Linkage
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Lance Fortnow’s pet peeve (\(\mathbb{M}\)): A speaker who cites their own work using only their initial instead of their name.
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Useless Map Projection #6: The Asymmetric Monstrosity Part sinusoidal, part Mollweide, part Equal Earth, part cylindrical equal-area, all ugly.
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Escher’s Circle Limit III, animated, on a pseudosphere. There might be a seam around the back side that you can’t see, or there might be an anholonomy where the more times you walk around the more spiralized the motion of the fish becomes.
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On faith, religion, conjectures and Schubert calculus (\(\mathbb{M}\)), Igor Pak. This blog post is related to Pak’s new preprint “Positivity of Schubert Coefficients” with Colleen Robichaux, arXiv:2412.18984. The preprint combines an unproved assumption from mathematical analysis (the Generalized Riemann Hypothesis) with one from complexity theory (the Miltersen–Vinodchandran Assumption on the non-existence of subexponential circuits for nondeterministic exponential time problems) to prove that, under those assumptions, the positivity of Schubert coefficients can be verified in \(\mathsf{NP}\). The blog post explains how Pak went from a position of disbelieving in derandomization (lacking the faith in \(\mathsf{P}=\mathsf{BPP}\) that many complexity theorists hold) to a position where it is reasonable to make an assumption that implies the ability to derandomize interactive proofs (\(\mathsf{NP}=\mathsf{AM}\)).
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Mathematically incorrect footballs in the Bing animated children’s TV series, with a new and creative method of being incorrect: use the correct truncated icosahedron geometry but then color two pentagons and six hexagons black instead of all the pentagons.
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Why is zero plural? (\(\mathbb{M}\), via). Grammatically plural, when used as an adjective, that is. Sadly, the answers describe that zero things are plural in English without pointing to any historical linguistics scholarship on why that came to be.
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What do all those different squiggly equal signs mean (\(\mathbb{M}\))? A Wikipedia discussion on the symbol \(\simeq\) (called “asymptotically equal to” by the Unicode consortium, unicode U+2243) led me here, where I learn that it is commonly used for homotopy equivalence or maybe equivalence of categories. In my experience the more common symbol for asymptotic equivalence is \(\sim\). It’s unfortunate that Unicode named these by a specific (and in this case dubious) choice of semantics than by a name that could convey a wider meaning. I guess the LaTeX name “simeq” also conveys the idea of a meaning “similar or equal” but “similar” can mean a lot of things. The link also discusses several other such symbols.
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My paper “What is…treewidth” has appeared in the Notices of the AMS for its February issue (\(\mathbb{M}\)). It is a very high-level survey of the subject, necessarily limited by the format to have not much technical depth and not many references. If you already know what treewidth is and use it in your work, you might not learn much of use from it. My focus instead was on explaining to mathematicians (and computer scientists) in related fields why they might want to learn it. If you work in sparse numerical linear algebra, Bayesian inference, control theory, game theory, low dimensional topology, network science, or algebraic geometry, then this paper overviews applications of treewidth in your area, and if you don’t work in those areas then you might still learn something from it about those applications. And if you’re still wondering “what is treewidth” after reading this post, then the survey does also cover the (many) ways of defining it and using it in graph structure theory and graph algorithms. But for this post, I’ll try to do it in a single sentence: it’s a way of describing arbitrary graphs as “thickened trees” and a measure of how much thickening is needed.
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Big specialized forums have started backdating millions of LLM-generated posts. “Now you cannot be sure a reply from 2009 on some forum for physics or maps or flower or drill enthusiasts haven’t been machine-generated and totally wrong.”
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NSF cancels all grant panels in response to White House orders (\(\mathbb{M}\), see also). Scott Aaronson and Lance Fortnow on the ongoing situation at the NSF or, as Scott calls it, “the American science funding catastrophe” (\(\mathbb{M}\)). And a multi-part post by Terry Tao on how nonlinear systems have the property that once you break them it becomes difficult or impossible to get back to the state they used to be in.
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Grid bracing (\(\mathbb{M}\)). This is a mathematical abstraction of problems of making a grid-like structure rigid by adding cross-bracing, of a type that should be familiar to anyone who has played with building blocks, built houses of cards, or tried to assemble a backless bookshelf. It forms a bridge between combinatorics and structural engineering in providing a solution to these problems based purely on graph connectivity analysis rather than numerical simulation. Now a Good Article on Wikipedia.
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The local-global conjecture for Apollonian circle packings is false (\(\mathbb{M}\)). An hour-long talk by Katherine Stange on Apollonian gaskets with integer curvatures, and on the patterns of integers that do not occur in these gaskets. There are some obvious ways for some integers to be omitted (all but one negative number, certain patterns mod 24, and a finite number of exceptional curvatures that correspond to circles too big to fit among the other circles of the gasket) but Stange and her students found a new obstacle involving quadratic reciprocity. For instance, there exist gaskets that avoid all the square curvatures.