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In certain dialects of Chinese, Japanese, Korean, and Vietnamese, the word for ‘four’ sounds very similar to the word for ‘death’1. Consequently, the number 4 is considered by many people in East Asian nations to be unlucky. It is not unusual for buildings in that region to skip the number 4 when labeling floors, much in the same way 13th floors are omitted in some parts of the world2. In Hong Kong, at least one skyscraper avoids the proper numbering for floors 40-49. Four is the smallest positive non-prime number3. It is the only natural number where one can get the same result by multiplying its square roots (2×2), or adding them (2+2). Four happens to be the only number that has the same number of letters as its actual value4. The four color theorem tells us that four is an adequate number of colors for any two dimensional map–no two bordering regions would need to share a color. Four is the number of bonds that a carbon atom can make, which is why life can exist, a quality known as tetravalency. Fear of the number four is known as tetraphobia, and anyone suffering from it has almost certainly stopped reading by now, or at least uttered some four-letter words. It’s no secret that direct donations to Damn Interesting have been on a downward trend in recent years, so we are aiming to diversify. To that end, we’ve made something new, and it’s called Omiword. Continue reading ▶
Edmund Lawall must have felt cursed. He’d brought his family to New York in the late 1800s to carry on his father’s business as a pharmacist, but fate—or perhaps the city itself—seemed determined to drive him back out again. Lawall’s health had been in decline since their arrival, and his wife’s kidney disease had worsened, despite all of the tinctures and patent medicines available to his turn-of-the-century expertise. Not long after that, his business partner had been revealed as a crook, sending Lawall scrambling into bankruptcy court to convince the judge that his pharmacy had nothing to do with shady real estate dealings. Then, in the midst of the bankruptcy proceedings, an anonymous woman had staggered into Lawall’s drug store, collapsed on the floor, and died of unknown causes. Likely no one could have saved her, but it wasn’t exactly a ringing endorsement of the pharmaceutical services available at the corner of Eighth Street and Avenue C. None of that compared, however, to the morning of 27 June 1906, when a disheveled man in a medical coat burst through the narrow glass doors of the pharmacy, begging for protection. He was immediately followed by a young man with a revolver, and an angry crowd screaming in Yiddish. Lawall didn’t speak the language, but there was no mistaking the young man’s intent as he strode purposely forward and raised his gun to the doctor’s head. It was a grim, but recognizable tableau: the young man’s stance and grip were confident, clearly marking him as a budding gangster. The behavior of the crowd, on the other hand, made no sense at all. Innocent bystanders tended to run away from gang violence, yet the pushcart vendors and housewives surrounding the apparent holdup were not frightened, or even appealing for mercy. They were shouting, quite insistently, for the doctor’s execution. And everything seemed to indicate they were going to get what they wanted. In the weeks that followed, blame would be pointed in nearly every direction—because at that moment, unbeknownst to Lawall, similar scenes were playing out all over the neighborhood, with other doctors, teachers, reporters, and even utility workers being assaulted by hordes of people howling at them in Yiddish. By the time it was over, the incident would be measured as one of the largest riots ever in New York City, and the confrontation at Lawall’s Pharmacy would be mentioned only in passing, if at all. Another name, however, would be repeated over and over again: Adeline E. Simpson, the principal of Public School No. 110. Continue reading ▶
Iceland is known to the rest of the world as the land of Vikings and volcanos, an island caught between continents at the extremities of the map. Remote and comparatively inhospitable, it was settled only as long ago as the 9th century, and has seen little additional in-migration since. Even today, more than 90 percent of Iceland’s 390,000 residents can trace their ancestry back to the earliest permanent inhabitants, a Nordic-Celtic mix. The tradition of the Norse sagas lives on in the form of careful record-keeping about ancestry—and a national passion for genealogy. In other words, it is not the place to stumble upon old family mysteries. But growing up in the capital city of Reykjavík in the 1950s, neurologist Dr. Kári Stefánsson heard stories that left him curious. Stefánsson’s father had come from Djúpivogur, an eastern coastal town where everyone still spoke of a Black man who had moved there early in the 19th century. “Hans Jónatan”, they called him—a well-liked shopkeeper who had arrived on a ship, married a spirited woman from a local farm, and became a revered member of the community. The local census did record a man by the name of Hans Jónatan, born in the Caribbean, who was working at the general store in Djúpivogur in the 19th century—but that was all. No images of the man had survived, and his time in Iceland was well before any other humans with African ancestry are known to have visited the island. If tiny, remote Djúpivogur did have a Black man arrive in the 19th century, the circumstances must have been unusual indeed. It was an intriguing puzzle—and solid grounds for a scientific investigation. Given the amount of homogeneity in the baseline Icelandic population, the genetic signature of one relative newcomer with distinct ancestry might still stand out across a large sample of his descendants. Geneticists thus joined locals and history scholars, and they pieced together a story that bridged three continents. Continue reading ▶
