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1. Stable homotopy refinement of quantum annular homology

   https://doi.org/10.1112/S0010437X20007721  

   Akhmechet, Rostislav; Krushkal, Vyacheslav; Willis, Michael (April 2021, Compositio Mathematica)

   null (Ed.)

   We construct a stable homotopy refinement of quantum annular homology, a link homology theory introduced by Beliakova, Putyra and Wehrli. For each $$r\geq ~2$$ we associate to an annular link $$L$$ a naive $$\mathbb {Z}/r\mathbb {Z}$$ -equivariant spectrum whose cohomology is isomorphic to the quantum annular homology of $$L$$ as modules over $$\mathbb {Z}[\mathbb {Z}/r\mathbb {Z}]$$ . The construction relies on an equivariant version of the Burnside category approach of Lawson, Lipshitz and Sarkar. The quotient under the cyclic group action is shown to recover the stable homotopy refinement of annular Khovanov homology. We study spectrum level lifts of structural properties of quantum annular homology. 

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2. Symmetries of the cyclic nerve

   https://doi.org/10.1016/j.aim.2025.110170  

   Ayala, David; Mazel-Gee, Aaron; Rozenblyum, Nick (April 2025, Advances in Mathematics)

   We undertake a systematic study of the Hochschild homology, i.e. (the geometric realization of) the cyclic nerve, of -categories (and more generally of category-objects in an ∞-category), as a version of factorization homology. In order to do this, we codify -categories in terms of quiver representations in them. By examining a universal instance of such Hochschild homology, we explicitly identify its natural symmetries, and construct a non-stable version of the cyclotomic trace map. Along the way we give a unified account of the cyclic, paracyclic, and epicyclic categories. We also prove that this gives a combinatorial description of the case of factorization homology as presented in [4], which parametrizes -categories by solidly 1-framed stratified spaces. 

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3. Spanning Configurations and Representation Stability

   https://doi.org/10.37236/11136  

   Pawlowski, Brendan; Ramos, Eric; Rhoades, Brendon (January 2023, The Electronic Journal of Combinatorics)

   Let $$V_1, V_2, V_3, \dots $$ be a sequence of $$\mathbb {Q}$$-vector spaces where $$V_n$$ carries an action of $$\mathfrak{S}_n$$. Representation stability and multiplicity stability are two related notions of when the sequence $$V_n$$ has a limit. An important source of stability phenomena arises when $$V_n$$ is the $$d^{th}$$ homology group (for fixed $$d$$) of the configuration space of $$n$$ distinct points in some fixed topological space $$X$$. We replace these configuration spaces with moduli spaces of tuples $$(W_1, \dots, W_n)$$ of subspaces of a fixed complex vector space $$\mathbb {C}^N$$ such that $$W_1 + \cdots + W_n = \mathbb {C}^N$$. These include the varieties of spanning line configurations which are tied to the Delta Conjecture of symmetric function theory. 

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4. Convex cocompactness for Coxeter groups

   https://doi.org/10.4171/jems/1539  

   Danciger, Jeffrey; Guéritaud, François; Kassel, Fanny; Lee, Gye-Seon; Marquis, Ludovic (February 2025, Journal of the European Mathematical Society)

   We investigate representations of Coxeter groups into\mathrm{GL}(n,\mathbb{R})as geometric reflection groups which are convex cocompact in the projective space\mathbb{P}(\mathbb{R}^{n}). We characterize which Coxeter groups admit such representations, and we fully describe the corresponding spaces of convex cocompact representations as reflection groups, in terms of the associated Cartan matrices. The Coxeter groups that appear include all infinite word hyperbolic Coxeter groups; for such groups, the representations as reflection groups that we describe are exactly the projective Anosov ones. We also obtain a large class of nonhyperbolic Coxeter groups, thus providing many examples for the theory of nonhyperbolic convex cocompact subgroups in\mathbb{P}(\mathbb{R}^{n})developed by Danciger–Guéritaud–Kassel (2024). 

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5. Splicing knot complements and bordered Floer homology

   https://doi.org/10.1515/crelle-2014-0064  

   Hedden, Matthew; Levine, Adam Simon (January 2016, Journal für die reine und angewandte Mathematik (Crelles Journal))

   null (Ed.)

   Abstract We show that the integer homology sphere obtained by splicing two nontrivial knot complements in integer homology sphere L-spaces has Heegaard Floer homology of rank strictly greater than one. In particular, splicing the complements of nontrivial knots in the 3-sphere never produces an L-space. The proof uses bordered Floer homology. 

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