Just A Different Way Of Thinking About The Fabric Of Reality

phys.org  |  By shining a laser pulse sequence inspired by the Fibonacci numbers at atoms inside a quantum computer, physicists have created a remarkable, never-before-seen phase of matter. The phase has the benefits of two time dimensions despite there still being only one singular flow of time, the physicists report July 20 in Nature.

This mind-bending property offers a sought-after benefit: Information stored in the phase is far more protected against errors than with alternative setups currently used in quantum computers. As a result, the information can exist without getting garbled for much longer, an important milestone for making quantum computing viable, says study lead author Philipp Dumitrescu.

The approach's use of an "extra" time dimension "is a completely different way of thinking about phases of matter," says Dumitrescu, who worked on the project as a research fellow at the Flatiron Institute's Center for Computational Quantum Physics in New York City. "I've been working on these theory ideas for over five years, and seeing them come actually to be realized in experiments is exciting."

The best way to understand their approach is by considering something else ordered yet non-repeating: "quasicrystals." A typical crystal has a regular, repeating structure, like the hexagons in a honeycomb. A quasicrystal still has order, but its patterns never repeat. (Penrose tiling is one example of this.) Even more mind-boggling is that quasicrystals are crystals from higher dimensions projected, or squished down, into lower dimensions. Those higher dimensions can even be beyond physical space's three dimensions: A 2D Penrose tiling, for instance, is a projected slice of a 5-D lattice.

Dumitrescu spearheaded the study's theoretical component with Andrew Potter of the University of British Columbia in Vancouver, Romain Vasseur of the University of Massachusetts, Amherst, and Ajesh Kumar of the University of Texas at Austin. The experiments were carried out on a quantum computer at Quantinuum in Broomfield, Colorado, by a team led by Brian Neyenhuis.

The workhorses of the team's quantum computer are 10 atomic ions of an element called ytterbium. Each ion is individually held and controlled by electric fields produced by an ion trap, and can be manipulated or measured using laser pulses.

Each of those atomic ions serves as what scientists dub a quantum bit, or "qubit." Whereas traditional computers quantify information in bits (each representing a 0 or a 1), the qubits used by quantum computers leverage the strangeness of quantum mechanics to store even more information. Just as Schrödinger's cat is both dead and alive in its box, a qubit can be a 0, a 1 or a mashup—or "superposition"—of both. That extra information density and the way qubits interact with one another promise to allow quantum computers to tackle computational problems far beyond the reach of conventional computers.

There's a big problem, though: Just as peeking in Schrödinger's box seals the cat's fate, so does interacting with a qubit. And that interaction doesn't even have to be deliberate. "Even if you keep all the atoms under tight control, they can lose their quantumness by talking to their environment, heating up or interacting with things in ways you didn't plan," Dumitrescu says. "In practice, experimental devices have many sources of error that can degrade coherence after just a few laser pulses."

The challenge, therefore, is to make qubits more robust. To do that, physicists can use "symmetries," essentially properties that hold up to change. (A snowflake, for instance, has rotational symmetry because it looks the same when rotated by 60 degrees.) One method is adding time symmetry by blasting the atoms with rhythmic laser pulses. This approach helps, but Dumitrescu and his collaborators wondered if they could go further. So instead of just one time symmetry, they aimed to add two by using ordered but non-repeating laser pulses.

More Of The Less...,

aeon  |  Calling these numbers imaginary came later, in the 1600s, when the philosopher René Descartes argued that, in geometry, any structure corresponding to imaginary numbers must be impossible to visualise or draw. By the 1800s, thinkers such as Carl Friedrich Gauss and Leonhard Euler included imaginary numbers in their studies. They discussed complex numbers made up of a real number added to an imaginary number, such as 3+4i, and found that complex-valued mathematical functions have different properties than those that only produce real numbers.

Yet, they still had misgivings about the philosophical implications of such functions existing at all. The French mathematician Augustin-Louis Cauchy wrote that he was ‘abandoning’ the imaginary unit ‘without regret because we do not know what this alleged symbolism signifies nor what meaning to give to it.’

In physics, however, the oddness of imaginary numbers was disregarded in favour of their usefulness. For instance, imaginary numbers can be used to describe opposition to changes in current within an electrical circuit. They are also used to model some oscillations, such as those found in grandfather clocks, where pendulums swing back and forth despite friction. Imaginary numbers are necessary in many equations pertaining to waves, be they vibrations of a plucked guitar string or undulations of water along a coast. And these numbers hide within mathematical functions of sine and cosine, familiar to many high-school trigonometry students.

