Linkage

Finding a stationary point of a smooth function (as can be done, eventually, by gradient descent) is complete for \(\mathsf{PPAD}\cap\mathsf{PLS}\) (\(\mathbb{M}\), via) as Fearnley, Goldberg, Hollender, and Savani write in STOC 2021 best paper “The complexity of gradient descent”. Gradient descent finds local minima quickly in many cases, but can be slow to navigate shallow labyrinthine optimization landscapes; this result suggests that there are unlikely to be much better shortcuts.

Dyadic rational (\(\mathbb{M}\)), the fractions whose denominators are powers of two. These are among the earliest concepts of fractions to develop in schoolchildren, frequently used in systems of weights and measures, and (because they are the ones with finite binary representations) central to computing. Now a Good Article on Wikipedia.

Pentagonum pentagonorum (\(\mathbb{M}\)). The space of shapes of convex equiangular pentagons itself forms a right equilateral pentagon in a hyperbolic plane of self-crossing pentagon shapes, and its tiling by edge-reflections corresponds to moves that add or remove a “bowtie” on a pentagon edge. Other choices of fixed angles give other right hyperbolic pentagons, parameterized by a pentagon of pentagons of pentagons. New “short story” by Danny Calegari in the Notices.

A deep math dive into why some infinities are bigger than others (\(\mathbb{M}\), via). Martin Goldstern and Jakob Kellner carefully explore the relations between certain infinite cardinalities, all more than countable but at most the cardinality of the continuum, defined using measure-zero and nowhere-dense subsets of the plane.

Australian Research Council unexpectedly nukes any applicants for Future Fellowships and Discovery Early Career Researcher Awards who had the temerity to cite “preprints”, anywhere in their proposals, defined as pre-peer-review versions of journal papers but interpreted more broadly to cover reviewed pre-publication versions, tech reports, white papers, etc: Times Higher Education, Guardian, Research Professional News, ARC Tracker (\(\mathbb{M}\)).

A tiling of the plane by center-hinged pairs of equilateral triangles, connected at their corners, expands into a different tiling by stars, hexagons, and diamonds (\(\mathbb{M}\), via).

Euclidean Geometry Street (\(\mathbb{M}\)), a street in Greece decorated with geometry diagram street art.

Two equal-length quadrilaterals from a tangential quadrilateral (\(\mathbb{M}\)): IMO 2021 Problem 4 explained with the help of some dynamic geometry software, by Russell Lim.

• The 3-vertex model in physics
• A relevant review on general topics in graph theory
• Household goods in China
• Sociolinguistics of sexuality
• Paths on meshes (maybe relevant?)
• Workplace trust
• Famous cases of asthma
• Arthropod DNA

Moral: AI is hard!

Among recent genetics papers using Excel spreadsheets for their data, roughly 30% contained autocorrect errors such as conversion of non-date text to dates, up from 20% in a 2016 study (\(\mathbb{M}\), via). “The lesson for researchers is clear: it’s past time to stop using Excel and learn to use more powerful software.”

Progress on universal covers for carpenter’s rules (\(\mathbb{M}\)), small-area diameter-1 shapes into which every polyline with segment lengths ≤ 1 can be folded. The new shape in this preprint by three Jiangs (at least one apparently a home-schooled high-schooler) kind of looks like the Starfleet emblem from Star Trek: a curvy triangle with a concave base and two convex sides. The sides are just involutes of the base; the harder part is finding the right base.

NP Complete 3rd Grade Math Problems (\(\mathbb{M}\)). A worksheet asks students to solve a subset sum problem with 279 valid solutions out of roughly 2 million possibilities. Maybe they expect the kids to code it on a computer?

My new favorite xkcd: some useful geometry formulas (\(\mathbb{M}\)).