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1. Trading networks and Hodge theory ^∗

   https://doi.org/10.1088/2399-6528/abd1c2  

   Schenck, Henry; Sowers, Richard; Song, Rui (January 2021, Journal of Physics Communications)

   Abstract The problem of analyzing interconnectedness is one of today’s premier challenges in understanding systemic risk. Connections can both stabilize networks and provide pathways for contagion. The central problem in such networks is establishing global behavior from local interactions. Jiang-Lim-Yao-Ye (Jianget al2011Mathematical Programming 1271203–244) recently introduced the use of theHodge decomposition(see Lim 2020SIAM Review62685–715 for a review), a fundamental tool from algebraic geometry, to construct global rankings from local interactions (see Barbarossaet al2018(2018 IEEE Data Science Workshop (DSW), IEEE)pp 51–5; Haruna and Fujiki 2016Frontiers in Neural Circuits1077; Jiaet al2019(Proc. of the XXV ACM SIGKDD International Conf. on Knowledge Discovery & Data Mining, pp 761–71 for other applications). We apply this to a study of financial networks, starting from the Eisenberg-Noe (Eisenberg and Noe 2001Management Science47236–249) setup of liabilities and endowments, and construct a network of defaults. We then use Jiang-Lim-Yao-Ye to construct a global ranking from the defaults, which yields one way of quantifying ‘systemic importance’. 

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2. Bonding, Thermodynamics, and Dissociation Dynamics of NiO⁺ and NiS⁺ Determined by Photofragment Imaging and Theory

   https://doi.org/10.1021/acs.jpca.1c05405  

   Lockwood, Schuyler P.; Chunga, Tala; Metz, Ricardo B. (September 2021, The Journal of Physical Chemistry A)
3. Hodge theory on ALG ^∗ manifolds

   https://doi.org/10.1515/crelle-2023-0016  

   Chen, Gao; Viaclovsky, Jeff; Zhang, Ruobing (May 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We develop a Fredholm theory for the Hodge Laplacian in weighted spaces on ALG ∗ manifolds in dimension four.We then give several applications of this theory.First, we show the existence of harmonic functions with prescribed asymptotics at infinity.A corollary of this is a non-existence result for ALG ∗ manifolds with non-negative Ricci curvature having group Γ = { e } \Gamma=\{e\} at infinity.Next, we prove a Hodge decomposition for the first de Rham cohomology group of an ALG ∗ manifold.A corollary of this is vanishing of the first Betti number for any ALG ∗ manifold with non-negative Ricci curvature.Another application of our analysis is to determine the optimal order of ALG ∗ gravitational instantons. 

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4. Probing the Electronic Manifold of MgCl with Millimeter-Wave Spectroscopy and Theory: (3) ² Σ ⁺ and (4) ² Σ ⁺ States

   https://doi.org/10.1021/acs.jpca.4c05458  

   Herman, T_J; Ravi, R.; Schuurman, M_S; DeYonker, N_J; Field, R_W; Ziurys, L_M (November 2024, The Journal of Physical Chemistry A)
5. Theory of anisotropic superfluid ⁴ He counterflow turbulence

   https://doi.org/10.1098/rsta.2021.0094  

   L'vov, Victor S.; Lvov, Yuri V.; Nazarenko, Sergey; Pomyalov, Anna (March 2022, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   We develop a theory of strong anisotropy of the energy spectra in the thermally driven turbulent counterflow of superfluid 4 He. The key ingredients of the theory are the three-dimensional differential closure for the vector of the energy flux and the anisotropy of the mutual friction force. We suggest an approximate analytic solution of the resulting energy-rate equation, which is fully supported by our numerical solution. The two-dimensional energy spectrum is strongly confined in the direction of the counterflow velocity. In agreement with the experiments, the energy spectra in the direction orthogonal to the counterflow exhibit two scaling ranges: a near-classical non-universal cascade dominated range and a universal critical regime at large wavenumbers. The theory predicts the dependence of various details of the spectra and the transition to the universal critical regime on the flow parameters. This article is part of the theme issue ‘Scaling the turbulence edifice (part 2)’. 

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