Notices by John Carlos Baez (johncarlosbaez@mathstodon.xyz), page 5

The Kervaire Invariant Problem is a famous problem in topology that remains open in just one case: the case of 126 dimensions! See the Wikipedia article for an explanation.

Apparently this one remaining case has been solved by Zhouli Xu, Weinan Lin and Guozhen Wang. Xu will reveal the answer tomorrow, Thursday May 30th, at the Princeton Algebraic Topology Seminar! I think a video will appear later online.

.....

Computing differentials in the Adams spectral sequence
May 30, 2024 - 01:00 - May 30, 2024 - 02:00
Zhouli Xu, University of California, San Diego

Online Talk

I will review classical methods computing differentials in the Adams spectral sequence, and then discuss some recent progress in joint work with Weinan Lin and Guozhen Wang. In particular, I will discuss the fate of h_6^2, resolving the Last Kervaire Invariant Problem in dimension 126.

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

Using his periodic table, Mendeleev predicted the properties of a lot of elements that hadn't been discovered yet. But he missed others - and more than half of his predicted elements were never found!

Two of the most interesting are newtonium and coronium. He predicted both to have atomic masses lighter than hydrogen!

Mysterious green spectal lines had been seen in the Sun's corona in 1869. Mendeleev was not the only one who thought it was a new element: coronium. It later turned out to be a highly ionized form of iron, Fe¹³⁺, meaning that 13 electrons get knocked off by the extremely high temperatures.

Even more interesting is newtonium. Back then, a lot of people thought light consisted of vibrations in a mysterious substance called the aether. In a 50-page paper called “An attempt towards the chemical conception of the aether”, Mendeleev hypothesized that the aether was a noble gas! A bunch of new noble gases had just been discovered, forcing a new column in his table. He dubbed this one newtonium, and he estimated its atomic mass as 0.17.

Details here:

• Gábor Lente, Where Mendeleev was wrong: predicted elements that have never been found, https://link.springer.com/article/10.1007/s40828-019-0092-5

Moral: you don't have to be right all the time.

When global warming gets bad enough to create millions of refugees, just imagine how nasty the EU and US will become. They may just ship people out to the desert to die.

Oh wait, you don't have to imagine! It's happening already:

"A year-long joint investigation by The Washington Post, Lighthouse Reports and a consortium of international media outlets shows how the European Union and individual European nations are supporting and financing aggressive operations by governments in North Africa to detain tens of thousands of migrants each year and dump them in remote areas, often barren deserts.

European funds have been used to train personnel and buy equipment for units implicated in desert dumps and human rights abuses, records and interviews show. Migrants have been pushed back into the most inhospitable parts of North Africa, exposing them to abandonment with no food or water, kidnapping, extortion, sale as human chattel, torture, sexual violence and, in the worst instances, death."

The full story is here: https://archive.is/1tEgl

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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

If I tell you the radii of the spheres 𝑎 and 𝑏 in this picture, can you figure out the radii 𝑟₁,...,𝑟₆ of the six spheres that touch them and snugly fit inside the big sphere? Can you at least do it if you know 𝑟₁?

Irisawa Shintarō Hiroatsu did it in 1822! He was a merchant who sold tea, textiles and ingredients for traditional Chinese medicine - and he had a hobby of solving math puzzles.

In 1932 his technique was rediscovered by a Nobel-prize-winning chemist, so it's often called Soddy’s Hexlet Theorem. But Hiroatsu did it earlier as part of a Japanese mathematical tradition called 𝑤𝑎𝑠𝑎𝑛 - and as part of this tradition, he donated a plaque containing this result to a shrine!

He wasn't the only one who did this sort of thing. This kind of plaque is called a 𝑠𝑎𝑛𝑔𝑎𝑘𝑢. These plaques were used to commemorate newly discovered solutions to hard math problems during the Edo Period from 1603 to 1868. There's a lot of interesting math in these 𝑠𝑎𝑛𝑔𝑎𝑘𝑢, and you can see some of them here:

• Abe Haruki, Japan’s “𝑊𝑎𝑠𝑎𝑛” mathematical tradition: surprising discoveries in an age of seclusion, https://www.nippon.com/en/japan-topics/c12801/

You can also learn more about the solution to the puzzle I gave! The most surprising thing is that the reciprocals of the opposite pairs of spheres in the "hexlet" of 6 spheres add up to the same number:

1/𝑟₁+1/𝑟₄ = 1/𝑟₂+1/𝑟₅ = 1/𝑟₃+1/𝑟₆

See also:

• Wikipedia, Soddy's hexlet, https://en.wikipedia.org/wiki/Soddy%27s_hexlet

This math is secretly all about conformal transformations, which map spheres to spheres... or planes!

