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1. Complexes from complexes: Finite element complexes in three dimensions

   https://doi.org/10.1090/mcom/4079  

   Chen, Long; Huang, Xuehai (February 2025, Mathematics of Computation)

   In the field of solving partial differential equations (PDEs), Hilbert complexes have become highly significant. Recent advances focus on creating new complexes using the Bernstein-Gelfand-Gelfand (BGG) framework, as shown by Arnold and Hu [Found. Comput. Math. 21 (2021), pp. 1739–1774]. This paper extends their approach to three-dimensional finite element complexes. The finite element Hessian, elasticity, and divdiv complexes are systematically derived by applying techniques such as smooth finite element de Rham complexes, the $t$- $n$decomposition, and trace complexes, along with related two-dimensional finite element analogs. The construction includes two reduction operations and one augmentation operation to address continuity differences in the BGG diagram, ultimately resulting in a comprehensive and effective framework for constructing finite element complexes, which have various applications in PDE solving. 

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2. Gluing complexes of sheaves

   https://doi.org/10.4171/DM/961  

   Olsson, Martin (September 2024, Documenta Mathematica)

   We prove two variations of the classical gluing result of Beilinson–Bernstein–Deligne. We recast the problem of gluing in terms of filtered complexes in the total topos of aD-topos, in the sense of SGA 4, and prove our results using the filtered derived category. 

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3. Singular Value Decomposition of Complexes

   https://doi.org/10.1137/18M1189270  

   Brake, Danielle A.; Hauenstein, Jonathan D.; Schreyer, Frank-Olaf; Sommese, Andrew J.; Stillman, Michael E. (January 2019, SIAM Journal on Applied Algebra and Geometry)
4. Topology of Augmented Bergman Complexes

   https://doi.org/10.37236/10739  

   Bullock, Elisabeth; Kelley, Aidan; Reiner, Victor; Ren, Kevin; Shemy, Gahl; Shen, Dawei; Sun, Brian; Zhang, Zhichun Joy (January 2022, The Electronic Journal of Combinatorics)

   The augmented Bergman complex of a matroid is a simplicial complex introduced recently in work of Braden, Huh, Matherne, Proudfoot and Wang.  It may be viewed as a hybrid of two well-studied pure shellable simplicial complexes associated to matroids: the independent set complex and Bergman complex. It is shown here that the augmented Bergman complex is also shellable, via two different families of shelling orders.  Furthermore, comparing the description of its homotopy type induced from the two shellings re-interprets a known convolution formula counting bases of the matroid. The representation of the automorphism group of the matroid on the homology of the augmented Bergman complex turns out to have a surprisingly simple description. This last fact is generalized to closures beyond those coming from a matroid. 

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5. Mixed Volumes of Normal Complexes

   https://doi.org/10.1007/s00454-024-00662-w  

   Nowak, Lauren; O’Melveny, Patrick; Ross, Dustin (June 2024, Discrete & Computational Geometry)