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1. Sensing the shape of a surface by tightly surface-bound filaments

   https://doi.org/10.1073/pnas.2526131122  

   Shi, Handuo; Nguyen, Jeffrey; Huang, Jordan_Alexander; Gitai, Zemer; Shaevitz, Joshua; Bratton, Benjamin_P; Gopinathan, Ajay; Grason, Gregory; Huang, Kerwyn_Casey (December 2025, Proceedings of the National Academy of Sciences)

   Understanding the mechanisms that dictate the localization of cytoskeletal filaments is crucial for elucidating cell shape regulation in prokaryotes. The actin homolog MreB plays a pivotal role in maintaining the shape of many rod-shaped bacteria such asEscherichia coliby directing cell-wall synthesis according to local curvature cues. However, the basis of MreB’s curvature-dependent localization has remained elusive. Here, we develop a biophysical model for the energetics of a filament binding to a surface that integrates the complex interplay between filament twist and bending and the two-dimensional surface geometry. Our model predicts that the spatial localization of a filament like MreB with substantial intrinsic twist is governed by both the mean and Gaussian curvatures of the cell envelope, which strongly covary in rod-shaped cells. Using molecular dynamics simulations to estimate the mechanical properties of MreB filaments, we show that their thermodynamic preference for regions with lower mean and Gaussian curvatures matches experimental observations for physiologically relevant filament lengths of ~50 nm. We find that the experimentally measured statistical curvature preference is maintained in the absence of filament motion and after a cycle of depolymerization, repolymerization, and membrane rebinding, indicating that equilibrium energetics can explain MreB localization. These findings provide critical insights into the physical principles underlying cytoskeletal filament localization and suggest design principles for synthetic shape-sensing nanomaterials. 

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2. The p-widths of a surface

   https://doi.org/10.1007/s10240-023-00141-7  

   Chodosh, Otis; Mantoulidis, Christos (June 2023, Publications mathématiques de l'IHÉS)

   Abstract The $$p$$ p -widths of a closed Riemannian manifold are a nonlinear analogue of the spectrum of its Laplace–Beltrami operator, which corresponds to areas of a certain min-max sequence of possibly singular minimal submanifolds. We show that the $$p$$ p -widths of any closed Riemannian two-manifold correspond to a union of closed immersed geodesics, rather than simply geodesic nets. We then prove optimality of the sweepouts of the round two-sphere constructed from the zero set of homogeneous polynomials, showing that the $$p$$ p -widths of the round sphere are attained by $$\lfloor \sqrt{p}\rfloor $$ ⌊ p ⌋ great circles. As a result, we find the universal constant in the Liokumovich–Marques–Neves–Weyl law for surfaces to be $$\sqrt{\pi }$$ π . En route to calculating the $$p$$ p -widths of the round two-sphere, we prove two additional new results: a bumpy metrics theorem for stationary geodesic nets with fixed edge lengths, and that, generically, stationary geodesic nets with bounded mass and bounded singular set have Lusternik–Schnirelmann category zero. 

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3. The Flex Divisor of a K3 Surface

   https://doi.org/10.1093/imrn/rnac089  

   Alexeev, Valery; Engel, Philip (May 2022, International Mathematics Research Notices)

   Abstract The flex divisor$$R_{\textrm flex}$$ of a primitively polarized K3 surface $(X,L)$ is, generically, the set of all points $$x\in X$$ for which there exists a pencil $$V\subset |L|$$ whose base locus is $$\{x\}$$. We show that if $L^2=2d$ then $$R_{\textrm flex}\in |n_dL|$$ with $$ \begin{align*} &n_d= \frac{(2d)!(2d+1)!}{d!^2(d+1)!^2} =(2d+1)C(d)^2,\end{align*}$$where $C(d)$ is the Catalan number. We also show that there is a well-defined notion of flex divisor over the whole moduli space $$F_{2d}$$ of polarized K3 surfaces. 

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4. The Morse Index of a Minimal Surface

   https://doi.org/10.1090/noti2291  

   Chodosh, Otis; Maximo, Davi (June 2021, Notices of the American Mathematical Society)
5. The degree of a tropical root surface of type A

   https://doi.org/10.2140/om.2024.1.243  

   Ardila-Mantilla, Federico; Cordero_Aguilar, Mont (January 2025, Orbita Mathematicae)

   We prove that the tropical surface of the root system A_{n−1} has degree n(n − 1)(n − 2)/2. 

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