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1. “Lagrangian disks” in M-theory

   https://doi.org/10.1007/JHEP11(2020)033  

   Franco, Sebastían; Gukov, Sergei; Lee, Sangmin; Seong, Rak-Kyeong; Sparks, James (November 2020, Journal of High Energy Physics)

   null (Ed.)

   A bstract While the study of bordered (pseudo-)holomorphic curves with boundary on Lagrangian submanifolds has a long history, a similar problem that involves (special) Lagrangian submanifolds with boundary on complex surfaces appears to be largely overlooked in both physics and math literature. We relate this problem to geometry of coassociative submanifolds in G 2 holonomy spaces and to Spin(7) metrics on 8-manifolds with T 2 fibrations. As an application to physics, we propose a large class of brane models in type IIA string theory that generalize brane brick models on the one hand and 2d theories T [ M 4 ] on the other. 

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2. Shape Analysis of Functional Data with Elastic Partial Matching

   https://doi.org/10.1109/TPAMI.2021.3130535  

   Bryner, Darshan; Srivastava, Anuj (November 2021, IEEE Transactions on Pattern Analysis and Machine Intelligence)

   Elastic Riemannian metrics have been used successfully for statistical treatments of functional and curve shape data. However, this usage suffers from a significant restriction: the function boundaries are assumed to be fixed and matched. Functional data often comes with unmatched boundaries, {\it e.g.}, in dynamical systems with variable evolution rates, such as COVID-19 infection rate curves associated with different geographical regions. Here, we develop a Riemannian framework that allows for partial matching, comparing, and clustering functions under phase variability {\it and} uncertain boundaries. We extend past work by (1) Defining a new diffeomorphism group G over the positive reals that is the semidirect product of a time-warping group and a time-scaling group; (2) Introducing a metric that is invariant to the action of G; (3) Imposing a Riemannian Lie group structure on G to allow for an efficient gradient-based optimization for elastic partial matching; and (4) Presenting a modification that, while losing the metric property, allows one to control the amount of boundary disparity in the registration. We illustrate this framework by registering and clustering shapes of COVID-19 rate curves, identifying basic patterns, minimizing mismatch errors, and reducing variability within clusters compared to previous methods. 

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3. Residually finite rationally p groups

   https://doi.org/10.1142/S0219199719500160  

   Koberda, Thomas; Suciu, Alexander I. (May 2020, Communications in Contemporary Mathematics)

   In this paper, we develop the theory of residually finite rationally [Formula: see text] (RFR[Formula: see text]) groups, where [Formula: see text] is a prime. We first prove a series of results about the structure of finitely generated RFR[Formula: see text] groups (either for a single prime [Formula: see text], or for infinitely many primes), including torsion-freeness, a Tits alternative, and a restriction on the BNS invariant. Furthermore, we show that many groups which occur naturally in group theory, algebraic geometry, and in 3-manifold topology enjoy this residual property. We then prove a combination theorem for RFR[Formula: see text] groups, which we use to study the boundary manifolds of algebraic curves [Formula: see text] and in [Formula: see text]. We show that boundary manifolds of a large class of curves in [Formula: see text] (which includes all line arrangements) have RFR[Formula: see text] fundamental groups, whereas boundary manifolds of curves in [Formula: see text] may fail to do so. 

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4. Holomorphic curves in the 6-pseudosphere and cyclic surfaces

   https://doi.org/10.1090/tran/9172  

   Collier, Brian; Toulisse, Jérémy (June 2024, Transactions of the American Mathematical Society)

   The space ${\mathbf{H}}^{4,2}$of vectors of norm $-<\#comment/>1$in ${\mathbb{R}}^{4,3}$has a natural pseudo-Riemannian metric and a compatible almost complex structure. The group of automorphisms of both of these structures is the split real form ${\mathsf{G}}_2^{\prime }$. In this paper we consider a class of holomorphic curves in ${\mathbf{H}}^{4,2}$which we call alternating. We show that such curves admit a so called Frenet framing. Using this framing, we show that the space of alternating holomorphic curves which are equivariant with respect to a surface group is naturally parameterized by certain ${\mathsf{G}}_2^{\prime }$-Higgs bundles. This leads to a holomorphic description of the moduli space as a fibration over Teichmüller space with a holomorphic action of the mapping class group. Using a generalization of Labourie’s cyclic surfaces, we then show that equivariant alternating holomorphic curves are infinitesimally rigid. 

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5. Evidence of Mixed Scaling for Mean Profile Similarity in the Stable Atmospheric Surface Layer

   https://doi.org/10.1175/JAS-D-22-0260.1  

   Heisel, Michael; Chamecki, Marcelo (August 2023, Journal of the Atmospheric Sciences)

   Abstract A new mixed scaling parameterZ=z/(Lh)^1/2is proposed for similarity in the stable atmospheric surface layer, wherezis the height,Lis the Obukhov length, andhis the boundary layer depth. In comparison with the parameterζ=z/Lfrom Monin–Obukhov similarity theory (MOST), the new parameterZleads to improved mean profile similarity for wind speed and air temperature in large-eddy simulations. It also yields the same linear similarity relation for CASES-99 field measurements, including in the strongly stable (but still turbulent) regime where large deviations from MOST are observed. Results further suggest that similarity for turbulent energy dissipation rate depends on bothZandζ. The proposed mixed scaling ofZand relevance ofhcan be explained by physical arguments related to the limit ofz-less stratification that is reached asymptotically above the surface layer. The presented evidence and fitted similarity relations are promising, but the results and arguments are limited to a small sample of idealized stationary stable boundary layers. Corroboration is needed from independent datasets and analyses, including for complex and transient conditions not tested here. 

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