Linkage for the start of the academic year

Equal-area right and isosceles triangles with their bases partitioning the radius of a circle and their apexes on the circle. What is the angle between them?

Joel Hamkins’ workshop on the mathematics of infinitary games (\(\mathbb{M}\)) consists of a sequence of 90-minute Zoom talks scattered through the fall, and kicked off with a talk by Davide Leonessi on infinite draughts (checkers). Hamkins writes: “The workshop talks will be run at a high level of sophistication, aimed for the most part at serious researchers currently working in this emerging area. Mathematicians, computer scientists, infinitary game theorists, all serious researchers are welcome.”

Simon Brendle wins the 2024 Breakthrough Prize in mathematics (\(\mathbb{M}\)) As Ian Agol writes, “he solved the Hsiang–Lawson conjecture, showing that the only embedded minimal surface in the (round) 3-sphere is the Clifford torus”.

In a lunchtime discussion about the greedy-algorithms part of the undergraduate algorithms course that all computer science majors take, one of my colleagues pointed out that the Gale–Shapley algorithm for stable matching defines an antimatroid (\(\mathbb{M}\)). If you consider the problem on employers and applicants, with an arbitrary unmatched employer sending a job offer to an applicant in each step, then every job offer that might be sent remains available to be sent until it eventually is sent, one of the defining properties of antimatroids. (The other is that triggering an offer letter depends only on what has already happened but not on what order it has happened: an offer is sent if the next-more-preferred candidate has already been sent an offer from the same employer and another offer they like better.) It is immediate consequence that you will always get the same matching regardless of how you choose which employer to send a letter in each step. (Usually this is instead proven as a consequence of the employer-optimality of the resulting matching.)

The same idea of having a pool of things that you can choose to do, with the ordering of choice irrelevant to the result, comes up frequently in other algorithms, and it would be easy to group other standard algorithms-course topics under the same framework (especially topological sorting). Sadly, I don’t know of any algorithms textbook that covers these topics in this way. Gale–Shapley is in the greedy chapter of Jeff Erickson’s algorithms text, but without discussion of the antimatroid property and without connection to other antimatroid problems.

When computers write proofs, what’s the point of mathematicians (\(\mathbb{M}\), via)? Video by Quanta magazine with Andrew Granville, with accompanying article “Why mathematical proof is a social compact.

Talk slides for my talk at Graph Drawing, “On the Biplanarity of Blowups” (\(\mathbb{M}\)). It won the best paper award for the theory track of the conference! I also presented a poster on “The Widths of Strict Outerconfluent Graphs”. For more in-depth posts on these papers see here and here respectively.

A coin flip by any other name (\(\mathbb{M}\)), or why random cyclic quadrilaterals contain the origin with probability \(\tfrac12\).

A tower of conjectures that rests upon a needle (\(\mathbb{M}\)). In Quanta, Jordana Cepelewicz explains how a seeming curiosity, the Kakeya needle problem on the minimum area within which a line segment can rotate, is connected to deep and fundamental questions in partial differential equations and harmonic analysis.

A faster direct sampling algorithm for equilateral closed polygons (\(\mathbb{M}\)), by Jason Cantarella, Henrik Schumacher, and Clayton Shonkwiler, improving their previous work on sampling a certain probability distribution on these polygons. As Shonkwiler writes, a key idea is to estimate the volume of “some weird polytope”, which turns out to have the same volume as a central slab of a hypercube, proven using toric symplectic geometry.

Optimal metric distortion for voting (\(\mathbb{M}\)). Blog post by Prasanna Ramakrishnan explaining the plurality veto voting rule of Kempe and Kızılkaya, which optimally approximates the minimum sum of voter-candidate distances in an unknown metric space.

US states mapped as a rectangular subdivision. Getting the Four Corners correct prevents California from being in the lower left corner where one would expect it. As Dan Drake writes, “topologists make for lousy cartographers.”

My school has posted a news story about my work at the recent Graph Drawing conference, “On the Biplanarity of Blowups” (\(\mathbb{M}\)) and its best-paper win at the conference.

After resigning en masse from American Institute of Mathematical Sciences’ Journal of Geometric Mechanics “following a dispute with their publisher over special issues and article volume” which “would have the effect of jeopardizing scientific integrity for the sake of financial gain” the editors have launched a new journal, Geometric Mechanics, with … World Scientific (\(\mathbb{M}\)). What are they thinking? As commercial publishers go, I have nothing against World Scientific, but for new journals intended to avoid financially-motivated distortion of their scientific integrity, diamond open access is the only way to go.

Lego Reuleaux triangle mechanisms (\(\mathbb{M}\)).