At the same time, in all these cases imaginary numbers are used as more of a bookkeeping device than a stand-in for some fundamental part of physical reality. Measurement devices such as clocks or scales have never been known to display imaginary values. Physicists typically separate equations that contain imaginary numbers from those that do not. Then, they draw some set of conclusions from each, treating the infamous i as no more than an index or an extra label that helps organise this deductive process. Unless the physicist in question is confronted with the tiny and cold world of quantum mechanics.

Quantum theory predicts the physical behaviour of objects that are either very small, such as electrons that make up electric currents in every wire in your home, or millions of times colder than the insides of your fridge. And it is chock-full of complex and imaginary numbers.

Imaginary numbers went from a problem seeking a solution to a solution that had just been matched with its problem

Emerging in the 1920s, only about a decade after Albert Einstein’s paradigm-shifting work on general relativity and the nature of spacetime, quantum mechanics complicated almost everything that physicists thought they knew about using mathematics to describe physical reality. One big upset was the proposition that quantum states, the fundamental way in which objects that behave according to the laws of quantum mechanics are described, are by default complex. In other words, the most generic, most basic description of anything quantum includes imaginary numbers.

In stark contrast to theories concerning electricity and oscillations, in quantum mechanics a physicist cannot look at an equation that involves imaginary numbers, extract a useful punchline, then forget all about them. When you set out to try and capture a quantum state in the language of mathematics, these seemingly impossible square roots of negative numbers are an integral part of your vocabulary. Eliminating imaginary numbers would highly limit how accurate of a statement you could make.

The discovery and development of quantum mechanics upgraded imaginary numbers from a problem seeking a solution to a solution that had just been matched with its problem. As the physicist and Nobel laureate Roger Penrose noted in the documentary series Why Are We Here? (2017): ‘[Imaginary numbers] were there all the time. They’ve been there since the beginning of time. These numbers are embedded in the way the world works at the smallest and, if you like, most basic level.’

Penrose Iconoclasm: The General Idea Is In The Geometrical Pictures

YOU - don't have to follow the detailed equations and everything. The celestial sphere is a Riemann sphere and the collection of light rays/points/events comprise a lorentz group in minkowski space. 

Points on the Riemann sphere are conformal comprising a circle on the sphere. A Riemann sphere is a complex curve or manifold, twistor space is therefore a complex space.

Quantum mechanics is based on complex numbers and amplitudes and thus we need a complex and non-local space-time manifold.

An entire light ray is a point, a simultaneous entity. A photon doesn't experience "time" and thus its mechanics are and can be non-local.

9:45 Stereographic projection on a Riemann spere with discussion of particle spin.

At 11:40 he shows the spinor calculus and breaks down the components of the spinor. Showing the light cone as a spinor in Riemann space, making things real and plain.

 At 12:40 he explains the twistor and its associated algebra.

At 40:00 he explains cohomology utilizing an impossible triangular solid - can the geometric structure be realized in 3-space

weylmann |  The notion of electron spin was first surmised in 1922 when the German physicists Otto Stern and WaltherGerlach noticed that a stream of silver atoms (each having a single electron in the outer 5s orbital) could be separated by a non-uniform magnetic field into two streams, ‘‘up’’ and ‘‘down.’’ At the time, no one knew what to make of this odd behavior, but three years later George Uhlenbeck and Samuel Goudsmit proposed that electrons could exist in two spin states,±1/2, each with the units of angular momentumħh=h/2π. 

Still, the concept of electron spin varied between a simplistic physical ‘‘spinning’’ of the electron about some axis (like a child’s top) and a kind of internal angular momentum. 

Today we look upon electrons and other spin-1/2 particles as havingan intrinsic angular momentum with no classical counterpart. This gave rise to the notion of such particles living in ‘‘spin space,’’ an abstract two-dimensional internal space that requires a description beyond that of scalars,vectors and tensors. Cartan’s spinor formalism was found to be appropriate for this description. 

Many attempts have been made over the years to explain spinors at an intuitive, elementary level, but the simplest approach remains an appeal to basic Lorentz group theory. This is rather a pity, because undergraduate students often express an aversion to group theory because of its mathematical nature. But the theory of Lorentz rotations and boosts is relatively simple, and it has the nice property of being relativistic from the start. 

