@johncarlosbaez@mathstodon.xyz
@johncarlosbaez@mathstodon.xyz avatar

johncarlosbaez

@johncarlosbaez@mathstodon.xyz

I'm a mathematical physicist who likes explaining stuff. Sometimes I work at the Topos Institute. Check out my blog! I'm also a member of the n-Category Café, a group blog on math with an emphasis on category theory. I also have a YouTube channel, full of talks about math, physics and the future.

This profile is from a federated server and may be incomplete. Browse more on the original instance.

julesh, to random
@julesh@mathstodon.xyz avatar

I'm a city kid and I've never seen any astronomy before... but the northern lights were so bright we could see them in central Glasgow

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@julesh - I failed to look out our north-facing windows in Edinburgh. 😿

johncarlosbaez, to random
@johncarlosbaez@mathstodon.xyz avatar

The rise of book bans in Florida made Lauren Groff start a new bookstore there. It 𝑠𝑝𝑒𝑐𝑖𝑎𝑙𝑖𝑧𝑒𝑠 in banned books. It's called The Lynx.

More here: https://archive.is/fD0uF

MartinEscardo, to random
@MartinEscardo@mathstodon.xyz avatar

It is just me? The following definition of category hurts my categorical instincts, because it uses object equality.

A category consists of

  1. A collection of objects.

  2. A collection of morphisms.

  3. Each morphism f has two assigned objects, its source s(f) and its target t(f).

  4. For each pair of morphisms f,g such that t(f)=s(g) there exists a specified morphism g ∘ f such that [it doesn't matter what]

  5. [Some axioms are satisfied.]

It is (4) that hurts my categorical instincts.

There is no reason to have "evilness" (in the categorical sense, rather than the emotional sense) built-in in the definition of category!

This definition is, for example, adopted by Freyd.

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@oantolin @MartinEscardo - indeed, most of us needed to learn any definition of category whatsoever - the nuances don't matter so much - before we could be attuned to the nuances of sameness (equality vs. isomorphism, etc.) and understand the problems of "evil", and appreciate why we'd want to avoid a definition of category that mentions equations between objects!

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@MartinEscardo @oantolin - I agree that it would be good to give people a nice definition of "category" the first time. But I don't see category theorists unwilling to give different styles of definition a fair chance. Maybe you're referring to non-category-theorists. I'd forgotten about them. 😳 I can easily imagine them sticking to the first definition of category they see, regarding it as sacrosanct.

johncarlosbaez, (edited )
@johncarlosbaez@mathstodon.xyz avatar

@antoinechambertloir @MartinEscardo @oantolin - There is a rather elaborate and carefully worked out esthetic of formulating definitions in categorical logic, aimed to ensure that everything works out as smoothly as possible, e.g. that everything you say about a functor is invariant under natural isomorphisms, and everything you say about a category is invariant under equivalence. Equations between objects can easily break these principles so we call them 'evil'; this then pressures us to take an approach where we don't need to check that the source of one morphism equals the target of the next.

A common approach is to make morphisms 'dependently typed', so that for each pair of objects (a,b) you have a set of morphisms hom(a,b). You never talk about the set of all morphisms, so you never mention source and target maps. Composition is not a single partially defined function, but instead a bunch of functions hom(a,b) × hom(b,c) → hom(a,c). So, you never need to check that the source of one morphism equals the target of another: it's impossible to even dream of composing morphisms unless you already know you can do it!

paulbalduf, to physics
@paulbalduf@mathstodon.xyz avatar

In , scattering amplitudes can be computed as sums of (very many) s. They contribute differently much, with most integrals contributing near the average (scaled to 1.0 in the plots), but a "long tail" of integrals that are larger by a significant factor.
We looked at patterns in these distributions, and one particularly striking one is that if instead of the Feynman integral P itself, you consider 1 divided by root of P, the distribution is almost Gaussian! To my knowledge, this is the first time anything like this has been observed. We only looked at one quantum field theory, the "phi^4 theory in 4 dimensions". It would be interesting to see if this is coincidence for this particular theory and class of Feynman integrals, or if it persists universally.
More background and relevant papers at https://paulbalduf.com/research/statistics-periods/

image/jpeg
image/jpeg

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@paulbalduf - that's a great observation! Have you written a paper about it? It's worthwhile, even if it's only true for the ϕ₄⁴ theory. Which class of Feynman diagrams were you considering? E.g. 4-point functions?

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@paulbalduf - thanks! I'll check out the paper and tell some people about it.

