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1. Families of metrics with positive scalar curvature on spectral sequence cobordisms

   https://doi.org/10.1142/S1793525324500109  

   Gong, Sherry (December 2025, Journal of Topology and Analysis)

   We study families of metrics on the cobordisms that underlie the differential maps in Bloom’s monopole Floer spectral sequence, a spectral sequence for links in [Formula: see text] whose [Formula: see text] page is the Khovanov homology of the link, and which abuts to the monopole Floer homology of the double branched cover of the link. The higher differentials in the spectral sequence count parametrized moduli spaces of solutions to Seiberg–Witten equations, parametrized over a family of metrics with asymptotic behavior corresponding to a configuration of unlinks with 1-handle attachments. For a class of configurations, we construct families of metrics with the prescribed behavior, such that each metric therein has positive scalar curvature. The positive scalar curvature implies that there are no irreducible solutions to the Seiberg–Witten equations and thus, when the spectral sequences are computed with these families of metrics, only reducible solutions must be counted. The class of configurations for which we construct these families of metrics includes all configurations that go into the spectral sequence for [Formula: see text] torus knots, and all configurations that involve exactly two 1-handle attachments. 

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2. Equivariant annular Khovanov homology

   https://doi.org/10.1142/S0218216523500025  

   Akhmechet, Rostislav (February 2023, Journal of Knot Theory and Its Ramifications)

   We construct an equivariant version of annular Khovanov homology via the Frobenius algebra associated with [Formula: see text]-equivariant cohomology of [Formula: see text]. Motivated by the relationship between the Temperley–Lieb algebra and annular Khovanov homology, we also introduce an equivariant analog of the Temperley–Lieb algebra. 

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3. Compact quantum metric spaces from free graph algebras

   https://doi.org/10.1142/S0129167X22500732  

   Aguilar, Konrad; Hartglass, Michael; Penneys, David (September 2022, International Journal of Mathematics)

   Starting with a vertex-weighted pointed graph [Formula: see text], we form the free loop algebra [Formula: see text] defined in Hartglass–Penneys’ article on canonical [Formula: see text]-algebras associated to a planar algebra. Under mild conditions, [Formula: see text] is a non-nuclear simple [Formula: see text]-algebra with unique tracial state. There is a canonical polynomial subalgebra [Formula: see text] together with a Dirac number operator [Formula: see text] such that [Formula: see text] is a spectral triple. We prove the Haagerup-type bound of Ozawa–Rieffel to verify [Formula: see text] yields a compact quantum metric space in the sense of Rieffel. We give a weighted analog of Benjamini–Schramm convergence for vertex-weighted pointed graphs. As our [Formula: see text]-algebras are non-nuclear, we adjust the Lip-norm coming from [Formula: see text] to utilize the finite dimensional filtration of [Formula: see text]. We then prove that convergence of vertex-weighted pointed graphs leads to quantum Gromov–Hausdorff convergence of the associated adjusted compact quantum metric spaces. As an application, we apply our construction to the Guionnet–Jones–Shyakhtenko (GJS) [Formula: see text]-algebra associated to a planar algebra. We conclude that the compact quantum metric spaces coming from the GJS [Formula: see text]-algebras of many infinite families of planar algebras converge in quantum Gromov–Hausdorff distance. 

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4. Edge-Cuts of Optimal Average Weights

   https://doi.org/10.1142/S0217595919400062  

   Payne, Scott; Fuller, Edgar; Zhang, Cun-Quan (April 2019, Asia-Pacific Journal of Operational Research)

   Let [Formula: see text] be a directed graph associated with a weight [Formula: see text]. For an edge-cut [Formula: see text] of [Formula: see text], the average weight of [Formula: see text] is denoted and defined as [Formula: see text]. An optimal edge-cut with average weight is an edge-cut [Formula: see text] such that [Formula: see text] is maximum among all edge-cuts (or minimum, symmetrically). In this paper, a polynomial algorithm for this problem is proposed for finding an optimal edge-cut in a rooted tree separating the root and the set of all leafs. This algorithm enables us to develop an automatic clustering method with more accurate detection of community output. 

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5. Local algorithms for maximum cut and minimum bisection on locally treelike regular graphs of large degree

   https://doi.org/10.1002/rsa.21149  

   El Alaoui, Ahmed; Montanari, Andrea; Sellke, Mark (May 2023, Random Structures & Algorithms)

   Abstract Given a graph of degree over vertices, we consider the problem of computing a near maximum cut or a near minimum bisection in polynomial time. For graphs of girth , we develop a local message passing algorithm whose complexity is , and that achieves near optimal cut values among all ‐local algorithms. Focusing on max‐cut, the algorithm constructs a cut of value , where is the value of the Parisi formula from spin glass theory, and (subscripts indicate the asymptotic variables). Our result generalizes to locally treelike graphs, that is, graphs whose girth becomes after removing a small fraction of vertices. Earlier work established that, for random ‐regular graphs, the typical max‐cut value is . Therefore our algorithm is nearly optimal on such graphs. An immediate corollary of this result is that random regular graphs have nearly minimum max‐cut, and nearly maximum min‐bisection among all regular locally treelike graphs. This can be viewed as a combinatorial version of the near‐Ramanujan property of random regular graphs. 

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