In addition, it neatly admits a formalism that underlies that of the 4-vectors it ordinarily applies to, which is where spinors come in.

 Even better, Lorentz theory confirms the intuitive notion that if a spinor represents half of a 4-vector (rather than the square root), then there should be two kinds of spinor: one comprising the upper half and another representing the lower half. This observation is critical, since a single two-component spinor can be shown to violate odd-even parity in quantum physics, and it takes two spinors acting together to preserve it. Thus, the Dirac bispinor—a four-component object consisting of two stacked spinors—fully preserves parity. 

Our approach will therefore be based on the Lorentz transformations of rotations and boosts. There are, however,a few subtleties that other, more advanced treatments either gloss over or assert by inference, and I will try toexplain these in a more straightforward and understandable manner as we go along. 

The Dirac Equation 

When it was discovered that electrons can exist in both+12and12spin states, the Austrian physicist Wolfgang Pauli suggested in 1927 that the scalar wave functionΨin Schrödinger’s equation be replaced by a two-component spinor, each component representing one of the electron’s allowed spin states. 

But while Pauli’s approach worked,it presented a problem having to do withparity; that is, the sign reversal operationΨ(x,t)→Ψ(−x,t)gaveinconsistent results in Pauli’s approach, violating the notion that Nature should be mirror-image invariant. 

However, in 1928, at the age of just 25, the great British mathematical physicist Paul Adrienne Maurice Diracmade a monumental discovery, perhaps the greatest discovery in all of modern physics. The student can look upthe details, but what Dirac did was essentially take the square root of the relativistic energy-mass equation

 E2=m2c4+c2p

He ended up with the set of four partial differential equations in (1) involving four4×4 matricesγμ(now calledthe Dirac gamma matrices), along with a new four-component wave functionΨa(x,t), where 

a=0,1,2,3. 

It was soon realized that Dirac’sΨwas abispinor, a four-component mathematical object consisting of theφRandφLWeyl spinors we identified earlier: 

Ψ=Ψ0Ψ1Ψ2Ψ3=ïφRφLò 

Dirac’s bispinor was found to preserve parity under the sign reversal operation 

Ψ(x,t)→Ψ(−x,t). 

Far more importantly, the spinorφReffectively represents the spin-up and spin-down components of an ordinary electron,while φL represents the spin-up and spin-down components of an anti-electron (known as a positron)—Dirac’swork thus predicted the existence of antimatter (the positron was subsequently discovered in 1932, for which the Caltech experimental physicist Carl Anderson won the Nobel Prize). 

Dirac’s relativistic electron equation alsoexplained electron spin as a form of intrinsic angular momentum called S. Thus, the angular momentum L of an electron alone is not conserved—instead, it is L+S that is conserved.Dirac’s equation and its underlying mathematics today represent the foundation of much of modern quantum field theory.

If Penrose Talm'bout Gravitational Waves, Then Get Your Surfboard Ready!!!

space  |  Penrose admits it's a wild suggestion, but believes that like all good scientific theories, it might be tested through experiment and observation. These tests stem from the idea that our aeon and the one preceding it were not completely isolated from one another. "Information does get through," he said. "It gets through in the form of a shock wave in our universe's initial dark matter." 

Dark matter, like dark energy, is a shadowy substance, this time needed to account for the way structures such as galaxies and clusters of galaxies formed in the early universe. According to Penrose's calculations, that shock wave would have had an effect on the cosmic microwave background (CMB), which is the leftover radiation from the Big Bang, released when the universe was under 400,000 years old. "You'd see rings in the CMB that are slightly warmer or cooler than the average temperature," he said.

The equations of CCC predict that a shock wave arriving from a previous aeon would have dragged matter into our universe. If that caused material to head toward us, we would see light from that region shunted to shorter wavelengths — an effect astronomers call blueshift. Equally, a region carried away from us by a CCC shock wave would be redshifted, meaning its wavelength would be stretched out. 

Blueshifted regions would appear hotter and redshifted areas cooler. It's these changes Penrose believes we'd see as rings in the cosmic microwave background. Multiple shockwaves might even have produced a series of concentric rings. "I asked whether anyone had looked for these rings in the sky," Penrose said.

Several years ago, it did seem as if those rings had been found, a veritable smoking gun for CCC. "Except nobody believed us. They said it must have been a fluke or something," Penrose said. 

"But those signatures have been confirmed by alternative groups," said Vahe Gurzadyan a physicist at the Yerevan Physics Institute in Armenia and Penrose's long time collaborator on CCC.