Thanks to @highergeometer for pointing this out!

Some wasps are called 'parasitoids' because they lay their eggs in still-living caterpillars. The eggs develop into larvae that eat the caterpillar from the inside and ultimately kill it.

But turnabout is fair play. Sometimes, other wasps called 'hyperparasitoids' lay their eggs in the larvae of these parasitoids!

The parasitized caterpillars also fight back. Their immune system detects the wasp's eggs, and they will do things like surround the eggs in a layer of tissue that chokes them.

But many parasitoid wasps have a trick to stop this. They deploy viruses that infect the caterpillar and affect its behavior in various ways - for example, slowing its immune response to the implanted eggs.

These viruses can become so deeply symbiotic with the wasps that their genetic code becomes part of the wasp's DNA. So every wasp comes *born* with the ability to produce these viruses. They're called 'polydnaviruses'.

In fact some wasps are symbiotic with *two kinds* of virus. One kind, on its own, would quickly kill the caterpillar - not good for the wasp. The other kind keeps the first kind under control.

And I'm immensely simplifying things here. There are over 25,000 species of parasitoid wasps, so there's a huge variety of things that happen, which scientists are just starting to understand! I had fun reading this:

• Marcel Dicke, Antonino Cusumano and Erik H. Poelman, Microbial symbionts of parasitoids,
Annual Review of Entomology, https://doi.org/10.1146/annurev-ento-011019-02493

Why such diversity? I think it's just that there are so many plants! So insect larvae like caterpillars naturally tend to feed on them... in turn providing a big food source for parasitoids, and so on.

Hardcore math tweet:

The complex numbers are nice in two ways. They are an 'algebraically closed' field, meaning that every polynomial equation with complex coefficients has a complex solution. And they are 'Cauchy complete' metric space, meaning that every Cauchy sequence converges.

We can get the complex numbers in two ways. We can start with the rational numbers and take their Cauchy completion. This gives us the real numbers. But these are not algebraically closed. So we can take their algebraic closure. The result is the complex numbers, which is still Cauchy complete.

Or, we can start with the rational numbers and take their algebraic closure. This gives us the 'algebraic numbers'. There's a way to define a nice metric on these, but the resulting metric space is not Cauchy complete. To fix that, we can take its Cauchy completion. The result is the complex numbers, which is still algebraically closed.

In the first route I used the usual metric on rational numbers. But what if we use one of the p-adic metrics?

We can start with the rational numbers and take their Cauchy completion using the p-adic metric. This gives us the p-adic numbers. But these are not algebraically closed. So then we can take their algebraic closure. There's a nice metric on it, but it's 𝑛𝑜𝑡 still Cauchy complete.

So we can take the Cauchy completion 𝑎𝑔𝑎𝑖𝑛. You may feel sort of pessimistic right around now... but this time the resulting field 𝑖𝑠 algebraically closed, and of course Cauchy complete by definition. So yay, we're done! 🎉

The weird part: the resulting field is isomorphic to the complex numbers equipped with a weird metric. Using the axiom of choice. 😬

WHAT MAKES PLANTS GO BAD?

Why do plants become carnivorous?

They do so only when it's very hard to get nutrients like nitrogen. They still get their energy from photosynthesis, and it costs a lot of energy for them to be carnivorous. So they only arise in unusual environments like acidic bogs that are nitrogen-poor but sunny. And some cease to be carnivorous in the winter, when there's less light.

There's a continuum of plants ranging from noncarnivorous to so-called "protocarnivorous" plants to fully carnivorous plants. Protocarnivorous plants trap and kill insects or other animals but don't have special enzymes to digest their prey. Some of these evolve to become carnivorous. But don't think of evolution as goal-directed: depending on changes in their environment, some evolve *away* from being carnivorous. It's mainly a matter of how easy it is to get nitrogen.

There's an interesting conflict in this game. Plants have many clever ways of trapping insects and forcing the insects to pollinate them. This is a perfect first step in becoming protocarnivorous. But this sets up a battle of competing forces: will evolution optimize the trap for pollination or for killing insects and feeding the plant?

This is called the 'pollinator-prey conflict'. Many carnivorous plants try to have it both ways! And some are good at releasing insects that pollinate the plant, while killing others.