It’s been a busy summer, and the large shortfall in donations last month has been demoralizing, so we’re taking a week off to rest and recuperate. The curated links section will be (mostly) silent, and behind the scenes we’ll be taking a brief break from our usual researching, writing, editing, illustrating, narrating, sound designing, coding, et cetera. We plan to return to normalcy on the 11th of September. (The word “normalcy” was not considered an acceptable alternative to “normality” until 14 May 1920, when then-presidential-candidate Warren G. Harding misused the mathematical term in a campaign speech, stating that America needed, “not nostrums, but normalcy.” He then integrated this error into his campaign slogan, “Return to Normalcy.” Also, the G in Warren G. Harding stood for “Gamaliel.”) While we are away, on 06 September 2023, Damn Interesting will be turning 18 years old. To celebrate, here are the first emojis to ever appear in the body of a Damn Interesting post: 🎂🎉🎁 If you become bored while we are away, you might try a little mobile game we’ve been working on called Wordwhile. It can be played alone, or with a friend. If you enjoy games like Scrabble and Wordle, you may find this one ENJOYABLE (75 points). Launch Wordwhile → And, as always, there are lots of ways to explore our back-catalog. View this post ▶
More in science
I have been away on vacation for the last week and a half, so I thought I would ease back into blogging with a light non-controversial topic – the Epstein Files. OK – let’s all take a collective deep breath. Everyone – and I mean everyone – I talk to about this issue has strong […] The post The Epstein Files Hubbub first appeared on NeuroLogica Blog.
In certain dialects of Chinese, Japanese, Korean, and Vietnamese, the word for ‘four’ sounds very similar to the word for ‘death’1. Consequently, the number 4 is considered by many people in East Asian nations to be unlucky. It is not unusual for buildings in that region to skip the number 4 when labeling floors, much in the same way 13th floors are omitted in some parts of the world2. In Hong Kong, at least one skyscraper avoids the proper numbering for floors 40-49. Four is the smallest positive non-prime number3. It is the only natural number where one can get the same result by multiplying its square roots (2×2), or adding them (2+2). Four happens to be the only number that has the same number of letters as its actual value4. The four color theorem tells us that four is an adequate number of colors for any two dimensional map–no two bordering regions would need to share a color. Four is the number of bonds that a carbon atom can make, which is why life can exist, a quality known as tetravalency. Fear of the number four is known as tetraphobia, and anyone suffering from it has almost certainly stopped reading by now, or at least uttered some four-letter words. It’s no secret that direct donations to Damn Interesting have been on a downward trend in recent years, so we are aiming to diversify. To that end, we’ve made something new, and it’s called Omiword. Continue reading ▶
Artificial intelligence software is designing novel experimental protocols that improve upon the work of human physicists, although the humans are still “doing a lot of baby-sitting.” The post AI Comes Up with Bizarre Physics Experiments. But They Work. first appeared on Quanta Magazine
I’ve spent the last week at CT2025, which has just come to a close. It was great getting to see so many old friends and meet so many new ones, and every time I go to a CT I’m reminded of just how much category theory there is in the world, as well as just how much I enjoy all of it! Right before this I was in Antwerp for some Noncommutative Geometry, where I learned a ton and met even more new friends! Then next week I go to Bonn for my third conference in a row. I’m trying to stay energetic, and thankfully I have a few days off between CT and QTMART to help me rest up! I want to write up a lot of things I learned over the last month, since I have a lot of new thoughts on noncommutative geometry, mirror symmetry, and deformation theory, all coming from just my time in Antwerp! I’ve also learned a lot at CT and talked to a lot of interesting people about interesting things, and I’m sure I’ll have even more to say after my time in Bonn. I think organizing all of those thoughts are going to take a while, though (if I end up writing them down at all), but today I have a quick observation inspired by a few lovely conversations I had with Clark Barwick at CT. One of the many questions I asked him was if there’s a conceptual reason the Sphere Spectrum (read: the homotopy groups of spheres) is so darn complicated. He