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John Baez

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Goodbye, good night, and good luck

I'll miss all of you. It was fun while it lasted. I hope to see you somewhere else down the road.

The death of G+ is a great example of how we shouldn't trust our social lives to companies that are really just treating us as resources to be exploited.

For a long time I've felt Google+ was dying. I tried to quit several times, but I only succeeded this year. I'm glad I did. I've still been posting things now and then - but only when I'm bored. Everything actually interesting that I posted here is or soon will be on my online diary:

and/or my blogs:

Unfortunately, the comments you wrote are not backed up.... and in the heyday of Google+, the comments were often the most interesting part.

If you want to keep track of me without looking at lots of different locations, you can read my stuff on Twitter:

But again, anything interesting that I post there is also on some site that I have more control over. This is what we gotta do, if we don't want to be mere sheep.

Ciao! Toodle-oo! Hasta la vista!
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Riverside math workshop

We're having a workshop with a bunch of cool math talks at U. C. Riverside:

• Riverside Mathematics Workshop for Excellence and Diversity Friday 19 October - Saturday 20 October, 2018. Organized by John Baez, Carl Mautner, José González and Chen Weitao.

You can register for it here:

This is the first of an annual series of workshops to showcase and celebrate excellence in research by women and other under-represented groups for the purpose of fostering and encouraging growth in the U.C. Riverside mathematical community.

After tea at 3:30 p.m. on Friday there will be two plenary talks, lasting until 5:00. Catherine Searle will talk on "Symmetries of spaces with lower curvature bounds", and Edray Goins will give a talk called "Clocks, parking garages, and the solvability of the quintic: a friendly introduction to monodromy". There will then be a banquet in the Alumni Center 6:30 - 8:30 p.m.

On Saturday there will be coffee and a poster session at 8:30 a.m., and then two parallel sessions on pure and applied mathematics, with talks at 9:30, 10:30, 11:30, 1:00 and 2:00. Check out the abstracts!

(I'm especially interested in Christina Vasilakopoulou's talk on Frobenius and Hopf monoids in enriched categories, but she's my postdoc so I'm biased.)

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Applied Category Theory 2019

I'm helping organize ACT 2019, an applied category theory conference and school at Oxford, July 15-26, 2019. Here's the basic idea - more details will come later:

Dear All,

As part of a new growing community in Applied Category Theory, now with a dedicated journal Compositionality, a traveling workshop series SYCO, a forthcoming Cambridge U. Press book series Reasoning with Categories, and several one-off events including at NIST, we launch an annual conference+school series named Applied Category Theory, the coming one being at Oxford, July 15-19 for the conference, and July 22-26 for the school. The dates are chosen such that CT 2019 (Edinburgh) and the ACT 2019 conference (Oxford) will be back-to-back, for those wishing to participate in both.

There already was a successful invitation-only pilot, ACT 2018, last year at the Lorentz Centre in Leiden, also in the format of school+workshop.

For the conference, for those who are familiar with the successful QPL conference series, we will follow a very similar format for the ACT conference. This means that we will accept both new papers which then will be published in a proceedings volume (most likely a Compositionality special proceedings issue), as well as shorter abstracts of papers published elsewhere. There will be a thorough selection process, as typical in computer science conferences. The idea is that all the best work in applied category theory will be presented at the conference, and that acceptance is something that means something, just like in CS conferences. This is particularly important for young people as it will help them with their careers.

Expect a call for submissions soon, and start preparing your papers now!

The school in ACT 2018 was unique in that small groups of students worked closely with an experienced researcher (these were John Baez, Aleks Kissinger, Martha Lewis and Pawel Sobociński), and each group ended up producing a paper. We will continue with this format or a closely related one, with Jules Hedges and Daniel Cicala as organisers this year. As there were 80 applications last year for 16 slots, we may want to try to find a way to involve more students.

We are fortunate to have a number of private sector companies closely associated in some way or another, who will also participate, with Cambridge Quantum Computing Inc. and StateBox having already made major financial/logistic contributions.

On behalf of the ACT Steering Committee,

John Baez, Bob Coecke, David Spivak, Christina Vasilakopoulou
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Mind the gap!

People love to talk about twin primes. But a computer search has shown that among numbers less than a trillion, most common distance between successive primes is not 2 but 6. This goes on for quite a while longer…

… but Andrew Odlyzko, Michael Rubinstein and Marek Wolf have persuaded most experts that somewhere around x = 1.7427 ⋅ 10³⁵, the most common gap between consecutive primes less than x switches from 6 to 30:

• Andrew Odlyzko, Michael Rubinstein, and Marek Wolf, Jumping champions, Experimental Mathematics 8 (1999), 107–118. Available at

This is a nice example of how you may need to explore very large numbers to understand the true behavior of primes.

