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1. Extensions of polynomial plank covering theorems

   https://doi.org/10.1112/blms.12979  

   Glazyrin, Alexey; Karasev, Roman; Polyanskii, Alexander (December 2023, Bulletin of the London Mathematical Society)

   Abstract We prove a complex polynomial plank covering theorem for not necessarily homogeneous polynomials. As the consequence of this result, we extend the complex plank theorem of Ball to the case of planks that are not necessarily centrally symmetric and not necessarily round. We also prove a weaker version of the spherical polynomial plank covering conjecture for planks of different widths. 

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2. Thin spectra and singular continuous spectral measures for limit‐periodic Jacobi matrices

   https://doi.org/10.1002/mana.202100561  

   Damanik, David; Fillman, Jake; Wang, Chunyi (June 2023, Mathematische Nachrichten)

   Abstract This paper investigates the spectral properties of Jacobi matrices with limit‐periodic coefficients. We show that generically the spectrum is a Cantor set of zero Lebesgue measure, and the spectral measures are purely singular continuous. For a dense set of limit‐periodic Jacobi matrices, we show that the spectrum is a Cantor set of zero lower box counting dimension while still retaining the singular continuity of the spectral type. We also show how results of this nature can be established by fixing the off‐diagonal coefficients and varying only the diagonal coefficients, and, in a more restricted version, by fixing the diagonal coefficients to be zero and varying only the off‐diagonal coefficients. We apply these results to produce examples of weighted Laplacians on the multidimensional integer lattice having purely singular continuous spectral type and zero‐dimensional spectrum. 

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3. Spatiotemporal Persistent Homology for Dynamic Metric Spaces

   https://doi.org/10.1007/s00454-019-00168-w  

   Kim, Woojin; Mémoli, Facundo (February 2020, Discrete & Computational Geometry)

   Characterizing the dynamics of time-evolving data within the framework of topological data analysis (TDA) has been attracting increasingly more attention. Popular instances of time-evolving data include flocking/swarming behaviors in animals and social networks in the human sphere. A natural mathematical model for such collective behaviors is a dynamic point cloud, or more generally a dynamic metric space (DMS). In this paper we extend the Rips filtration stability result for (static) metric spaces to the setting of DMSs. We do this by devising a certain three-parameter “spatiotemporal” filtration of a DMS. Applying the homology functor to this filtration gives rise to multidimensional persistence module derived from the DMS. We show that this multidimensional module enjoys stability under a suitable generalization of the Gromov–Hausdorff distance which permits metrization of the collection of all DMSs. On the other hand, it is recognized that, in general, comparing two multidimensional persistence modules leads to intractable computational problems. For the purpose of practical comparison of DMSs, we focus on both the rank invariant or the dimension function of the multidimensional persistence module that is derived from a DMS. We specifically propose to utilize a certain metric d for comparing these invariants: In our work this d is either (1) a certain generalization of the erosion distance by Patel, or (2) a specialized version of the well-known interleaving distance. In either case, the metric d can be computed in polynomial time. 

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4. Surface Gluing with Signs and Gradings in Decategorified Heegaard Floer Theory

   https://doi.org/10.1093/imrn/rnad238  

   Manion, Andrew (October 2023, International Mathematics Research Notices)

   Abstract A previous result about the decategorified bordered (sutured) Heegaard Floer invariants of surfaces glued together along intervals, generalizing the decategorified content of Rouquier and the author’s higher-tensor-product-based gluing theorem in cornered Heegaard Floer homology, was proved only over $${\mathbb{F}}_2$$ and without gradings. In this paper we add signs and prove a graded version of the interval gluing theorem over $${\mathbb{Z}}$$, enabling a more detailed comparison of these aspects of decategorified Heegaard Floer theory with modern work on non-semisimple 3d TQFTs in mathematics and physics. 

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5. THERE IS NO FINITELY ISOMETRIC KRIVINE'S THEOREM

   Kilbane, James; Ostrovskii, Mikhail I (January 2018, Houston journal of mathematics)

   null (Ed.)

   We prove that for every p in (1, infinity) different from 2, there exist a Banach space X isomorphic to l_p and a fin ite subset U in l_p, such that U is not isometric to a subset of X. This result shows that the fi nite isometric version of the Krivine theorem (which would be a strengthening of the Krivine theorem (1976)) does not hold. 

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