BartoszMilewski, to random
@BartoszMilewski@mathstodon.xyz avatar

I'm struggling with the definition of the category of elements--the direction of morphisms. Grothendieck worked with presheaves (C^{op} \to \mathbf{Set}), with a morphism ((a, x) \to (b, y)) being an an arrow (a \to b) in (C). The question is, what is it for co-presheaves? Is it (b \to a)? nLab defines it as (a \to b) and doesn't talk about presheaves. Emily Riehl defines both as (a \to b), which makes one wonder what it is for (𝐶ᵒᵖ)ᵒᵖ→𝐒𝐞𝐭 , not to mention (C^{op}\times C \to \mathbf{Set}).

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@BartoszMilewski - it's just a convention; either convention is possible. So in what sense are you struggling with it? Are you struggling to decide what you like best? You don't really need to pick a favorite.

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@BartoszMilewski - they both exist, neither is "right" by decree of god, so use the one that works for what you're doing... which may change next week.

johncarlosbaez, (edited ) to random
@johncarlosbaez@mathstodon.xyz avatar

Costa Talalaev gave me a fractal: a 3-dimensional Sierpinski gasket!

He made it using the 3d printer at the Hacklab, a makerspace here in Edinburgh. It was a bit hard to make since it's held together only at tiny spots. He had to build something with more plastic, a kind of scaffolding, and then tear that off. The end result is very light yet sturdy.

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@SamuelCraig - right, in the idealized limit it has zero volume!

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@bks - wow! Thanks!

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@codrusofathens - all this stuff is a lot of fun, indeed!

Unsolicited advice: fractals are just a tiny piece of the math landscape, so only do a math double major if you enjoy a lot of other math too. For example, to get serious about fractals you need to take advanced calculus and then real analysis, which are tough but to me utterly delightful subjects - and much bigger than fractals in the grand scheme of things.

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@codrusofathens - numerical analysis is a natural road to a job. Number theory is beautiful but there are few jobs in it and it's insanely competitive. But one great thing about a double major is that you can study something just for the love of it.

I'm studying a lot of number theory these days, and the layers of conceptual depth are absolutely enthralling.

johncarlosbaez, to random
@johncarlosbaez@mathstodon.xyz avatar

This is an ORC - an "odd radio cluster". It's a faint circle of radio emissions surrounding a distant galaxy. 5 ORCs have been found. This one is about a million light years in diameter, roughly 10 times the size of a galaxy like ours. So it was probably formed by some sort of explosion, and took a long time to get this big. But the details remain a mystery!

Astronomers have just discovered that one ORC is emitting X-rays. So at least in that one ORC, the diffuse gas must be hot: about 8 million Celsius.

Here are some theories of what an ORC might be:

• it's a spherical shock wave from a cataclysmic event in the host galaxy, such as a merger of two supermassive black holes

• it's the shock wave formed by a 'starburst wind' created by a burst of star formation in the host galaxy

• it's the jet produced by a supermassive black hole, seen head on.

• it's a wormhole

As often the case, the most exciting theory is the least likely unless the others get ruled out.

But we can still have some fun. So watch an animated gif of what the birth of an ORC might look like!

(1/2)

johncarlosbaez, (edited )
@johncarlosbaez@mathstodon.xyz avatar

Here's a possible scenario for the formation of an ORC: some sort of explosion in the host galaxy, producing a shock wave that expanded outwards for a billion years.

For more on ORCs, try these:

• Wikipedia, Odd radio cluster, https://en.wikipedia.org/wiki/Odd_radio_circle

• Ray P. Norris, Evan Crawford and Peter Macgregor, Odd radio circles and their environment, Galaxies, https://www.mdpi.com/2075-4434/9/4/83

• Ashley Bazar, X-ray satellite XMM-Newton sees ‘Space Clover’ in a new light, NASA, https://www.nasa.gov/missions/xmm-newton/x-ray-satellite-xmm-newton-sees-space-clover-in-a-new-light/

(2/2)

dpiponi, to random
@dpiponi@mathstodon.xyz avatar

I have to admit I enjoy seeing familiar sci-fi plots appearing as papers. This one proposes that AIs cause civilizational collapse, explaining the Fermi "paradox".

https://arxiv.org/abs/2405.00042

"AI could represent a major threat to the future course of not only our technical civilisation but all technical civilisations"

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@dpiponi - funny that this is classified as "physics" on the arXiv. Okay, "popular physics".

johncarlosbaez, to random
@johncarlosbaez@mathstodon.xyz avatar

Last year, for the first time, 30% of electricity produced worldwide was from renewable sources. Wind and solar are growing. But notice that the biggest is hydroelectric, and it's going down! One reason is droughts in India, China, North America and Mexico. Climate change is causing droughts.