They give a sophisticated heuristic argument for their claim — not a rigorous proof. But they also checked the basic idea using Maple’s ‘probable prime’ function.

It takes work to check if a number is prime, but there’s a much faster way to check if it’s probably prime in a certain sense. Using this, they worked out the gaps between probable primes from 10³⁰ and 10³⁰+10⁷. They found that there are 5278 gaps of size 6 and just 5060 of size 30. They also worked out the gaps between probable primes from 10⁴⁰ and 10⁴⁰+10⁷ There were 3120 of size 6 and 3209 of size 30.

So, it seems that somewhere between 10³⁰ and 10⁴⁰, the number 30 replaces 6 as the most probable gap between successive primes! But their heuristic argument more precisely pins down the location where it switches.

Using the same heuristic argument, they argue that somewhere around 10⁴⁵⁰ , the number 30 ceases to be the most probable gap. The number 210 replaces 30 as the champion—and reigns for an even longer time.

Furthermore, they argue that this pattern continues forever, with the main champions being the ‘primorials’:


2⋅3 = 6

2⋅3⋅5 = 30



and so on.

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Sometimes you check just a few examples and decide something is always true. But sometimes even 9.8 × 10⁴² examples are not enough!!!

+Greg Egan and I came up with this shocker on Twitter after he explained some related integrals by the Borwein brothers. To see what's really going on, you can visit my blog:

In the comments on my blog, you'll see that Greg figured out the exact value of n for which the identity first fails! It's

n = 15,341,178,777,673,149,429,167,740,440,969,249,338,310,889

which is about 1.5 × 10⁴³. When I saw this I breathed a sigh of relief, because it meant my estimates were right.

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The 5/8 theorem

The vertices of this 4-dimensional shape are the quaternions

1, -1, i, -i, j, -j, k, -k

It's a 4d analogue of the regular octahedron!

These 8 points also form a group: the quaternion group. And it's one of the most commutative of noncommutative groups!

The quaternion group has 2 elements, ±1, that commute with everything. The rest commute with 4 elements each: for example i commutes with ±1 and ±i.

So, 1/4 of the elements commute with all the elements. 3/4 of the elements commute with only half the elements. Thus the chance that two elements commute is

1/4 × 1 + 3/4 × 1/2 = 5/8

If you randomly choose two elements of a finite group, what's the probability that they commute? For the quaternion group we've just seen it's 5/8.

But what's the biggest this probability can be, for any noncommutative group?

It's 5/8. And I explain why here:

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Noether's Theorem

I'm visiting the Centre for Quantum Technologies (in Singapore). Tomorrow I'm giving a talk on Noether's theorem relating symmetries and conserved quantities. It's the 100th anniversary of her paper on this! But we still haven't gotten to the bottom of it.

Noether showed that in a theory of physics obeying the "principle of least action", any 1-parameter family of transformations preserving the action gives a conserved quantity. This video is an easy intro if you don't want to get into the mathematical details:

But Noether's theorem takes different guises in other approaches to physics, and my talk focuses on the algebraic approach using Poisson brackets or commutators. I argue that this explains the role of complex numbers in quantum theory!

You can see my slides at the link!

By the way, I mainly hang out on Twitter these days, not G+:

The trick, I discovered, is to post series of tweets, so I can actually say something mildly interesting. 280 characters sucks. But a lot more people are there, than here.

Noether’s Theorem
Noether’s Theorem
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After Quanta magazine wrote about octonions they decided to write about on quaternions - and they interviewed me!

Did you know the 'dot product' and 'cross product' were originally two parts of a single thing, the quaternion product?

Quaternions were invented by Hamilton in 1843. They became a mandatory examination topic in Dublin, and in some US universities they were the only advanced math taught!

But then came Josiah Willard Gibbs, the first person to get a Ph.D. in engineering in the US, who chopped the quaternion into its scalar and vector parts. After a long battle, quaternions lost to vectors. But they're still worth learning about!

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Big news! European organizations spending €7.6 billion on research annually will require every paper they fund to be freely available from the moment of publication, starting in 2020!

Crush the rip-off journals! They are howling with outrage: their business model is to get taxpayers to fund science research, get the research papers for free, then sell them back to scientists and the taxpayers at enormous profits.
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Those he commands move only in command,

Nothing in love. Now does he feel his title

Hang loose about him, like a giant’s robe

Upon a dwarfish thief.

- William Shakespeare
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