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1. C*-Framework for Higher-Order Bulk-Boundary Correspondences

   https://doi.org/10.1007/s00220-025-05415-1  

   Polo_Ojito, Danilo; Prodan, Emil; Stoiber, Tom (October 2025, Communications in Mathematical Physics)

   A typical crystal is a finite piece of a material which may be invariant under some point symmetry group. If it is a so-called intrinsic higher-order topological insulator or superconductor, then it displays boundary modes at hinges or corners protected by the crystalline symmetry and the bulk topology. We explain the mechanism behind such phenomena using operator K-theory. Specifically, we derive a groupoid C ∗ -algebra that (1) encodes the dynamics of the electrons in the infinite size limit of a crystal; (2) remembers the boundary conditions at the crystal’s boundaries, and (3) admits a natural action by the point symmetries of the atomic lattice. The filtrations of the groupoid’s unit space by closed subsets that are invariant under the groupoid and point group actions supply equivariant cofiltrations of the groupoid C ∗ -algebra. We show that specific derivations of the induced spectral sequences in twisted equivariant K-theories enumerate all non-trivial higher-order bulk-boundary correspondences. 

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2. Model structures on finite total orders

   https://doi.org/10.1007/s00209-023-03287-6  

   Balchin, Scott; Ormsby, Kyle; Osorno, Angélica M.; Roitzheim, Constanze (July 2023, Mathematische Zeitschrift)

   Abstract We initiate the study of model structures on (categories induced by) lattice posets, a subject we dub homotopical combinatorics . In the case of a finite total order [ n ], we enumerate all model structures, exhibiting a rich combinatorial structure encoded by Shapiro’s Catalan triangle. This is an application of previous work of the authors on the theory of $$N_\infty $$ N ∞ -operads for cyclic groups of prime power order, along with new structural insights concerning extending choices of certain model structures on subcategories of [ n ]. 

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3. Realizing the s-Permutahedron via Flow Polytopes

   González_D'León, Rafael_S; Morales, Alejandro_H; Philippe, Eva; Tamayo_Jiménez, Daniel; Yip, Martha (July 2024, Séminaire Lotharingien de Combinatoire)

   In 2019, Ceballos and Pons introduced the s-weak order on s-decreasing trees, for any weak composition s. They proved its lattice structure and conjectured that it could be realized as the 1-skeleton of a polyhedral subdivision of a zonotope of dimension n-1. We answer their conjecture in the case where s is a (strict) composition by providing three geometric realizations of the s-permutahedron. The first one is the dual graph of a triangulation of a flow polytope of high dimension. The second, obtained using the Cayley trick, is the dual graph of a fine mixed subdivision of a sum of hypercubes that has the conjectured dimension. The third, obtained using tropical geometry, is the 1-skeleton of a polyhedral complex for which we can provide explicit coordinates of the vertices and whose support is a permutahedron as conjectured. 

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4. Combinatorial Results on Barcode Lattices

   https://doi.org/10.1007/s11083-024-09670-0  

   Bouquet, Alex; Vindas-Meléndez, Andrés_R (July 2024, Order)

   Abstract A barcode is a finite multiset of intervals on the real line. Jaramillo-Rodriguez (2023) previously defined a map from the space of barcodes with a fixed number of bars to a set of multipermutations, which presented new combinatorial invariants on the space of barcodes. A partial order can be defined on these multipermutations, resulting in a class of posets known as combinatorial barcode lattices. In this paper, we provide a number of equivalent definitions for the combinatorial barcode lattice, show that its Möbius function is a restriction of the Möbius function of the symmetric group under the weak Bruhat order, and show its ground set is the Jordan-Hölder set of a labeled poset. Furthermore, we obtain formulas for the number of join-irreducible elements, the rank-generating function, and the number of maximal chains of combinatorial barcode lattices. Lastly, we make connections between intervals in the combinatorial barcode lattice and certain classes of matchings. 

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5. Mechanical properties of additively manufactured variable lattice structures of Ti6Al4V

   https://doi.org/doi.org/10.1016/j.msea.2021.140925  

   Traxel, Kellen; Groden, Cory; Valladares, Jesus; Bandyopadhyay, Amit (March 2021, Materials science engineering)

   null (Ed.)

   Engineered micro- and macro-structures via additive manufacturing (AM) or 3D-Printing can create structurally varying properties in a part, which is difficult via traditional manufacturing methods. Herein we have utilized powder bed fusion-based selective laser melting (SLM) to fabricate variable lattice structures of Ti6Al4V with uniquely designed unit cell configurations to alter the mechanical performance. Five different configurations were designed based on two natural crystal structures – hexagonal closed packed (HCP) and body centered cubic (BCC). Under compressive loading, as much as 74% difference was observed in compressive strength, and 71% variation in elastic modulus, with all samples having porosities in a similar range of 53 to 65%, indicating the influence of macro-lattice designs alone on mechanical properties. Failure analysis of the fracture surfaces helped with the overall understanding of how configurational effects and unit cell design influences mechanical properties of these samples. Our work highlights the ability to leverage advanced manufacturing techniques to tailor the structural performance of multifunctional components. 

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