would be useful. 

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4. Topological k-Metrics

   https://doi.org/10.4230/LIPIcs.SoCG.2024.13  

   Barkan, Willow; Bennett, Huck; Nayyeri, Amir (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Mulzer, Wolfgang; Phillips, Jeff M (Ed.)

   Metric spaces (X, d) are ubiquitous objects in mathematics and computer science that allow for capturing pairwise distance relationships d(x, y) between points x, y ∈ X. Because of this, it is natural to ask what useful generalizations there are of metric spaces for capturing "k-wise distance relationships" d(x_1, …, x_k) among points x_1, …, x_k ∈ X for k > 2. To that end, Gähler (Math. Nachr., 1963) (and perhaps others even earlier) defined k-metric spaces, which generalize metric spaces, and most notably generalize the triangle inequality d(x₁, x₂) ≤ d(x₁, y) + d(y, x₂) to the "simplex inequality" d(x_1, …, x_k) ≤ ∑_{i=1}^k d(x_1, …, x_{i-1}, y, x_{i+1}, …, x_k). (The definition holds for any fixed k ≥ 2, and a 2-metric space is just a (standard) metric space.) In this work, we introduce strong k-metric spaces, k-metric spaces that satisfy a topological condition stronger than the simplex inequality, which makes them "behave nicely." We also introduce coboundary k-metrics, which generalize 𝓁_p metrics (and in fact all finite metric spaces induced by norms) and minimum bounding chain k-metrics, which generalize shortest path metrics (and capture all strong k-metrics). Using these definitions, we prove analogs of a number of fundamental results about embedding finite metric spaces including Fréchet embedding (isometric embedding into 𝓁_∞) and isometric embedding of all tree metrics into 𝓁₁. We also study relationships between families of (strong) k-metrics, and show that natural quantities, like simplex volume, are strong k-metrics. 

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5. Some integral geometry problems for wave equations

   https://doi.org/10.1088/1361-6420/ac77b1  

   Wang, Yiran (June 2022, Inverse Problems)

   Abstract We consider the Cauchy problem and the source problem for normally hyperbolic operators on the Minkowski spacetime, and study the determination of solutions from their integrals along light-like geodesics. For the Cauchy problem, we give a new proof of the stable determination result obtained by Vasy and Wang (2021 Commun. Math. Phys. 384 503–32). For the source problem, we obtain stable determination for sources with space-like singularities. Our proof is based on the microlocal analysis of the normal operator of the light ray transform composed with the parametrix for strictly hyperbolic operators. 

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