August math linkdump

• 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.

 

 

 

Corn and sweet potato chowder

Dinner tonight, cobbled together from various recipes found online:

4 cobs sweet corn

1 medium sweet potato

1 onion

2 cloves garlic

2 scallions

1 red pepper

1 jalapeno

1/4 c butter

1/4 c flour

4 c whole milk

salt, pepper, cumin

Recipe:  Preheat oven to 450.  Scrape corn kernels off the cobs.  Melt butter in pan, add flour, cook until it is roux.  Add a little more butter if needed and saute diced onion and garlic about 5 min until soft.  Add milk and kernel-less cobs.  Remove ribs and seeds from jalapeno and add it whole.  This is going to simmer about 30 mins. and meanwhile you are cutting up the sweet potato and red pepper and scallion and roasting them with the corn kernels until everything is slightly charred and smoky.  That being done, take some of the sweet potatoes and puree them with some soup to make a nice orange-brown paste.  Throw out the cobs and the jalapenos and put the sweet potato paste, red peppers, corn, and scallions in the soup.  Heat through, season with salt, pepper, cumin to taste.

Notes:  It’s not clear to me that the jalapeno added anything.  Also, it was too thick; next time I might skip the roux.

Update:  Skipped the roux, dropped the jalapeno, added a chopped/seeded Anaheim to the red pepper, even better.

Soup looked like this: