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1. Free fermions revisited

   https://doi.org/10.1142/S0219498826502695  

   Milas, Antun; Penn, Michael (June 2025, Journal of Algebra and Its Applications)

   Free fermion vertex superalgebras are discussed from the point of view of Urod vertex algebras [T. Arakawa, T. Creutzig and B. Feigin, Urod algebras and translation of [Formula: see text]-algebras, Forum Mathematics Sigma, Vol. 10 (Cambridge University Press, 2022) and M. Bershtein, B. Feigin and A. Litvinov, Coupling of two conformal field theories and Nakajima–Yoshioka blow-up equations, preprint (2013), arXiv:1310.7281]. We present all finite decompositions of the [Formula: see text]-fermion vertex algebra via Virasoro and [Formula: see text] superconformal vertex algebras. We also present decompositions of higher rank fermion algebras using affine [Formula: see text]-algebras, and a conjecture on the existence of the “square root” of the [Formula: see text] fermion algebra. 

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2. 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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3. Bartnik mass via vacuum extensions

   https://doi.org/10.1142/S0129167X19400068  

   Miao, Pengzi; Xie, Naqing (December 2019, International Journal of Mathematics)

   We construct asymptotically flat, scalar flat extensions of Bartnik data [Formula: see text], where [Formula: see text] is a metric of positive Gauss curvature on a two-sphere [Formula: see text], and [Formula: see text] is a function that is either positive or identically zero on [Formula: see text], such that the mass of the extension can be made arbitrarily close to the half area radius of [Formula: see text]. In the case of [Formula: see text], the result gives an analog of a theorem of Mantoulidis and Schoen [On the Bartnik mass of apparent horizons, Class. Quantum Grav. 32(20) (2015) 205002, 16 pp.], but with extensions that have vanishing scalar curvature. In the context of initial data sets in general relativity, the result produces asymptotically flat, time-symmetric, vacuum initial data with an apparent horizon [Formula: see text], for any metric [Formula: see text] with positive Gauss curvature, such that the mass of the initial data is arbitrarily close to the optimal value in the Riemannian Penrose inequality. The method we use is the Shi–Tam type metric construction from [Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature, J. Differential Geom. 62(1) (2002) 79–125] and a refined Shi–Tam monotonicity, found by the first named author in [On a localized Riemannian Penrose inequality, Commun. Math. Phys. 292(1) (2009) 271–284]. 

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4. Metric Thickenings and Group Actions

   https://doi.org/10.1142/S1793525320500569  

   Adams, Henry; Heim, Mark; Peterson, Chris (January 2021, Journal of Topology and Analysis)

   null (Ed.)

   Let [Formula: see text] be a group acting properly and by isometries on a metric space [Formula: see text]; it follows that the quotient or orbit space [Formula: see text] is also a metric space. We study the Vietoris–Rips and Čech complexes of [Formula: see text]. Whereas (co)homology theories for metric spaces let the scale parameter of a Vietoris–Rips or Čech complex go to zero, and whereas geometric group theory requires the scale parameter to be sufficiently large, we instead consider intermediate scale parameters (neither tending to zero nor to infinity). As a particular case, we study the Vietoris–Rips and Čech thickenings of projective spaces at the first scale parameter where the homotopy type changes. 

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5. Computing cohomology groups that classify bundles of strongly self-absorbing C∗-algebras

   https://doi.org/10.1142/S1793525324500110  

   Dadarlat, Marius; McClure, James E; Pennig, Ulrich (March 2024, Journal of Topology and Analysis)

   Locally trivial bundles of [Formula: see text]-algebras with fiber [Formula: see text] for a strongly self-absorbing [Formula: see text]-algebra [Formula: see text] over a finite CW-complex [Formula: see text] form a group [Formula: see text] that is the first group of a cohomology theory [Formula: see text]. In this paper, we compute these groups by expressing them in terms of ordinary cohomology and connective [Formula: see text]-theory. To compare the [Formula: see text]-algebraic version of [Formula: see text] with its classical counterpart we also develop a uniqueness result for the unit spectrum of complex periodic topological [Formula: see text]-theory. 

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