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1. Hodge Representations

   https://doi.org/10.1017/exp.2020.55  

   Han, Xiayimei; Robles, Colleen (January 2020, Experimental Results)

   Clingher, Adrian (Ed.)

   Abstract Green–Griffiths–Kerr introduced Hodge representations to classify the Hodge groups of polarized Hodge structures, and the corresponding Mumford–Tate subdomains. We summarize how, given a fixed period domain $$ \mathcal{D} $$ , to enumerate the Hodge representations and corresponding Mumford–Tate subdomains $$ D \subset \mathcal{D} $$ . The procedure is illustrated in two examples: (i) weight two Hodge structures with $$ {p}_g={h}^{2,0}=2 $$ ; and (ii) weight three CY-type Hodge structures. 

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2. Hodge and Teichmüller

   https://doi.org/10.3934/jmd.2022007  

   Kahn, Jeremy; Wright, Alex (January 2022, Journal of Modern Dynamics)

   We consider the derivative \begin{document}$$ D\pi $$\end{document} of the projection \begin{document}$$ \pi $$\end{document} from a stratum of Abelian or quadratic differentials to Teichmüller space. A closed one-form \begin{document}$$ \eta $$\end{document} determines a relative cohomology class \begin{document}$$ [\eta]_\Sigma $$\end{document}, which is a tangent vector to the stratum. We give an integral formula for the pairing of \begin{document}$$ D\pi([\eta]_\Sigma) $$\end{document} with a cotangent vector to Teichmüller space (a quadratic differential). We derive from this a comparison between Hodge and Teichmüller norms, which has been used in the work of Arana-Herrera on effective dynamics of mapping class groups, and which may clarify the relationship between dynamical and geometric hyperbolicity results in Teichmüller theory. 

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3. Hodge Laplacian of Brain Networks

   https://doi.org/10.1109/TMI.2022.3233876  

   Anand, D. Vijay; Chung, Moo K. (May 2023, IEEE Transactions on Medical Imaging)
4. Hodge-Decomposition of Brain Networks

   https://doi.org/10.1109/ISBI56570.2024.10635590  

   Anand, D Vijay; Chung, Moo K (May 2024, Proceedings)
5. Evolutionary de Rham-Hodge method

   https://doi.org/10.3934/dcdsb.2020257  

   Chen, Jiahui; Zhao, Rundong; Tong, Yiying; Wei, Guo-Wei (January 2021, Discrete & Continuous Dynamical Systems - B)

   null (Ed.)