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1. Pointwise Asymptotics for Orthonormal Polynomials at the Endpoints of the Interval Via Universality

   Lubinsky, Doron (January 2019, International mathematics research notices)

   We show that universality limits and other bounds imply pointwise asymptotics for orthonormal polynomials at the endpoints of the interval of orthonormality. As a consequence, we show that if μ is a regular measure supported on [−1, 1], and in a neighborhood of 1, μ is absolutely continuous, while for some α > −1, μ (t) = h (t)(1 − t) α, where h (t) → 1 as t → 1−, then the corresponding orthonormal polynomials {pn} satisfy the asymptotic limn→∞pn1 − z22n2pn (1) = J∗α (z)J∗α (0) uniformly in compact subsets of the plane. Here J∗α (z) = Jα (z) /zα is the normalized Bessel function of order α. These are by far the most general conditions for such endpoint asymptotics 

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2. Cubic Planar Graphs and Legendrian Surface Theory

   https://doi.org/http://dx.doi.org/10.4310/ATMP.2018.v22.n5.a5  

   Treumann, David; Zaslow, Eric (May 2018, Advances in theoretical and mathematical physics)

   We study Legendrian surfaces determined by cubic planar graphs. Graphs with distinct chromatic polynomials determine surfaces that are not Legendrian isotopic, thus giving many examples of non-isotopic Legendrian surfaces with the same classical invariants. The Legendrians have no exact Lagrangian fillings, but have many interesting non-exact fillings. We obtain these results by studying sheaves on a three-ball with microsupport in the surface. The moduli of such sheaves has a concrete description in terms of the graph and a beautiful embedding as a holomorphic Lagrangian submanifold of a symplectic period domain, a Lagrangian that has appeared in the work of Dimofte–Gabella–Goncharov. We exploit this structure to find conjectural open Gromov–Witten invariants for the non-exact filling, following Aganagic–Vafa. 

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3. Towards Scalable Spectral Embedding and Data Visualization via Spectral Coarsening

   https://doi.org/10.1145/3437963.3441767  

   Zhao, Zhiqiang; Zhang, Ying; Feng, Zhuo (March 2021, ACM International Conference on Web Search and Data Mining)

   null (Ed.)

   This paper proposes a scalable multilevel framework for the spectral embedding of large undirected graphs. The proposed method first computes much smaller yet sparse graphs while preserving the key spectral (structural) properties of the original graph, by exploiting a nearly-linear time spectral graph coarsening approach. Then, the resultant spectrally-coarsened graphs are leveraged for the development of much faster algorithms for multilevel spectral graph embedding (clustering) as well as visualization of large data sets. We conducted extensive experiments using a variety of large graphs and datasets and obtained very promising results. For instance, we are able to coarsen the "coPapersCiteseer" graph with 0.43 million nodes and 16 million edges into a much smaller graph with only 13K (32X fewer) nodes and 17K (950X fewer) edges in about 16 seconds; the spectrally-coarsened graphs allow us to achieve up to 1,100X speedup for multilevel spectral graph embedding (clustering) and up to 60X speedup for t-SNE visualization of large data sets. 

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4. Infinite flags and Schubert polynomials

   https://doi.org/10.1017/fms.2024.99  

   Anderson, David (January 2025, Forum of Mathematics, Sigma)

   Abstract We study Schubert polynomials using geometry of infinite-dimensional flag varieties and degeneracy loci. Applications include Graham-positivity of coefficients appearing in equivariant coproduct formulas and expansions of back-stable and enriched Schubert polynomials. We also construct an embedding of the type C flag variety and study the corresponding pullback map on (equivariant) cohomology rings. 

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5. Topological Pooling on Graphs

   https://doi.org/10.1609/aaai.v37i6.25866  

   Chen, Yuzhou; Gel, Yulia R. (June 2023, Proceedings of the AAAI Conference on Artificial Intelligence)

   Graph neural networks (GNNs) have demonstrated a significant success in various graph learning tasks, from graph classification to anomaly detection. There recently has emerged a number of approaches adopting a graph pooling operation within GNNs, with a goal to preserve graph attributive and structural features during the graph representation learning. However, most existing graph pooling operations suffer from the limitations of relying on node-wise neighbor weighting and embedding, which leads to insufficient encoding of rich topological structures and node attributes exhibited by real-world networks. By invoking the machinery of persistent homology and the concept of landmarks, we propose a novel topological pooling layer and witness complex-based topological embedding mechanism that allow us to systematically integrate hidden topological information at both local and global levels. Specifically, we design new learnable local and global topological representations Wit-TopoPool which allow us to simultaneously extract rich discriminative topological information from graphs. Experiments on 11 diverse benchmark datasets against 18 baseline models in conjunction with graph classification tasks indicate that Wit-TopoPool significantly outperforms all competitors across all datasets. 

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