January 23, 2020
Emily Gets A Huge Prize
Posted by Tom Leinster
Café host Emily Riehl has just been awarded a $250,000 prize by her university!! Johns Hopkins gives one President’s Frontier Award every year across the whole university, and the 2020 one has gone to Emily. Up to now it’s usually been given to biological and medical researchers, but when Emily came along they had to make an exception and give it to a mathematician. The award has the goal of “supporting exceptional scholars … who are on the cusp of transforming their fields.”.
Congratulations, Emily! Obviously you’re too modest to announce it yourself here, but someone had to.
You can read all about it here, including the delightful description of how the news was sprung on her:
When Riehl arrived at what she thought was a meeting with a department administrator, she says it was “a complete shock” to find JHU President Ronald J. Daniels, Provost Sunil Kumar, other university leaders, and many colleagues poised to surprise her.
January 17, 2020
Random Permutations (Part 13)
Posted by John Baez
Last time I started talking about the groupoid of ‘finite sets equipped with permutation’, $\mathsf{Perm}$. Remember:
- an object $(X,\sigma)$ of $\mathsf{Perm}$ is a finite set $X$ with a bijection $\sigma \colon X \to X$;
- a morphism $f \colon (X,\sigma) \to (X',\sigma')$ is a bijection $f \colon X \to X'$ such that $\sigma' = f \sigma f^{-1}$.
In other words, $\mathsf{Perm}$ is the groupoid of finite $\mathbb{Z}$-sets. It’s also equivalent to the groupoid of covering spaces of the circle having finitely many sheets!
Today I’d like to talk about another slightly bigger groupoid. It’s very pretty, and I think it will shed light on a puzzle we saw earlier: the mysterious connection between random permutations and Poisson distributions.
I’ll conclude with a question for homotopy theorists.
January 16, 2020
Codensity Monads
Posted by Tom Leinster
Yesterday I gave a seminar at the University of California, Riverside, through the magic of Skype. It was the first time I’ve given a talk sitting down, and only the second time I’ve done it in my socks.
The talk was on codensity monads, and that link takes you to the slides. I blogged about this subject lots of times in 2012 (1, 2, 3, 4), and my then-PhD student Tom Avery blogged about it too. In a nutshell, the message is:
Whenever you meet a functor, ask what its codensity monad is.
This should probably be drilled into learning category theorists as much as better-known principles like “whenever you meet a functor, ask what adjoints it has”. But codensity monads took longer to be discovered, and are saddled with a forbidding name — should we just call them “induced monads”?
In any case, following this principle quickly leads to many riches, of which my talk was intended to give a taste.
January 11, 2020
Random Permutations (Part 12)
Posted by John Baez
This time I’d like to repackage some of the results in Part 11 in a prettier way. I’ll describe the groupoid of ‘finite sets equipped with a permutation’ in terms of Young diagrams and cyclic groups. Taking groupoid cardinalities, this description will give a well-known formula for the probability that a random permutation belongs to any given conjugacy class!
January 10, 2020
Quotienting Out The Degenerate
Posted by Tom Leinster
This is a quick, off-the-cuff, conceptual question. Hopefully, it has an easy answer.
Often in algebra, we want to quotient out by a set of elements that we regard as trivial or degenerate. That’s almost a tautology: any time we take a quotient, the elements quotiented out are by definition treated as negligible. And often the situation is mathematically trivial too, as when we quotient by the kernel of a homomorphism.
But some examples of quotienting by degenerates are slightly more subtle. The two I have in mind are:
the definition of exterior power;
the definition of normalized chain complex.
I’d like to know whether there’s a thread connecting the two.
December 30, 2019
Counting Nilpotents: A Short Paper
Posted by Tom Leinster
Inspired by John’s recent series of posts on random permutations, I started thinking about random operators on vector spaces, and nilpotent operators, and Cayley’s tree formula, and, especially, Joyal’s wonderful proof of Cayley’s formula that led him (I guess) to create the equally wonderful theory of species.
Blog posts and comments are often rambling and discursive. That’s part of the fun of it: we think out loud, we try out ideas, we stumble ignorantly through things that others have done better before us, we make mistakes, we refine our ideas, and we learn how to communicate those ideas more efficiently. My own posts on this topic (1, 2, 3) are no exception.
But short sharp accounts are also good! So I wrote a 4.5-page paper containing the thing I think is new. It’s a new proof of the old theorem that when you choose at random a linear operator on a vector space of finite cardinality $N$, the probability of it being nilpotent is $1/N$. And this proof is a linear analogue of Joyal’s proof of Cayley’s formula.
December 29, 2019
Compositionality: First Issue
Posted by John Baez
Yay! The first volume of Compositionality has been published! You can read it here:
https://compositionality-journal.org
“Compositionality” is about how complex things can be assembled out of simpler parts. Compositionality is a journal for research using compositional ideas, most notably of a category-theoretic origin, in any discipline. Example areas include but are not limited to: computation, logic, physics, chemistry, engineering, linguistics, and cognition.
