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1. Spectral Data for Spin Higgs Bundles

   https://doi.org/10.1093/imrn/rny296  

   Mukhopadhyay, Swarnava; Wentworth, Richard (January 2019, International Mathematics Research Notices)

   null (Ed.)

   Abstract In this paper we determine the spectral data parametrizing Higgs bundles in a generic fiber of the Hitchin map for the case where the structure group is the special Clifford group with fixed Clifford norm. These are spin and “twisted” spin Higgs bundles. The method used relates variations in spectral data with respect to the Hecke transformations for orthogonal bundles introduced by Abe. The explicit description also recovers a result from the geometric Langlands program, which states that the fibers of the Hitchin map are the dual abelian varieties to the corresponding fibers of the moduli spaces of projective orthogonal Higgs bundles (in the even case) and projective symplectic Higgs bundles (in the odd case). 

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2. Perverse-Hodge complexes for Lagrangian fibrations

   https://doi.org/10.46298/epiga.2023.9617  

   Shen, Junliang; Yin, Qizheng (August 2023, Épijournal de Géométrie Algébrique)

   Perverse-Hodge complexes are objects in the derived category of coherentsheaves obtained from Hodge modules associated with Saito's decompositiontheorem. We study perverse-Hodge complexes for Lagrangian fibrations andpropose a symmetry between them. This conjectural symmetry categorifies the"Perverse = Hodge" identity of the authors and specializes to Matsushita'stheorem on the higher direct images of the structure sheaf. We verify ourconjecture in several cases by making connections with variations of Hodgestructures, Hilbert schemes, and Looijenga-Lunts-Verbitsky Lie algebras. 

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3. Convergence of the self‐dual U (1)‐Yang–Mills–Higgs energies to the (n−2)$(n-2)$‐area functional

   https://doi.org/10.1002/cpa.22150  

   Parise, Davide; Pigati, Alessandro; Stern, Daniel (January 2024, Communications on Pure and Applied Mathematics)

   Abstract Given a hermitian line bundle on a closed Riemannian manifold , the self‐dual Yang–Mills–Higgs energies are a natural family of functionalsdefined for couples consisting of a section  and a hermitian connection ∇ with curvature . While the critical points of these functionals have been well‐studied in dimension two by the gauge theory community, it was shown in [52] that critical points in higher dimension converge as (in an appropriate sense) to minimal submanifolds of codimension two, with strong parallels to the correspondence between the Allen–Cahn equations and minimal hypersurfaces. In this paper, we complement this idea by showing the Γ‐convergence of to (2π times) the codimension two area: more precisely, given a family of couples with , we prove that a suitable gauge invariant Jacobian converges to an integral ‐cycle Γ, in the homology class dual to the Euler class , with mass . We also obtain a recovery sequence, for any integral cycle in this homology class. Finally, we apply these techniques to compare min‐max values for the ‐area from the Almgren–Pitts theory with those obtained from the Yang–Mills–Higgs framework, showing that the former values always provide a lower bound for the latter. As an ingredient, we also establish a Huisken‐type monotonicity result along the gradient flow of . 

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4. Signed permutohedra, delta‐matroids, and beyond

   https://doi.org/10.1112/plms.12592  

   Eur, Christopher; Fink, Alex; Larson, Matt; Spink, Hunter (March 2024, Proceedings of the London Mathematical Society)

   Abstract We establish a connection between the algebraic geometry of the type permutohedral toric variety and the combinatorics of delta‐matroids. Using this connection, we compute the volume and lattice point counts of type generalized permutohedra. Applying tropical Hodge theory to a new framework of “tautological classes of delta‐matroids,” modeled after certain vector bundles associated to realizable delta‐matroids, we establish the log‐concavity of a Tutte‐like invariant for a broad family of delta‐matroids that includes all realizable delta‐matroids. Our results include new log‐concavity statements for all (ordinary) matroids as special cases. 

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5. Rich representations and superrigidity

   https://doi.org/10.1017/etds.2024.136  

   BALDI, GREGORIO; MILLER, NICHOLAS; STOVER, MATTHEW; ULLMO, EMMANUEL (August 2025, Ergodic Theory and Dynamical Systems)

   Abstract We investigate and compare applications of the Zilber–Pink conjecture and dynamical methods to rigidity problems for arithmetic real and complex hyperbolic lattices. Along the way, we obtain new general results about reconstructing a variation of Hodge structure from its typical Hodge locus that may be of independent interest. Applications to Siu’s immersion problem are also discussed, the most general of which only requires the hypothesis that infinitely many closed geodesics map to proper totally geodesic subvarieties under the immersion. 

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