The scientists point to the fact that a team of Polish and Canadian researchers confirmed the presence of the rings to a confidence level of 99.7%. However, there are still many doubters. Gurzadyan remains steadfast. "These structures are real – there is no doubt that our calculations are reliable and correct," he said. Still, Penrose has been exploring other approaches that might further support the pair's claims about CCC and a time before the Big Bang.

The transition between aeons would do something more fundamental that just create a shock wave in our dark matter and rings in the cosmic microwave background. "A new material, the dominant material in the universe, is created at the crossover," Penrose said. He regards that new material as the initial form of dark matter itself. 

"But in order that it doesn't build up from aeon to aeon, it has to decay," he said. He calls these initial dark matter particles erebons after Erebos, the Greek god of darkness.

On average it would take 100 billion years for an erebon to decay, but there are some that will have decayed in the 14-billion-year history of our universe. Crucially, as they decay, Penrose says erebons dump all their energy into gravitational waves.

Probative Cognitions: You Betting Against Penrose's Pluripotent Theories?!?!?!?!

nautil.us  |  Once you start poking around in the muck of consciousness studies, you will soon encounter the specter of Sir Roger Penrose, the renowned Oxford physicist with an audacious—and quite possibly crackpot—theory about the quantum origins of consciousness. He believes we must go beyond neuroscience and into the mysterious world of quantum mechanics to explain our rich mental life. No one quite knows what to make of this theory, developed with the American anesthesiologist Stuart Hameroff, but conventional wisdom goes something like this: Their theory is almost certainly wrong, but since Penrose is so brilliant (“One of the very few people I’ve met in my life who, without reservation, I call a genius,” physicist Lee Smolin has said), we’d be foolish to dismiss their theory out of hand.

Penrose, 85, is a mathematical physicist who made his name decades ago with groundbreaking work in general relativity and then, working with Stephen Hawking, helped conceptualize black holes and gravitational singularities, a point of infinite density out of which the universe may have formed. He also invented “twistor theory,” a new way to connect quantum mechanics with the structure of spacetime. His discovery of certain geometric forms known as “Penrose tiles”—an ingenious design of non-repeating patterns—led to new directions of study in mathematics and crystallography.

The breadth of Penrose’s interests is extraordinary, which is evident in his recent book Fashion, Faith and Fantasy in the New Physics of the Universe—a dense 500-page tome that challenges some of the trendiest but still unproven theories in physics, from the multiple dimensions of string theory to cosmic inflation in the first moment of the Big Bang. He considers these theories to be fanciful and implausible.

Penrose doesn’t seem to mind being branded a maverick, though he disputes the label in regard to his work in physics. But his theory of consciousness pushes the edges of what’s considered plausible science and has left critics wondering why he embraces a theory based on so little evidence.

Most scientists regard quantum mechanics as irrelevant to our understanding of how the brain works. Still, it’s not hard to see why Penrose’s theory has gained attention. Artificial intelligence experts have been predicting some sort of computer brain for decades, with little to show so far. And for all the recent advances in neurobiology, we seem no closer to solving the mind-brain problem than we were a century ago. Even if the human brain’s neurons, synapses and neurotransmitters could be completely mapped—which would be one of the great triumphs in the history of science—it’s not clear that we’d be any closer to explaining how this 3-pound mass of wet tissue generates the immaterial world of our thoughts and feelings. Something seems to be missing in current theories of consciousness. The philosopher David Chalmers has speculated that consciousness may be a fundamental property of nature existing outside the known laws of physics. Others—often branded “mysterians”—claim that subjective experience is simply beyond the capacity of science to explain.

 

Singularities and Spinors Were Just Another Day At The Office - Twistors Are Penrose's Big Bet

phys.org  |  In the 1960s, in an attempt to understand quantum gravity, physicist Roger Penrose proposed such a radical alternative. In Penrose's twistor theory, geometric points are replaced by twistors—entities that most closely resemble stretched, light ray-like shapes. Within this twistor space, Penrose discovered a highly efficient way to represent fields that travel at the speed of light, such as electromagnetic and gravitational fields. Reality, however, is composed of more than fields—any theory needs also to account for the interactions between fields, such as the electric force between charges, or, in the more complicated case of General Relativity, gravitational attraction resulting from the energy of the field itself. However, including the interactions of General Relativity into this picture has proven a formidable task.

