- Algebraists eat corn row by row, analysts eat corn circle by circle. Yep, I eat down the rows like a typewriter. Why? Because it is the right way.
- This short paper by Johan de Jong and Wei Ho addresses an interesting question I’d never thought about; does a Brauer-Severi variety over a field K contain a genus-1 curve defined over K? They show the answer is yes in dimensions up to 4 (3 and 4 being the new cases.) In dimension 1, this just asks about covers of Brauer-Severi curves by genus 1 curves; I remember this kind of situation coming up in Ekin Ozman’s thesis, where certain twists of modular curves end up being covers of Brauer-Severi curves. Which Brauer-Severi varieties are split by twisted modular curves?
- Always nice to see a coherent description of the p-adic numbers in the popular press; and George Musser delivers the goods in Scientific American, in the context of recent work in cosmology by Harlow, Shenker, Stanford, and Susskind. Two quibbles: first, if I understood Susskind’s talk on this stuff correctly, the point is to model things by an infinite regular tree. The fact that when the out-degree is a prime power this happens to look like the Bruhat-Tits tree is in some sense tangential, though very useful for getting an intuitive picture of what’s going on — as long as your intuition is already p-adic, of course! Second quibble is that Musser seems to suggest at the end that p-adic distances can’t get arbitrarily small:
On top of that, distance is always finite. There are no p-adic infinitesimals, or infinitely small distances, such as the dx and dy you see in high-school calculus. In the argot, p-adics are “non-Archimedean.” Mathematicians had to cook up a whole new type of calculus for them.
Prior to the multiverse study, non-Archimedeanness was the main reason physicists had taken the trouble to decipher those mathematics textbooks. Theorists think that the natural world, too, has no infinitely small distances; there is some minimal possible distance, the Planck scale, below which gravity is so intense that it renders the entire notion of space meaningless. Grappling with this granularity has always vexed theorists. Real numbers can be subdivided all the way down to geometric points of zero size, so they are ill-suited to describing a granular space; attempting to use them for this purpose tends to spoil the symmetries on which modern physics is based.