Month: October 2018

Surreal birthdays and their arithmetic

~ David Roberts ~ Leave a comment

Edit: now published as Practically surreal: Surreal arithmetic in Julia, SoftwareX Volume 9, January–June 2019, Pages 293-298, doi:10.1016/j.softx.2019.03.005

This is the title of a new paper (arXiv:1810.10373) by one of my colleagues, Matthew Roughan. Surreal numbers are a fascinating construction due to John H. Conway: they contain all the real numbers, but in fact every ordered field is a subfield of the (proper class-sized) field of surreal numbers. Strictly speaking, the surreal numbers are, like the rationals, given by equivalence classes of surreal number forms, where each form is a pair of sets of surreal numbers, called the left class and the right class. Surreal numbers are generated by transfinite induction and one can give, for each number form, the smallest ordinal at which that form is generated. This is known as the birthday of the form. An interesting question to ask what relation the birthday of x+y, for instance, has to the birthdays of x and y separately, and if there is a general expression. This is the sort of question that the paper answers, for addition, negation, subtraction and multiplication of surreal numbers corresponding to dyadic rational numbers, which are precisely the surreal numbers that admit a finite form. Or equivalently, the surreals one can get after only a (arbitrary) finite number of stages.
Surreal number form DAGs for 3/4 and 1/2
Representations of the surreal numbers 3/4 and 1/2 using DAGs from Matthew Roughan’s SurrealNumbers.jl package (re-use license: https://github.com/mroughan/SurrealNumbers.jl/blob/master/LICENSE.md)

Matt has also created a Julia package to calculate and verify certain properties that were apparently until now unchecked, due to the complexity of calculating with surreal number forms. You can fork it on GitHub and have a play!

Update

~ David Roberts ~ Leave a comment

I have updated my notes on Mochizuki’s recent Report on discussions…, new copy available at my previous blog post. I’m more than happy to discuss these in the comments (and would welcome some feedback, especially from people who are more expert in arithmetic geometry).

Burritos for the hungry mathematician

~ David Roberts ~ 2 Comments

This is just to provide a link to this literary, culinary and mathematic-y masterpiece

Ed Morehouse, Burritos for the Hungry Mathematician, 2015 (pdf)

Abstract: The advent of fast-casual Mexican-style dining establishments, such as Chipotle and Qdoba, has greatly improved the productivity of research mathematicians and theoretical computer scientists in recent years. Still, many experience confusion upon encountering burritos for the first time.
Numerous burrito tutorials (of varying quality) are to be found on the Internet. Some describe a burrito as the image of a crêpe under the action of the new-world functor. But such characterizations merely serve to reindex the confusion contravariantly. Others insist that the only way to really understand burritos is to eat many different kinds of burrito, until the common underlying concept becomes apparent.
It has been recently remarked by Yorgey [9] that a burrito can be regarded as an instance of a universally-understood concept, namely, that of monad. It is this characterization that we intend to explicate here. To wit, a burrito is just a strong monad in the symmetric monoidal category of food, what’s the problem?

Possibly useful documents on IUTT

~ David Roberts ~ Leave a comment

Just a short post so that I can find these again. Chung Pang Mok has some notes on the first two IUTT papers that distill Mochizuki’s wordiness down to what seems like the essential minimum. They are pdf scans of hand-written notes (of high quality):

Together they are 86 pages, but this is shorter than the original, at least, with no ‘motivating’ paragraphs.