We're in a race against time. But at least we're running.

https://apnews.com/article/renewable-energy-climate-solar-wind-fossil-fuels-2718fce0ed37232dc25dbf46fff87955

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@doc - The article I cited says that the sudden recent drop in % hydro is due to droughts, but you're right that even if hydro power remained constant its % would drop as other forms of power production rise.

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@benjohn wrote: "If people can see this trend, it might both give hope, inspiration of a joint venture, and help them prepare for how life might differ."

Agreed! Already the Washington Post is whining about how California has more solar power than they know what to do with - sometimes. But it's not a "problem": it's an opportunity.

Since I'm a worrier, I note that only 20% of world energy production is used for electricity, and a lot of the other uses are harder to decarbonize! For example about 25% is used for transportation. We can convert cars and trains to electric, trucks are harder but doable, ships are still harder, and airplanes are still harder: we could use biofuels for those to make them carbon-neutral, theoretically. And so on... we need to do everything.

And all this decarbonization is just part of grappling with the Anthropocene: how we're now in control of all the biogeochemical cycles on Earth, whether we like it or not.

johncarlosbaez, (edited )
@johncarlosbaez@mathstodon.xyz avatar

@adrianmorales - There are lots of problems, not "the problem". I'm in Scotland now, so I've seen how gas prices have skyrocketed here. I'm not surprised it's driving people into poverty. Global warming is another huge problem.

Meanwhile, the price of electricity here sometimes goes negative on windy days. Same in California. The Washington Post even claimed that's a problem! But I don't think that is a serious problem.

chrisamaphone, to random
@chrisamaphone@hci.social avatar

full professor implies the existence of new, crescent, half, and gibbous professors

(it’s rude to ask waxing or waning though)

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@jcreed - I'm
enjoying being an undistinguished professor.

@aadmaa @chrisamaphone

johncarlosbaez, (edited ) to random
@johncarlosbaez@mathstodon.xyz avatar

Tolstoy: "Happy families are all alike; every unhappy family is unhappy in its own way."

Mathematics: "Real tori are all alike; every complex torus is complex in its own way."

To be precise, a 'n-dimensional real torus' is a real manifold of the form V/Λ where V is an n-dimensional real vector space and Λ ⊆ V is a lattice of rank n in this vector space. They are all isomorphic.

An 'n-dimensional complex torus' is a complex manifold of the form V/Λ where V is an n-dimensional complex vector space and Λ ⊆ V is a lattice of rank 2n in this vector space. These are not all isomorphic, because there are different ways the lattice can get along with multiplication by i. For example we might have iΛ = Λ or we might not.

And so, it's possible to write a whole book - and indeed a fascinating one - on complex tori. For example a 1-dimensional complex torus is an elliptic curve, and there are whole books just about those.

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@wnj - I have been enjoying 𝐶𝑜𝑚𝑝𝑙𝑒𝑥 𝐴𝑏𝑒𝑙𝑖𝑎𝑛 𝑉𝑎𝑟𝑖𝑒𝑡𝑖𝑒𝑠 by Herbert Lange and Christina Birkenhake - not easy, but full of beautiful material. Complex abelian varieties are the 'best' complex tori. The same authors have an earlier book 𝐶𝑜𝑚𝑝𝑙𝑒𝑥 𝑇𝑜𝑟𝑖, but for some stupid reason I haven't looked at that yet.

https://link.springer.com/book/10.1007/978-1-4612-1566-0

johncarlosbaez,
@johncarlosbaez@mathstodon.xyz avatar

@berber - Yes, everything you said is right. It's not surprising. Perhaps what's surprising is that there's such a rich and beautiful theory of complex tori. This especially true of the nicest ones, the 'abelian varieties', which are complex algebraic varieties.

https://en.wikipedia.org/wiki/Abelian_variety

  • All
  • Subscribed
  • Moderated
  • Favorites
  • •
  • Leos
  • cubers
  • magazineikmin
  • everett
  • thenastyranch
  • Youngstown
  • slotface
  • hgfsjryuu7
  • ngwrru68w68
  • rosin
  • kavyap
  • khanakhh
  • PowerRangers
  • DreamBathrooms
  • anitta
  • mdbf
  • InstantRegret
  • ethstaker
  • Durango
  • osvaldo12
  • tacticalgear
  • vwfavf
  • tester
  • GTA5RPClips
  • cisconetworking
  • modclub
  • normalnudes
  • provamag3
  • All magazines
    •