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1. How to morph graphs on the torus

   Chambers, Erin Wolf (January 2021, Proceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms)

   null (Ed.)

   We present the first algorithm to morph graphs on the torus. Given two isotopic essentially 3-connected embeddings of the same graph on the Euclidean flat torus, where the edges in both drawings are geodesics, our algorithm computes a continuous deformation from one drawing to the other, such that all edges are geodesics at all times. Previously even the existence of such a morph was not known. Our algorithm runs in O(n1+ω/2) time, where ω is the matrix multiplication exponent, and the computed morph consists of O(n) parallel linear morphing steps. Existing techniques for morphing planar straight-line graphs do not immediately generalize to graphs on the torus; in particular, Cairns' original 1944 proof and its more recent improvements rely on the fact that every planar graph contains a vertex of degree at most 5. Our proof relies on a subtle geometric analysis of 6-regular triangulations of the torus. We also make heavy use of a natural extension of Tutte's spring embedding theorem to torus graphs. 

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2. Elliptic modular graph forms. Part I. Identities and generating series

   https://doi.org/10.1007/JHEP03(2021)151  

   D’Hoker, Eric; Kleinschmidt, Axel; Schlotterer, Oliver (March 2021, Journal of High Energy Physics)

   null (Ed.)

   A bstract Elliptic modular graph functions and forms (eMGFs) are defined for arbitrary graphs as natural generalizations of modular graph functions and forms obtained by including the character of an Abelian group in their Kronecker-Eisenstein series. The simplest examples of eMGFs are given by the Green function for a massless scalar field on the torus and the Zagier single-valued elliptic polylogarithms. More complicated eMGFs are produced by the non-separating degeneration of a higher genus surface to a genus one surface with punctures. eMGFs may equivalently be represented by multiple integrals over the torus of combinations of coefficients of the Kronecker-Eisenstein series, and may be assembled into generating series. These relations are exploited to derive holomorphic subgraph reduction formulas, as well as algebraic and differential identities between eMGFs and their generating series. 

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3. The algebraic structure of hyperbolic graph braid groups

   https://doi.org/10.1142/S0218196724500528  

   Appiah, Benjamin; Dani, Pallavi; Ge, Wayne; Hudson, Christopher; Jain, Saumya; Lemoine, Matthew; Murphy, Jake; Murray, Justin; Pandikkadan, Adithyan; Schreve, Kevin; et al (May 2025, International Journal of Algebra and Computation)

   Genevois recently classified which graph braid groups are word hyperbolic. In the 3-strand case, he asked whether all such word hyperbolic groups are actually free; this reduced to checking two infinite classes of graphs: sun and pulsar graphs. We prove that 3-strand braid groups of sun graphs are free. On the other hand, it was known to experts that 3-strand braid groups of most pulsar graphs contain surface subgroups. We provide a simple proof of this and prove an additional structure theorem for these groups. 

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4. Random Algebraic Graphs and Their Convergence to ErdőS–Rényi

   https://doi.org/10.1002/rsa.21276  

   Bangachev, Kiril; Bresler, Guy (January 2025, Random Structures & Algorithms)

   ABSTRACT A random algebraic graph is defined by a group with a uniform distribution over it and a connection with expectation satisfying . The random graph with vertex set is formed as follows. First, independent variables are sampled uniformly from . Then, vertices are connected with probability . This model captures random geometric graphs over the sphere, torus, and hypercube; certain instances of the stochastic block model; and random subgraphs of Cayley graphs. The main question of interest to the current paper is: when is a random algebraic graph statistically and/or computationally distinguishable from ? Our results fall into two categories. (1) Geometric. We focus on the case and use Fourier‐analytic tools. We match and extend the following results from the prior literature: For hard threshold connections, we match for , and for ‐Lipschitz connections we extend the results of when to the non‐monotone setting. (2) Algebraic. We provide evidence for an exponential statistical‐computational gap. Consider any finite group and let be a set of elements formed by including each set of the form independently with probability Let be the distribution of random graphs formed by taking a uniformly random induced subgraph of size of the Cayley graph . Then, and are statistically indistinguishable with high probability over if and only if . However, low‐degree polynomial tests fail to distinguish and with high probability over when 

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5. Mass Supply from Io to Jupiter’s Magnetosphere

   https://doi.org/10.1007/s11214-025-01137-x  

   Roth, Lorenz; Blöcker, Aljona; de_Kleer, Katherine; Goldstein, David; Lellouch, Emmanuel; Saur, Joachim; Schmidt, Carl; Strobel, Darrell_F; Tao, Chihiro; Tsuchiya, Fuminori; et al (February 2025, Space Science Reviews)

   Abstract Since the Voyager mission flybys in 1979, we have known the moon Io to be both volcanically active and the main source of plasma in the vast magnetosphere of Jupiter. Material lost from Io forms neutral clouds, the Io plasma torus and ultimately the extended plasma sheet. This material is supplied from Io’s upper atmosphere and atmospheric loss is likely driven by plasma-interaction effects with possible contributions from thermal escape and photochemistry-driven escape. Direct volcanic escape is negligible. The supply of material to maintain the plasma torus has been estimated from various methods at roughly one ton per second. Most of the time the magnetospheric plasma environment of Io is stable on timescales from days to months. Similarly, Io’s atmosphere was found to have a stable average density on the dayside, although it exhibits lateral (longitudinal and latitudinal) and temporal (both diurnal and seasonal) variations. There is a potential positive feedback in the Io torus supply: collisions of torus plasma with atmospheric neutrals are probably a significant loss process, which increases with torus density. The stability of the torus environment may be maintained by limiting mechanisms of either torus supply from Io or the loss from the torus by centrifugal interchange in the middle magnetosphere. Various observations suggest that occasionally (roughly 1 to 2 detections per decade) the plasma torus undergoes major transient changes over a period of several weeks, apparently overcoming possible stabilizing mechanisms. Such events (as well as more frequent minor changes) are commonly explained by some kind of change in volcanic activity that triggers a chain of reactions which modify the plasma torus state via a net change in supply of new mass. However, it remains unknown what kind of volcanic event (if any) can trigger events in torus and magnetosphere, whether Io’s atmosphere undergoes a general change before or during such events, and what processes could enable such a change in the otherwise stable torus. Alternative explanations, which are not invoking volcanic activity, have not been put forward. We review the current knowledge on Io’s volcanic activity, atmosphere, and the magnetospheric neutral and plasma environment and their roles in mass transfer from Io to the plasma torus and magnetosphere. We provide an overview of the recorded events of transient changes in the torus, address several contradictions and inconsistencies, and point out gaps in our current understanding. Lastly, we provide a list of relevant terms and their definitions. 

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