So can we express in twistor language a full-fledged quantum gravitational theory, perhaps simpler than General Relativity, but with both fields and interactions fully taken into account? Yes, according to Neiman.

Neiman's model builds on higher spin gravity, a model developed by Mikhail Vasiliev in the 1980s and '90s. Higher spin gravity can be thought of as the "smaller cousin" of String Theory, "too simple to reproduce General Relativity, but very instructive as a playground for ideas," as Neiman puts it. In particular, it is perfectly suited for exploring possible bridges between holography and twistor theory.

On one hand, as discovered by Igor Klebanov and Alexander Polyakov in 2001, higher spin gravity, just like string theory, can be described holographically. Its behavior within space can be captured completely in terms of a boundary at infinity. On the other hand, its equations contain twistor-like variables, even if these are still tied to particular points in ordinary space.

From these starting points, Neiman's paper takes an additional step, constructing a mathematical dictionary that ties together the languages of holography and twistor theory.

"The underlying math that makes this story tick is all about square roots," writes Neiman. "It's about identifying subtle ways in which a geometric operation, such as a rotation or reflection, can be done 'halfway.' A clever square root is like finding a crack in a solid wall, opening it in two, and revealing a new world."

Using square roots in this way has a longstanding history in math and physics. In fact, the intrinsic shape of all matter particles—such as electrons and quarks—as well as twistors, is described by a square root of ordinary directions in space. In a subtle technical sense, Neiman's method for connecting space, its boundary at infinity, and twistor space, boils down to taking such a square root again.

Neiman hopes that his proof of concept can pave the way toward a quantum theory of gravity that does not rely on a boundary at infinity.

"It will take a lot of creativity to uncover the code of the world," says Neiman. "And there's joy in fumbling around for it."

Probative Cognition: Impossible Patterns And Materials That Aren't Supposed To Exist

oakridger  |  “If nuclear fission can occur naturally near the surface, why wouldn’t it occur deep in the earth?” Hollenbach asked.

Using SCALE nuclear safety analysis codes, Hollenbach simulated a georeactor that would function as a fast neutron breeder reactor and have an energy output (under 3 terawatts) that would enable it to heat the planet’s core and power its magnetic field for 4.5 billion years, the widely accepted age of Earth. The initial density and relative abundance of uranium isotopes that he assumed for his simulation were based on what is determined to be present in a certain kind of meteor that is almost oxygen-free (as Earth was during its formation).

By absorbing neutrons, the uranium isotope U-235 would break into lighter elements more readily than the much more abundant uranium isotope U-238, releasing considerable heat energy and neutrons that will trigger more fission, or self-sustaining chain reactions. Free neutrons absorbed by U-238 nuclei can cause the formation of plutonium-239, another nuclear fuel. This process, known as breeding, can significantly extend the lifetime of a nuclear reactor. 

Hollenbach’s calculations also generated data on the fission products that would result from uranium fission deep within Earth, as well as from radioactive decay. He showed that two helium isotopes, He-3 and He-4, would be produced in the same relative proportion by georeactors as helium isotopes found in basalt extruded from volcanic lavas in Hawaii and Iceland. Because helium is a light noble gas that does not react with other materials, it could migrate from a georeactor to hot spots on Earth’s surface. “The only way helium is produced on the Earth is through fission or decay of heavy elements,” he said.

When helium was first discovered in the 1960s on Earth’s surface, it was assumed that helium gas in space was trapped in the surface during Earth’s formation. Hollenbach said trapped helium would have outgassed during Earth’s molten stage. He found that the ratio of He-3 to He-4 at the surface corresponds to the ratio calculated to be produced by deep-Earth fission. It’s not the same as the ratio of helium isotopes formed in the air by cosmic rays (which is up to 34 times lower). 

Because most fission products are lighter and less dense than nuclear fuel in a georeactor, Hollenbach said, they most likely migrate away from a georeactor after accumulating there. As a result, the georeactor’s energy output will stop decreasing and start to rise again.

The Earth’s magnetic field varies in strength and has flipped its polarity over millions of years. These changes, he suggested, could be explained by georeactors that turn on and off. 

“The cyclic nature of geomagnetic field reversals and periodic high volcanic and plate tectonic activity indicate a varied power source,” he said.

If beryllium-10 and certain noble gases were discovered in deep mantle magmas and volcanic lavas and if anti-neutrinos could be detected, such evidence would help validate the georeactor model, he added. A FORNL participant suggested that Hollenbach confer with researchers at the IceCube Neutrino Observatory in Antarctica. 