gave me an answer that’s obvious in hindsight, but which totally rearranged the way I think about things: I think I internalized a while ago that “free” constructions are fairly concrete. After all, you look at the syntax of whatever object you’re interested in, quotient out by the relations you want to be true and you’re done! Plus, mapping out of a free thing is as simple as possible, since it’s a left adjoint! All you have to do is find a (usually simpler) map from your generating set to a structure of interest and let the magic of category theory build your (usually more complicated) map for you… Of course, this view is heavily influenced by the kind of free structures I have experience with, and the kinds of questions I was asking about them. I was thinking about free groups and monoids, which you can study with word combinatorics, free (dg-)algebras on (graded) vector spaces, which look like polynomials, free categories on graphs, free $k$-linear or dg-categories on categories, and free cauchy completions of these, all of which come from just looking at paths, linear combinations, concentrating things in degree $0$, or working with twisted closures to add shifts and cones and whatnot1. I was thinking about relatively free constructions like the universal enveloping algebra, with its PBW-basis, or the right angled artin group attached to a (reflexive, simple) graph… All of these constructions feel like friends to me, in part because I know how to compute with them. Why would the sphere spectrum – the free spectrum on a single point – be so different? The point is that I’ve internalized these constructions as being tractable because I’m usually mapping out of them, in the direction the category theory encourages. I’m also usually relying on serious “normal form” theorems that make computing with these things tractable, or I’m doing fairly simple combinatorics with my generating set before arguing that these extend in some obvious way to things defined on the whole free object. All of these constructions become much less friendly when you start mapping into them, or asking more difficult questions about their internal structure. In hindsight, I’ve even personally struggled with tons of questions about free structures in my research! Free groups are extremely interesting from basically any perspective, with deep questions about their first order theory (Tarski’s Problem), the combinatorics relating their generating sets (The Andrews-Curtis Conjecture), or the coarse geometry of their outer automorphisms. Free cauchy completions are obviously complicated when you want to understand them on their own terms! If you take an algebra $A$ and view it as a one-object dg-category, then its cauchy completion1 is its whole category of perfect complexes! An enormous chunk of representation theory is, in that lens, dedicated to nothing more than the study of a certain, complicated, free construction! I’ve personally given up2 on a problem about relatively free constructions in right angled artin groups! These interpolate between free and free-abelian groups, and geometric group theory is teeming with interesting open problems about raags. For instance, can you understand, at the level of the underlying graphs, when one raag will embed into another? I thought about this off and on for a year before I started working seriously with my current advisor, and I made almost no progress at all. I also spent some time working with the adjunctions It’s interesting to try and construct these explicitly, and to understand the essential images of the left adjoints. This amounts to understanding which essentially algebraic theories are actually algebraic, and which algebraic theories are actually props. One of the big difficulties here is that we have a relatively free construction which adds relations rather than just operations. Adding new operations tends to be a fairly mild thing to do – consider the free algebra on an abelian group, which sends $A$ to its tensor algebra $\bigoplus_n A^{\otimes n}$ where it’s easy to recover the $A$ you started with. If instead we want to add new relations or axioms, for instance by freely sending a group $G$ to its abelianization $G \big / [G,G]$, then we lose lots of information in this construction. After some conversations with John Baez and Todd Trimble I came quite close to characterizing the image of the left adjoint between finite product categories and symmetric monoidal categories by factoring it into a “lossy” construction adding new axioms forcing the monoidal unit to be terminal and a much simpler construction which freely adds new operations corresponding to the product projections. I’ve had to put that project on hold while I focus on my thesis work, but I really want to come back and finish it soon. Of course as soon as you’re interested in logic, you have to accept that free things are complicated! The freest version of any theory lives in its classifying category, where truth and provability coincide. Then proving anything at all about the free model gives immediate understanding about all other models of that theory! This is already true for groups, whose classifying finite-product category is just the category of finitely generated free groups and homomorphisms. We don’t usually think about it because the kinds of statements you