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January 26, 2023

Mathematics for Humanity

Posted by John Baez

I mentioned this earlier, but now it’s actually happening! I hope you can think of good workshops and apply to run them in Edinburgh.

Continue reading “Mathematics for Humanity” …
Posted at 9:42 PM UTC | Permalink | Followups (10)

January 23, 2023

Question on Condensed Matter Physics

Posted by John Baez

The tenfold way is a mathematical classification of Hamiltonians used in condensed matter physics, based on their symmetries. Nine kinds are characterized by choosing one of these 3 options:

  • antiunitary time-reversal symmetry with T 2=1T^2 = 1, with T 2=−1T^2 = -1, or no such symmetry.

and one of these 3 options:

  • antiunitary charge conjugation symmetry with C 2=1C^2 = 1, with C 2=−1C^2 = -1, or no such symmetry.

(Charge conjugation symmetry in condensed matter physics is usually a symmetry between particles - e.g. electrons or quasiparticles of some sort - and holes.)

The tenth kind has unitary “SS” symmetry, a symmetry that simultaneously reverses the direction of time and interchanges particles and holes. Since it is unitary and we’re free to multiply it by a phase, we can assume without loss of generality that S 2=1S^2 = 1.

What are examples of real-world condensed matter systems of all ten kinds?

Continue reading “Question on Condensed Matter Physics” …
Posted at 7:43 PM UTC | Permalink | Followups (9)

January 17, 2023

The Tenfold Way (Part 8)

Posted by John Baez

Last time I explained a wonderful one-to-one correspondence between the 10 Morita equivalence classes of Clifford algebras and Cartan’s 10 infinite families of compact symmetric spaces.

This time I’ll do something different. I’ll explain a wonderful one-to-one correspondence between the 10 Morita equivalence classes of Clifford algebras and Cartan’s 10 infinite families of compact symmetric spaces.

Yes, it’s different! Not only will the details of the construction look very different, it gives a different correspondence! And I hope you can help me figure out what’s going on.

I thank Claude Schochet for pointing out that these two constructions don’t match.

Continue reading “The Tenfold Way (Part 8)” …
Posted at 11:05 PM UTC | Permalink | Followups (4)

January 16, 2023

The Tenfold Way (Part 7)

Posted by John Baez

Last time I reviewed a bit of Bott periodicity. Now I want to start leading up to a question about it. It will take a while.

So, this time, I will explain a wonderful one-to-one correspondence between the 10 Morita equivalence classes of Clifford algebras and Cartan’s 10 infinite families of compact symmetric spaces.

Then, next time, I will explain a wonderful one-to-one correspondence between the 10 Morita equivalence classes of Clifford algebras and Cartan’s 10 infinite families of compact symmetric spaces.

Unfortunately these correspondences are not the same! And my question will be: why? Or, perhaps better: what’s the relationship between them?

Continue reading “The Tenfold Way (Part 7)” …
Posted at 12:15 AM UTC | Permalink | Followups (3)

January 15, 2023

The Tenfold Way (Part 6)

Posted by John Baez

I’ve been studying Bott periodicity on and off since 1979, when I did a term paper on Clifford algebras in an undergrad course on group representation theory and physics taught by Valentine Bargmann. He was 71 at the time. Some of the students mocked him for being a bit slow — but if we’d known he’d been Einstein’s assistant from 1937 to 1946, we might have shown him a bit more respect, and asked him what working with Einstein was like!

I still have that term paper somewhere. Now I’m getting a bit slow, and I still don’t understand Bott periodicity quite as well as I want. So I have some questions. But in this part I’ll mainly just explain a bunch of stuff.

Continue reading “The Tenfold Way (Part 6)” …
Posted at 11:17 PM UTC | Permalink | Followups (12)

Total Freedom

Posted by John Baez

Wow! I just learned an objective reason why sets and vector spaces are special!

Of course we all know math relies heavily on set theory and linear algebra. And if you know category theory, you can say various things about why the categories Set\mathsf{Set} and Vect\mathsf{Vect} are particularly convenient frameworks for calculation. But I’d never known a theorem that picks out these categories, and just a few others.

Briefly: these are categories of algebraic gadgets where all the objects are free!

We could call these ‘totally free’ algebraic gadgets.

Continue reading “Total Freedom” …
Posted at 5:59 PM UTC | Permalink | Followups (18)

January 6, 2023

Topos Institute Positions

Posted by John Baez

The Topos Institute is doing some remarkable work in applying category theory to real-world problems. And they’re growing!

They want to hire a Finance and Operations Manager and a Research Software Engineer. For more information, go here.

And if you’re a grad student, you definitely want to check out their summer research positions! For more information on those, go here. Applications for these are due February 15th, 2023.

Posted at 10:25 PM UTC | Permalink | Post a Comment

January 4, 2023

A Curious Integral

Posted by John Baez

On Mathstodon, Robin Houston pointed out a video where Oded Margalit claimed that it’s an open problem why this integral:

∫ 0 ∞cos(2x)∏ n=1 ∞cos(xn)dx \displaystyle{ \int_0^\infty\cos(2x)\prod_{n=1}^\infty\cos\left(\frac{x}{n} \right) d x }

is so absurdly close to π8\frac{\pi}{8}, but not quite equal.

Continue reading “A Curious Integral” …
Posted at 4:26 PM UTC | Permalink | Followups (14)