When researchers at Microsoft released a list of the 40 jobs most likely to be affected by generative artificial intelligence (gen AI), few outsiders would have expected to see “mathematician” among them. Yet according to speakers at this year’s Heidelberg Laureate Forum (HLF), which connects early-career researchers with distinguished figures in mathematics and computer science, computers are already taking over many tasks formerly performed by human mathematicians – and the humans have mixed feelings about it.

One of those expressing disquiet is Yang-Hui He, a mathematical physicist at the London Institute for Mathematical Sciences. In general, He is extremely keen on AI. He’s written a textbook about the use of AI in mathematics, and he told the audience at an HLF panel discussion that he’s been peddling machine-learning techniques to his mathematical physics colleagues since 2017.

More recently, though, He has developed concerns about gen AI specifically. “It is doing mathematics so well without any understanding of mathematics,” he said, a note of wonder creeping into his voice. Then, more plaintively, he added, “Where is our place?”

AI advantages

Some of the things that make today’s gen AI so good at mathematics are the same as the ones that made Google’s DeepMind so good at the game of Go. As the theoretical computer scientist Sanjeev Arora pointed out in his HLF talk, “The reason it’s better than humans is that it’s basically tireless.” Put another way, if the 20^th-century mathematician Alfréd Rényi once described his colleagues as “machines for turning coffee into theorems”, one advantage of 21^st-century AI is that it does away with the coffee.

Arora, however, sees even greater benefits. In his view, AI’s ability to use feedback to improve its own performance – a technique known as reinforcement learning – is particularly well-suited to mathematics.

In the standard version of reinforcement learning, Arora explains, the AI model is given a large bank of questions, asked to generate many solutions and told to use the most correct ones (as labelled by humans) to refine its model. But because mathematics is so formalized, with answers that are so verifiably true or false, Arora thinks it will soon be possible to replace human correctness checkers with AI “proof assistants”. Indeed, he’s developing one such assistant himself, called Lean, with his colleagues at Princeton University in the US.

Humans in the loop?

But why stop there? Why not use AI to generate mathematical questions as well as producing and checking their solutions? Indeed, why not get it to write a paper, peer review it and publish it for its fellow AI mathematicians – which are, presumably, busy combing the literature for information to help them define new questions?

Arora clearly thinks that’s where things are heading, and many of his colleagues seem to agree, at least in part. His fellow HLF panellist Javier Gómez-Serrano, a mathematician at Brown University in the US, noted that AI is already generating results in a day or two that would previously have taken a human mathematician months. “Progress has been quite quick,” he said.

The panel’s final member, Maia Fraser of the University of Ottawa, Canada, likewise paid tribute to the “incredible things that are possible with AI now”.  But Fraser, who works on mathematical problems related to neuroscience, also sounded a note of caution. “My concern is the speed of the changes,” she told the HLF audience.

The risk, Fraser continued, is that some of these changes may end up happening by default, without first considering whether humans want or need them. While we can’t un-invent AI, “we do have agency” over what we want, she said.

So, do we want a world in which AI mathematicians take humans “out of the loop” entirely? For He, the benefits may outweigh the disadvantages. “I really want to see a proof of the Riemann hypothesis,” he said,  to ripples of laughter. If that means that human mathematicians “become priests to oracles”, He added, so be it.

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In this week’s special CEO Series edition of Space Minds, we're at the World Space Business Week in Paris. In today's episode, SpaceNews editor Mike Gruss talks with Mina Mitry, CEO of Kepler Communications.

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The first-ever “space–time crystal” has been created in the US by Hanqing Zhao and Ivan Smalyukh at the University of Colorado Boulder. The system is patterned in both space and time and comprises a rigid lattice of topological solitons that are sustained by steady oscillations in the orientations of liquid crystal molecules.

In an ordinary crystal atomic or molecular structures repeat at periodic intervals in space. In 2012, however, Frank Wilczek suggested that systems might also exist with quantum states that repeat at perfectly periodic intervals in time – even as they remain in their lowest-energy state.

First observed experimentally in 2017, these time crystals are puzzling to physicists because they spontaneously break time–translation symmetry, which states that the laws of physics are the same no matter when you observe them. In contrast, a time crystal continuously oscillates over time, without consuming energy.

A space–time crystal is even more bizarre. In addition to breaking time–translation symmetry, such a system would also break spatial symmetry, just like the repeating molecular patterns of an ordinary crystal. Until now, however, a space–time crystal had not been observed directly.

Rod-like molecules

In their study, Zhao and Smalyukh created a space–time crystal in the nematic phase of a liquid crystal. In this phase the crystal’s rod-like molecules align parallel to each other and also flow like a liquid. Building on computer simulations, they confined the liquid crystal between two glass plates coated with a light-sensitive dye.

“We exploited strong light–matter interactions between dye-coated, light-reconfigurable surfaces, and the optical properties of the liquid crystal,” Smalyukh explains.

When the researchers illuminated the top plate with linearly polarized light at constant intensity, the dye molecules rotate to align perpendicular to the direction of polarization. This reorients nearby liquid crystal molecules, and the effect propagates deeper into the bulk. However, the influence weakens with depth, so that molecules farther from the top plate are progressively less aligned.