prove in equational logic aren’t very deep. But if you look instead at the classifying topos for groups and ask geometric questions suddenly you’re able to do a lot more, and the game becomes much harder! Perhaps this is clearest in the semantics of programming languages, where the free model (often called the “term model” in this context3) is the programming language, and checking whether two terms are equal in this free model literally amounts to evaluating two programs and seeing if their respective values agree. Because of this, many important structural results about a programming language (such as canonicity) can be proven by building another model whose semantics you understand, and then producing a section of the unique map from the free model. Also coming from logic are various lattices, whose free models can be quite intricate. Famously the free modular lattice on $3$ generators has $28$ elements, while the free modular lattice on $4$ generators is infinite! Indeed, this lattice has an undecidable word problem4, so that no program can tell whether two descriptions of its elements are the same or not! Heyting algebras are extremely important for semantics of intuitionistic logic, yet already the free heyting algebra on one generator is infinite, and the free heyting algebra on two generators is famously complicated. The study of free heyting algebras is still ongoing and seems quite difficult (at least as an outsider). See, for instance, Almeida’s recent preprint Colimits and Free Constructions of Heyting Algebras through Esakia Duality. With all this in mind, it shouldn’t be surprising at all that the sphere spectrum is so complicated! It’s the free spectrum on a point, and as such the only “relations” it will have are those that hold in all spectra! But of course spectra should obviously be complicated – They control all possible (co)homology theories for all spaces! So in this sense one should expect the internal structure of the sphere spectrum to be quite complicated, since any simplification would persist to something true of all cohomology theories. Of course, it’s easy to be complicated without being interesting, and I still think it’s a bit of a miracle that the sphere spectrum should have all this intricate structure inside it. As I understand it, much of Chromatic Homotopy Theory came from trying to explain patterns in the homotopy groups of spheres, and this subject is now as famously intimidating to outsiders5 as it is famously fascinating once you put in the work to become an insider6. Thanks for reading all! This really was a quick one for once, since I already had a lot of these examples floating around in my head. I really had all of the tools to realize that free things are obviously complicated in general, especially their “internal structure” that doesn’t ride the coattails of the universal property, but for some reason I just didn’t put it together until my conversation with Clark. It’s always dangerous to say what I’m thinking about writing about, but I at least have one more short post planned from my time in Antwerp, and maybe a longer one too if I have the energy. I’m doing a ton of writing right now, since I have two half-finished papers that I want to submit by the end of the summer. I think I’ll be able to get it done, but between writing these and going to conferences it’s been a tiring month. It’s tiring in a fun way, though, and I really feel like I’ve been productive in a way that I haven’t felt in a little while. I’m excited to start crossing a lot of these projects off my long-term-todo-list, especially since I already have three more projects I want to start! Regardless, I hope you’re having a more restful summer than I am! Stay safe, all, and we’ll talk soon ^_^. Emily Roff, taken at the top of the main tower in Brno) Actually it’s not completely obvious to me that the cauchy completion of a dg-category should be its idempotent triangulated closure… The cauchy completion will certainly be idempotent complete and triangulated, but in the well-named paper Cauchy Completeness for DG-Categories, Nicolić, Street, and Tendas show that to be cauchy complete you also need to be closed under “cokernels of protosplit chain maps”… I think this is some kind of split idempotent condition? But I haven’t read the paper closely enough to know for sure. If you want to be guaranteed to be correct, instead of “cauchy completion” you can say “the idempotent closure of the twisted closure”. That’s still a free construction and it still gives you the derived category in the special case your dg-category is a ring. ↩ ↩2 At least for now ↩ Pun intended ↩ Which is made more interesting by the fact that if you look at the class of lattices coming from lattices of subgroups of abelian groups, the corresponding free lattice on $4$ generators does have solvable word problem! This is remarkable since (as I understand it) modular lattices are called that because they look like lattices of submodules (in particular, sublattices of a lattice of subgroups of an abelian group). This is apparently proven in Herrmann’s On the Equational Theory of Submodule Lattices, and I learned all this from Ralph Freese the comments of this n-Category Cafe post. ↩ Such as myself. ↩ Which it seems like I might start doing soon, for a project I’m not ready to talk about yet. ↩