December 26, 2019
Mathematical Images
Posted by Tom Leinster
Many mathematicians like the drawings of Escher, or sculptures of surfaces, or colourful plots of fractals and other mathematical phenomena. My own taste seems to be a bit different. I’m not sure how to describe it, but over the years I’ve amassed on my computer a small collection of “mathematical” images, in some rather loose sense of the word. Every time I save one I tell myself I might use it in a blog post one day, but every time I write a blog post, I forget.
Since I never use them, and since it’s Christmas, I thought I’d present them all here instead in one big gallery. Some illustrate some mathematical concept, some refer to the experience of being a mathematician, and a couple are silly visual puns. But many just ring a bell in a part of my mind that seems to me to be connected with the activity of doing mathematics.
I’m afraid I have no idea who any of them were created by (not me!); all have been downloaded from the internet at some point. If you want to find out, I’d suggest a reverse image search.
Merry Christmas! And don’t take the images too seriously — this is just for fun.
Schrödinger’s Unified Field Theory
Posted by John Baez
Erwin Schrödinger fled Austria during World War II. In 1940 he found a position in the newly founded Dublin Institute for Advanced Studies. This allowed him to think again. He started publishing papers on unified field theories in 1943, based on earlier work of Eddington and Einstein. He was trying to unify gravity, electromagnetism and a scalar ‘meson field’… all in the context of classical field theory, nothing quantum.
Then he had a new idea. He got very excited about it, and January of 1947 he wrote:
At my age I had completely abandoned all hope of ever again making a really big important contribution to science. It is a totally unhoped-for gift from God. One could become a believer or superstitious [gläubig oder abergläubig], e.g., could think that the Old Gentleman had ordered me specifically to go to Ireland to live in 1939, the only place in the world where a person like me would be able to live comfortably and without any direct obligations, free to follow all his fancies.
He even thought he might get a second Nobel prize.
He called a press conference… and the story of how it all unraveled is a bit funny and a bit sad. But what was his theory, actually?
December 24, 2019
Random Permutations (Part 11)
Posted by John Baez
I think I’m closing in on a good understanding of random permutations using species and groupoid cardinality. Today I want to use this approach to state and prove a categorified version of the Cycle Length Lemma, which is Lemma 2.1 here:
This is a grand generalization of the result in Part 10.
Applied Category Theory 2020 - Adjoint School
Posted by John Baez
Like last year and the year before, there will be a school associated to this year’s conference on applied category theory! If you’re trying to get into applied category theory, this is the best possible way.
The school will consist of online meetings from February to June 2020, followed by a research week June 29–July 3, 2020 at MIT in Cambridge Massachusetts. The conference follows on July 6–10, 2020, and if you attend the school you should also go to the conference.
The deadline to apply is January 15 2020; apply here.
December 20, 2019
A Joyal-Type Proof Of The Number Of Nilpotents
Posted by Tom Leinster
Let $V$ be a finite set. What’s the probability that a random endofunction of $V$ is eventually constant — that is, becomes constant after you iterate it enough times? It’s $1/\#V$. That’s basically Cayley’s tree formula, Joyal’s beautiful proof of which I showed you recently.
Now suppose that $V$ has the structure of a vector space. What’s the probability that a random linear operator on $V$ is eventually constant — or as one usually says, nilpotent? It’s still $1/\#V$. I recently showed you a proof of this using generating functions, but it gave no indication of why the answer is as simple as it is. There was some miraculous cancellation.
Today I’ll show you a much more direct proof of the probability of getting a nilpotent, or equivalently the number of nilpotents. The argument shadows Joyal’s proof of Cayley’s formula.
December 18, 2019
Random Permutations (Part 10)
Posted by John Baez
Groupoids generalize sets in an obvious way: while elements of a set can only be the same or different, objects of a groupoid can be ‘the same’ — that is, isomorphic — in more than one way.
Less obviously, groupoids also let us generalize the concept of cardinality. While the cardinality of a finite set is a natural number, the cardinality of a finite groupoid can be any nonnegative rational number!
This suggests that probabilities in combinatorics could secretly be cardinalities of groupoids. Indeed, Poisson distributions naturally show up this way. For a good self-contained introduction, see:
- Qiaochu Yuan, Groupoid cardinality, Annoying Precision, November 8, 2012.
Now I want to explain how groupoid cardinality can be used to prove some important (and well-known) facts about random permutations.
December 13, 2019
Applied Category Theory Postdocs at NIST
Posted by John Baez
Here is an advertisement for postdoc positions in applied category theory at the National Institute of Standards and Technology.
December 11, 2019
Random Permutations (Part 9)
Posted by John Baez
In our quest to understand the true nature of random permutations, Part 7 took us into a deeper stratum: their connection to Poisson distributions. I listed a few surprising facts. For example, these are the same:
- The probability that $N$ raindrops land on your head in a minute, if on average one lands on your head every $k$ minutes.
- The probability that a random permutation of a huge finite set has $N$ cycles of length $k$.
Here the raindrops are Poisson distributed, and ‘huge’ means I’m taking a limit as the size of a finite set goes to infinity.
Now let’s start trying to understand what’s going on here! Today we’ll establish the above connection between raindrops and random permutations by solving this puzzle:
Puzzle 12. Treat the number of cycles of length $k$ in a random permutation of an $n$-element set as a random variable. What do the moments of this random variable approach as $n \to \infty$?
First, let me take a moment to explain moments.