Most of us take it for granted that there are three dimensions, perhaps four if we count time. But for over 200 years, mathematicians and scientists have proposed further dimensions. In some standard versions of contemporary physics eleven dimensions are now proposed. But might the notion of additional dimensions be an empty idea that derails physics? Richard Feynman argued that proponents of extra dimensions.
It has puzzled scientists for years whether and how bacteria, that live from dissolved organic matter in marine waters, can carry out N2 fixation. It was assumed that the high levels of oxygen combined with the low amount of dissolved organic matter in the marine water column would prevent the anaerobic and energy consuming N2 fixation.
Already in the 1980s it was suggested that aggregates, so-called “marine snow particles,” could possibly be suitable sites for N2 fixation, and this was recently confirmed. Still, it has been an open question why the bacteria carrying out this N2 fixation can be found worldwide in the ocean. Moreover, the global magnitude and the distribution of the activity have been unknown… until now.
In a new study, researchers from the Leibniz Centre for Tropical Marine Research in Germany, Technical University of Denmark, and the University of Copenhagen demonstrate, by use of mechanistic mathematical models, that bacteria attached to marine snow particles can fix N2 over a wide range of temperatures in the global oceans, from the tropics to the poles, and from the surface to the abyss.
We say a message is incoherent when we can’t make it out, or when it doesn’t make sense. A scribbled note, a drunken argument or a conversation taking place five tables down in a crowded cafe might all be incoherent. In general, “coherent” means the opposite—consistent, connected, clear.
In science, the word coherence takes on more specific, mathematical definitions, but they all get at a similar concept: Something is coherent if it can be understood, if it forms a unified whole and if those first two qualities persist.
Scientists originally developed the concept of coherence to understand and describe the wave-like behavior of light. Since then, the concept has been generalized to other systems involving waves, such as acoustic, electronic and quantum mechanical systems.
Physicists have found a simple and effective way to skip over an energy level in a three-state system, potentially leading to increased quantum computational power with fewer qubits.
Nearly a century ago, Lev Landau, Clarence Zener, Ernst Stückelberg, and Ettore Majorana found a mathematical formula for the probability of jumps between two states in a system whose energy is time-dependent. Their formula has since had countless applications in various systems across physics and chemistry.
Now physicists at Aalto University’s Department of Applied Physics have shown that the jump between different states can be realized in systems with more than two energy levels via a virtual transition to an intermediate state and by a linear chirp of the drive frequency. This process can be applied to systems where it is not possible to modify the energy of the levels.
Elon Musk’s AI startup xAI has introduced Grok 3, the latest version of its chatbot model, which Musk describes as the most advanced AI system yet.
XAI claims Grok 3 outperforms rival AI models from Alphabet’s Google Gemini, DeepSeek’s V3, Anthropic’s Claude, and OpenAI’s GPT-4o in benchmarks for math, science, and coding.
A strange molecular pattern, first mistaken for an error, led researchers to an unexpected discovery: molecules forming non-repeating structures similar to the einstein tiling problem.
This phenomenon, driven by chirality and energy balance, could pave the way for novel insights into molecular physics.
At the crossroads of mathematics and tiling lies the einstein problem—a puzzle that, despite its name, has nothing to do with Albert Einstein. The question is simple yet profound: Can a single shape tile an infinite surface without ever creating a repeating pattern? In 2022, English amateur mathematician David Smith discovered such a shape, known as a “proto-tile.”
The words “optimal” and “optimize” derive from the Latin “optimus,” or “best,” as in “make the best of things.” Alessio Figalli, a mathematician at the university ETH Zurich, studies optimal transport: the most efficient allocation of starting points to end points. The scope of investigation is wide, including clouds, crystals, bubbles and chatbots.
Dr. Figalli, who was awarded the Fields Medal in math that is motivated by concrete problems found in nature. He also likes the discipline’s “sense of eternity,” he said in a recent interview. “It is something that will be here forever.” (Nothing is forever, he conceded, but math will be around for “long enough.”) “I like the fact that if you prove a theorem, you prove it,” he said. “There’s no ambiguity, it’s true or false. In a hundred years, you can rely on it, no matter what.”
The study of optimal transport was introduced almost 250 years ago by Gaspard Monge, a French mathematician and politician who was motivated by problems in military engineering. His ideas found broader application solving logistical problems during the Napoleonic Era — for instance, identifying the most efficient way to build fortifications, in order to minimize the costs of transporting materials across Europe.
In order to bridge the yawning gulf between the humanities and the sciences we must turn to an unexpected field: mathematics.
A team of mathematicians and statisticians from the University of Wisconsin-La Crosse, the University of Tennessee and Valparaiso University, all in the U.S., has found new evidence that wolves had ample time to self-domesticate and evolve into modern dogs. In their study published in the journal Proceedings of the Royal Society B, the group developed a computer simulation showing the evolution process.
Prior research has suggested that the process of self-domesticating and then slowly evolving into modern dogs would have taken too long. Additionally, researchers believe that humans and dogs have been living in close proximity for approximately 30,000 years and that for the past 15,000 years, humans have been breeding them to perform certain tasks. But what happened in the first 15,000 years is less clear.
Some have suggested that humans may have begun encouraging the friendliest wolves to hang around by adopting their puppies, finding their presence advantageous. Others have suggested that wolves moved ever closer to groups of humans for access to leftover food. But this evolution, others have noted, would take more than 15,000 years to reach the point where humans began breeding them.
Standing at the intersection between mathematics and the tiler’s trade is the so-called einstein problem. Despite its name, this mathematical question has nothing to do with the Nobel Prize winner Albert Einstein. It asks: Can you seamlessly tile an endless surface with a single shape (an “einstein”) in such a way that the resulting pattern is never repeated? Such a “proto-tile” was first discovered in 2022 by the English amateur mathematician David Smith.
Empa researcher Karl-Heinz Ernst is neither a mathematician nor a tiler. As a chemist, he researches the crystallization of molecules on metal surfaces. He never expected to deal with the einstein problem in his professional life—until his doctoral student Jan Voigt approached him with the unusual results of an experiment.
When a certain molecule crystallized on a silver surface, instead of the expected regular structure, irregular patterns were formed that never seemed to repeat themselves. Even more surprising: Each time he repeated the experiment, different aperiodic patterns emerged.