The game gets more complicated, too. One species of tree frog in Borneo specializes in eating insects caught in pitcher plants. And there's an ant that dives into pitcher plants to eat insects caught there - and also lubricates the top of the plant to make it easier for insects to fall in!

The CEO of Exxon just said "we've waited too long" to tackle climate change. He blames "society" and "activists".

Yes, Exxon. Yes, Exxon. Yes, Exxon: the corporation that for decades has been spending millions to slow progress on climate change, despite its own research showing the problem was urgent.

He said:

"We've waited too long to open the aperture on the solution sets in terms of what we need, as a society, to start reducing emissions.... Frankly, society, and the activist — the dominant voice in this discussion — has tried to exclude the industry that has the most capacity and the highest potential for helping with some of the technologies."

What is this — some sort of dark, twisted joke?

The interview is here:

https://web.archive.org/web/20240228015207/https://fortune.com/2024/02/27/exxon-ceo-darren-woods-interview-pay-the-price-for-net-zero/

https://www.salon.com/2024/02/29/exxon-ceo-to-world-climate-isnt-our-fault-now-pay-the-price/

@Gargron @guysoft - and while you're at it, can you fix it to 22,000,003 BC?

I decided to see how my research has changed over time by making word clouds based on Google Scholar.

Here's what I wrote about from 1985 to 1995. At this early stage of my career I was doing traditional mathematical physics. I got tenure based on my work on quantum and classical field theory - for example, scattering and complete integrability for classical field theories like Yang-Mills theory.

After I got tenure, I got excited about how knot theory is connected to quantum gravity.

By 1995 things had changed....

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@roger - or maybe they'll have survived an era where they, too, needed to change to take responsibility for their ecosystem.

@pjakobs - I agree: humanity has enormous potential, we're our own worst enemy but we need to keep in mind how great we can be.

@phiofx - LIGO changed our views on astrophysics: it turns out there are lots of black holes 30-60 times the mass of our Sun, and we still don't where they came from. But not our views on gravity or cosmology. LISA, on the other hand, may measure the gravitational background radiation, which was produced about 10⁻³⁸ seconds after the Big Bang! This probes a realm we've never accessed before, and it could shed new light on quantum gravity.

@dbattistella - we're similarly impressed by the stone heads of Easter Island: they died out after they cut down all the trees on the island.

@dom - we don't laugh at Neolithic stone circles like Stonehenge. At least I don't. I find them impressive.

@jsdodge - in aome ways it's less absurd to do this in space. LIGO is so sensitive it detects logging in nearby forests.

@jw4ke - I used to work on quantum gravity and particle physics. Now I'm trying to work on epidemiology and carbon emissions, since I don't feel I have luxury to whatever I enjoy.

@CascadeTommy - I've been amazed by this experiment ever since I heard about it. Back in 1999 I wrote this:

"The idea is to orbit 3 satellites in an equilateral triangle with sides 5 million kilometers long, and constantly measure the distance between them to an accuracy of a tenth of an angstrom - 10⁻¹¹ meters - using laser interferometry. The big distances would make it possible to detect gravitational waves with frequencies of .0001 to .1 hertz, much lower than the frequencies for which the ground-based detectors are optimized. The plan involves a really cool technical trick to keep the satellites from being pushed around by solar wind and the like: each satellite will have a free-falling metal cube floating inside it, and if the satellite gets pushed to one side relative to this mass, sensors will detect this and thrusters will push the satellite back on course.

I don't think LISA has been funded yet, but if all goes well, it may fly within 10 years or so. Eventually, a project called LISA 2 might be sensitive enough to detect gravitational waves left over from the early universe - the gravitational analogue of the cosmic microwave background radiation!

The microwave background radiation tells us about the universe when it was roughly 10⁵ years old, since that's when things cooled down enough for most of the hydrogen to stop being ionized, making it transparent to electromagnetic radiation. In physics jargon, that's when electromagnetic radiation "decoupled". But the gravitational background radiation would tell us about the universe when it was roughly 10⁻³⁸ seconds old, since that's when gravitational radiation decoupled."

https://math.ucr.edu/home/baez/week143.html

@Archnemysis - Not 38 seconds, 10⁻³⁸ seconds. As others have said in different ways, that means the gravitational background radiation was released about 0.0000000000000000000000000000000000000001 seconds after the Big Bang! Check out my timeline of the very early history of the Universe to put this in context:

https://math.ucr.edu/home/baez/timeline.html#bang