I'm a mathematical physicist interested in saving the planet.

14 added to My Authors

Jan 19 • 10 tweets • 3 min read

I've been explaining how spaces with dimension is a multiple of 4 are special. Using these ideas you can build a 4d topological manifold that can't be made smooth - starting from the Dynkin diagram of E8! But we won't get that far today.

(1/n) For today, 'manifold' will mean 'compact smooth oriented manifold'. If such a manifold M has dimension n = 2k, its deRham cohomology comes with a bilinear map

b: Hᵏ(M) × Hᵏ(M) → R

which is symmetric when k itself is even:

b(x,y) = b(y,x)

Jan 18 • 6 tweets • 3 min read

Without special relativity, mercury would melt at about 80 °C.

(1/n) The transition metals on the far right are weird compared to the rest because the d subshell is completely full at this point. Zinc is hard, cadmium is soft and mercury is actually liquid.

(2/n)

Jan 16 • 6 tweets • 3 min read

Some of these crazy stars pulse in brightness as fast as every 90 seconds! Waves of ionizing iron pulse through their thin surface atmosphere.

What's up with these weird stars?

(1/n) Sometimes a red giant loses most of its outer hydrogen... nobody knows why... leaving just a thin layer of hydrogen over its helium core.

We get a star with at most 1/4 the diameter of the Sun, but really hot.

It's the blue-hot heart of a red giant, stripped bare.

(2/n)

Jan 13 • 5 tweets • 4 min read

Of course you'd instantly be fried by gamma rays of arbitrarily high frequency, but still.

This is also the color of a typical neutron star. They're so hot they look the same.

This was worked out by @gro_tsen.

(1/n) @gro_tsen As a blackbody gets hotter and hotter, its spectrum approaches the classical "Rayleigh-Jeans law".

That is, its true spectrum as predicted by quantum mechanics - the "Planck law" - approaches the classical prediction over a larger and larger range of frequencies.

(2/n)

Jan 11 • 6 tweets • 2 min read

It's so weird some astronomers have wondered if it's a nuclear waste dump for an alien civilization.

(1/n) The problem is, how do you get a lot of short-lived radioactive elements into the upper atmosphere of a big hot star? They're usually formed by merging neutron stars. So I'll make my own wacky guess: Przybylski's star zipped through the remains of such an event.

(2/n)

Jan 8 • 12 tweets • 5 min read

Who would have guessed that 9d spaces are more like 17d spaces than 10d spaces? Just saying this makes me feel like a mad scientist.

(1/n) Here's the basic idea. If you have two smooth 1d curves drawn on a 2d plane, they can intersect - and if they do, usually moving them slightly won't stop them from intersecting. How does this generalize to higher dimensions?

(2/n)

Jan 6 • 10 tweets • 4 min read

d3qi0qp55mx5f5.cloudfront.net/cpost/i/docs/a…

It tries to figure out who these people are.

(1/n) First, note that 10 million people is a lot. Even if you limit yourself to the small fraction of these people who are also white male gun owners who have military training since they're veterans, that's 360,000 people - almost the size of the entire US National Guard.

(2/n)

Jan 2 • 9 tweets • 4 min read

I have a crush on the 𝗕𝗼𝗹𝘇𝗮 𝗰𝘂𝗿𝘃𝗲:

y² = x⁵ - x

The space of complex solutions of this equation, plus a point at infinity, is the 2-holed Riemann surface with the biggest possible symmetry group! Its universal cover looks like this:

(1/n) More precisely you get the complex version of the Bolza curve, with its correct conformal structure, by gluing together opposite sides of this patch of the hyperbolic plane.

(2/n)

Dec 13, 2021 • 7 tweets • 3 min read

People want to test this at the Extreme Light Infrastructure.

(1/n) This is an enormous electric field, able to accelerate a proton from rest to Large Hadron Collider energies in just 5 micrometers!

Light-on-light scattering has been seen at the LHC by colliding lead ions. But people want to do it with lasers.

(2/n)

atlas.cern/updates/briefi…

Dec 12, 2021 • 6 tweets • 3 min read

Second heaviest: tennessine.

Combine them: get oganesson tetratenneside, OgTn₄ - so heavy that special relativity makes it weird!

Alas, it decays radioactively in less than a millisecond.

(1/n) It's purely theoretical, since both elements are artificially made in tiny amounts, and short-lived. But we can study the compounds they make using computer calculations. And it seems OgTn₄ should be possible, shaped like the tetrahedron at left.