Hollenback asserted that an even better understanding of georeactors could be attained through simulations using advanced software on today’s supercomputers — if funding for such a study is available.

 

The World Finally Caught Up With And Properly Acknowledged My Manz Sir Roger Penrose

bbc  |  UK-born mathematical physicist Sir Roger, from the University of Oxford, demonstrated that black holes were an inevitable consequence of Albert's Einstein's general theory of relativity.

Reacting to the win, he told the BBC: "It was an extreme honour and great pleasure to hear the news this morning, in a slightly unusual way - I had to get out of my shower to hear it."

Among scientific awards, he said, this is "the prime one".

Penrose receives half of this year's prize, with the other half being shared by Genzel and Ghez. Prof Ghez is only the fourth woman to win the physics prize, out of more than 200 laureates since 1901.

The other female recipients are Marie Curie (1903), Maria Goeppert-Mayer (1963) and Donna Strickland (2018).

"The history of black holes goes way back in time to the end of the 18th Century. Then, through Einstein's general relativity, we had the tools to describe these objects for real," said Ulf Danielsson, a member of the Nobel Committee.

But the mathematics of black holes was incredibly complex. Many researchers believed they were nothing more than mathematical artefacts, existing only on paper. It took researchers decades to realise they could persist in the real world.

"That's what Roger Penrose did," said Danielsson. "He understood the mathematics, he introduced new tools and then could actually prove this is a process you can naturally expect to happen - that a star collapses and turns into a black hole."

Sir Roger explained: "People were very sceptical at the time, it took a long time before black holes were accepted... their importance is, I think, only partially appreciated." 

Penrose was born in 1931 in Colchester and comes from a distinguished scientific family. He is the son of the psychiatrist and geneticist Lionel Penrose and Margaret Leathes, who was the daughter of a well-known English physiologist. His brother Jonathan is a chess grandmaster.

Sir Roger shied away from competition as a child and struggled in exams. He told BBC Radio 4's The Life Scientific programme in 2016: "I was good at maths, but I didn't necessarily do very well in my tests." However, he added: "The teacher realised if he gave me enough time, I would do well.

"I think I had to do all my arithmetic working it out from first principles," he chuckled, adding: "I just was slow, and I'm slow at writing."

In the 1950s, he came up with the Penrose triangle, an impossible object which could be depicted in a perspective drawing but could not exist in reality. The triangle, along with other observations by Sir Roger and his father Lionel, influenced the Dutch artist MC Escher, who incorporated them into his artworks Waterfall, and Ascending and Descending.

Inspired by the British scientist Dennis Sciama, Penrose next applied his mathematical ability to physics. In 1965, he published a landmark paper in which he was able to show that a black hole always hides a singularity, a boundary where space and time ends.

 

 

When Our Most Powerful General Purpose Technology Inventories Our Most Probing Cognitions?

quantummagazine  |  Every day, dozens of like-minded mathematicians gather on an online forum called Zulip to build what they believe is the future of their field.

They’re all devotees of a software program called Lean. It’s a “proof assistant” that, in principle, can help mathematicians write proofs. But before Lean can do that, mathematicians themselves have to manually input mathematics into the program, translating thousands of years of accumulated knowledge into a form Lean can understand.

To many of the people involved, the virtues of the effort are nearly self-evident.

“It’s just fundamentally obvious that when you digitize something you can use it in new ways,” said Kevin Buzzard of Imperial College London. “We’re going to digitize mathematics and it’s going to make it better.”

Digitizing mathematics is a longtime dream. The expected benefits range from the mundane — computers grading students’ homework — to the transcendent: using artificial intelligence to discover new mathematics and find new solutions to old problems. Mathematicians expect that proof assistants could also review journal submissions, finding errors that human reviewers occasionally miss, and handle the tedious technical work that goes into filling in all the details of a proof.

But first, the mathematicians who gather on Zulip must furnish Lean with what amounts to a library of undergraduate math knowledge, and they’re only about halfway there. Lean won’t be solving open problems anytime soon, but the people working on it are almost certain that in a few years the program will at least be able to understand the questions on a senior-year final exam.

And after that, who knows? The mathematicians participating in these efforts don’t fully anticipate what digital mathematics will be good for.

“We don’t really know where we’re headed,” said Sébastien Gouëzel of the University of Rennes.