As light travels through this gradually twisting structure, its linear polarization is transformed, becoming elliptically polarized by the time it reaches the bottom plate. The dye molecules there become aligned with this new polarization, altering the liquid crystal alignment near the bottom plate. These changes propagate back upward, influencing molecules near the top plate again.

Feedback loop

This is a feedback loop, with the top and bottom plates continuously influencing each other via the polarized light passing through the liquid crystal.

“These light-powered dynamics in confined liquid crystals leads to the emergence of particle-like topological solitons and the space–time crystallinity,” Smalyukh says.

In this environment, particle-like topological solitons emerge as stable, localized twists in the liquid crystal’s orientation that do not decay over time. Like particles, the solitons move and interact with each other while remaining intact.

Once the feedback loop is established, these solitons emerge in a repeating lattice-like pattern. This arrangement not only persisted as the feedback loop continued, but is sustained by it. This is a clear sign that the system exhibits crystalline order in time and space simultaneously.

Accessible system

Having confirmed their conclusions with simulations, Zhao and Smalyukh are confident this is the first experimental demonstration of a space–time crystal. The discovery that such an exotic state can exist in a classical, room-temperature system may have important implications.

“This is the first time that such a phenomenon is observed emerging in a liquid crystalline soft matter system,” says Smalyukh. “Our study calls for a re-examining of various time-periodic phenomena to check if they meet the criteria of time-crystalline behaviour.”

Building on these results, the duo hope to broaden the scope of time crystal research beyond a purely theoretical and experimental curiosity. “This may help expand technological utility of liquid crystals, as well as expand the currently mostly fundamental focus of studies of time crystals to more applied aspects,” Smalyukh adds.

The research is described in Nature Materials.

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Physicists at Osaka Metropolitan University in Japan and the Korea Advanced Institute of Science and Technology (KAIST) claim to have observed the quantum counterpart of the classic Kelvin-Helmholtz instability (KHI), which is the most basic instability in fluids. The effect, seen in a quantum gas of ⁷Li atoms, produces a new type of exotic vortex pattern called an eccentric fractional skyrmion. The finding not only advances our understanding of complex topological quantum systems, it could also help in the development of next-generation memory and storage devices.

Topological defects occur when a system rapidly transitions from a disordered to an ordered phase. These defects, which can occur in a wide range of condensed matter systems, from liquid crystals and atomic gases to the rapidly cooling early universe, can produce excitations such as solitons, vortices and skyrmions.

Skyrmions, first discovered in magnetic materials, are swirling vortex-like spin structures that extend across a few nanometres in a material. They can be likened to 2D knots in which the magnetic moments rotate about 360° within a plane.

Eccentric fractional skyrmions contain singularities

Skyrmions are topologically stable, which makes them robust to external perturbations, and are much smaller than the magnetic domains used to encode data in today’s disk drives. That makes them ideal building blocks for future data storage technologies such as “racetrack” memories. Eccentric fractional skyrmions (EFSs), which had only been predicted in theory until now, have a crescent-like shape and contain singularities – points in which the usual spin structure breaks down, creating sharp distortions as it becomes unsymmetrical.

“To me, the large crescent moon in the upper right corner of Van Gogh’s ‘The Starry Night’ also looks exactly like an EFS,” says Hiromitsu Takeuchi at Osaka, who co-led this new study with Jae-Yoon Choi of KAIST. “EFSs carry half the elementary charge, which means they do not fit into traditional classifications of topological defects.”

The KHI is a classic phenomenon in fluids in which waves and vortices form at the interface between two fluids moving at different speeds. “To observe the KHI in quantum systems, we need a structure containing a thin superfluid interface (a magnetic domain wall), such as in a quantum gas of ⁷Li atoms,” says Takeuchi. “We also need experimental techniques that can skilfully control the behaviour of this interface. Both of these criteria have recently been met by Choi’s group.”

The researchers began by cooling a gas of ⁷Li atoms to near absolute zero temperatures to create a multi-component Bose-Einstein condensate – a quantum superfluid containing two streams flowing at different speeds. At the interface of these streams, they observed vortices, which corresponded to the predicted EFSs.

The behaviour of the KHI is universal

“We have shown that the behaviour of the KHI is universal and exists in both the classical and quantum regimes,” says Takeuchi. This finding could not only lead to a better understanding of quantum turbulence and the unification of quantum and classic hydrodynamics, it could also help in the development of technologies such as next-generation storage and memory devices and spintronics, an emerging technology in which magnetic spin is used to store and transfer information using much less energy than existing electronic devices.

“By further refining the experiment, we might be able to verify certain predictions (some of which were made as long ago as the 19th century) about the wavelength and frequency of KHI-driven interface waves in non-viscous quantum fluids, like the one studied in this work,” he adds.

“In addition to the universal finger pattern we observed, we expect structures like zipper and sealskin patterns, which are unique to such multi-component quantum fluids,” Takeuchi tells Physics World. “As well as experiments, it is necessary to develop a theory that more precisely describes the motion of EFSs, the interaction between these skyrmions and their internal structure in the context of quantum hydrodynamics and spontaneous symmetry breaking.”

The study is detailed in Nature Physics.

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