(2/n)

cen.acs.org/physical-chemi…

Dec 11, 2021 • 5 tweets • 2 min read

(1/n) James and I haven't talked much in about 10 years - I went off a different direction, and thought about climate change and ultimately applied category theory. But I'll give it a try today. I'll join him and Todd at 4 this afternoon and see what it's like.

(2/n)

Dec 6, 2021 • 12 tweets • 4 min read

Indeed any compact oriented 2-manifold, like these here, can be filled in - it's the boundary of some compact oriented 3-manifold.

The same thing is true one dimension up - but not two dimensions up!

(1/n) I say "oriented" because the Klein bottle is a compact 2-manifold that's not orientable, and it's not the boundary of any compact 3-manifold.

Basically the problem is that you can't tell what its "inside" would be.

(2/n )

Dec 4, 2021 • 4 tweets • 2 min read

It works for any function that obeys the quaternionic version of Cauchy's equation ∂f/∂x + i∂f/∂y = 0.

It was discovered by Rudolf Fueter in 1935.

(1/n) Quaternionic analysis isn't quite as beautiful as complex analysis, mainly because if you multiply two solutions of

∂f/∂x + i∂f/∂y + j∂f/∂z + k∂f/∂z = 0

you don't get another solution! 😢

But it ain't chopped liver, neither. It deserves a little love.

(2/n)

Nov 30, 2021 • 4 tweets • 2 min read

Why?

Because they're the first whose electron wavefunctions are described by 𝘲𝘶𝘢𝘥𝘳𝘢𝘵𝘪𝘤 functions of x,y, and z - not just linear or constant.

(1/n) More precisely: these waves involve a 'spherical harmonic', a function on the sphere that can be described using a polynomial in x,y,z. The jargon goes like this:

constant: s orbital

linear: p orbital

quadratic: d orbital

etc.

(2/n)

Nov 27, 2021 • 6 tweets • 3 min read

When you turn on a magnetic field, it automatically lines up at right angles to the field, and the electrons start moving around!

This current loop creates its own magnetic field (in purple).

(1/n) What does this current loop look like, exactly?

To understand this, you have to know that the 6 smeared out or "delocalized" electrons lie above and below the plane of the benzene molecule.

(They come from "p orbitals" of the carbon atoms, which point up & down.)

(2/n)

Nov 24, 2021 • 5 tweets • 2 min read

Ruth Bamford's plan: create a torus of plasma around the orbit of Mars' moon Phobos, carrying electric current!

(1/n) The electric current, going around a loop, could create a magnetic field that protects Mars from the solar wind... just as Earth's magnetic field protects us!

The solar wind hitting Mars now creates radiation 12,000 times that on Earth - not good for your health.

(2/n)

Nov 23, 2021 • 5 tweets • 2 min read

There 104 ways where the picture has 4 loops.

(1/n) By attaching another photon line to those pictures, Stefano Laporta found all 891 diagrams with 4 loops where an electron absorbs a photon. Here are examples.

An electron is a little magnet. He computed the effect of all 891 diagrams on the strength of that magnet.

(2/n)

Nov 22, 2021 • 4 tweets • 3 min read

(1/n)

johncarlosbaez.wordpress.com/2021/11/22/com… @Joe_DoesMath Owen blogged about this project early on, and this post explains the physics intuitions behind our work:

topos.site/blog/2021/09/c…

It's all about equilibrium thermodynamics, or "thermostatics". Thermostatic systems maximize entropy. Entropy is a concave function.

(2/n)

Nov 19, 2021 • 6 tweets • 3 min read

(1/n) If benzene looks like Kekulé first thought, there would be 4 ways to replace two hydrogens with chlorine! You could have two chlorines next to each other with a single bond between them as shown here... or a double bond.

But there aren't 4, just 3.

(2/n)

Nov 18, 2021 • 6 tweets • 3 min read

math.ucr.edu/home/baez/topo…

But since this is twitter let me say it even shorter... and more vaguely. A topos is a mathematical universe kinda like the one you grew up in, but maybe different. (1/n) Topos theory arose from the collision of Lawvere and Grothendieck, two great mathematicians with very different goals. Lawvere wanted to find foundations of mathematics more closely connected to actual practice, with the help of category theory. (2/n)

Nov 11, 2021 • 6 tweets • 3 min read

(1/n) The bacterium lives in bubbling hot mud in a volcano. It survives by metabolizing methane.

It can use any of the 4 lightest lanthanides to do this: lanthanum (Ln), cerium (Ce), praseodymium (Pr) and neodymium (Nd). These are chemically very similar.

(2/n)