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Fourier Transform: A Mathematical Revolution

2025-09-05

Fourier Transform: A Mathematical Revolution

This article recounts the discovery of the Fourier transform and its profound impact. In the early 19th century, French mathematician Joseph Fourier discovered a way to decompose any function into a set of fundamental waves – the Fourier transform. This not only sparked a mathematical revolution but also deeply influenced fields like physics and chemistry. From compressing files to enhancing audio signals, from studying tides to detecting gravitational waves, the Fourier transform is ubiquitous, even playing a crucial role in quantum mechanics. Its core idea is to decompose complex functions into simple sine and cosine waves, thereby simplifying problems; this is like breaking down a symphony into the sounds of individual instruments.

(www.quantamagazine.org)

Tech Fourier Transform

World Models: The Illusion and Reality of AGI

2025-09-03

World Models: The Illusion and Reality of AGI

The latest pursuit in AI research, especially in AGI labs, is the creation of a "world model" – a simplified representation of the environment within an AI system, like a computational snow globe. Leading figures like Yann LeCun, Demis Hassabis, and Yoshua Bengio believe world models are crucial for truly intelligent, scientific, and safe AI. However, the specifics of world models are debated: are they innate or learned? How do we detect their presence? The article traces the concept's history, revealing that current generative AI may rely not on complete world models, but on numerous disconnected heuristics. While effective for specific tasks, these lack robustness. Building complete world models remains crucial, promising solutions to AI hallucinations, improved reasoning, and greater interpretability, ultimately driving progress towards AGI.

(www.quantamagazine.org)

AI

Evolution's Surprise: Bursts of Change Rewrite the Story of Life

2025-09-02

Evolution's Surprise: Bursts of Change Rewrite the Story of Life

A new study challenges the traditional Darwinian view of gradual evolution, revealing bursts of rapid change in the history of life. Researchers used mathematical models to analyze evolutionary data from diverse organisms, including cephalopods, proteins, and human languages. They found that evolution isn't always slow and steady, but instead features concentrated periods of rapid evolution clustered at branching points in the evolutionary tree. This supports the punctuated equilibrium theory, suggesting species can remain stable for long periods before abruptly transforming into new species. The study offers a new perspective on the complexity and diversity of life's evolution.

(www.quantamagazine.org)

Tech punctuated equilibrium biological evolution

The Ten Martini Problem: A Quantum Leap in Mathematical Understanding

2025-08-26

The Ten Martini Problem: A Quantum Leap in Mathematical Understanding

Mathematicians Jitomirskaya and Avila famously solved the 'Ten Martini Problem,' proving a specific mathematical model concerning electron behavior. However, their proof had limitations, only applying to simplified scenarios. In more realistic situations, the proof broke down, and the beautiful mathematical patterns vanished. This changed in 2013 when physicists observed the patterns in a lab, prompting Jitomirskaya to seek a new mathematical explanation. In 2019, her collaborator Ge proposed a 'global theory' promising to solve this, offering a more elegant approach to understanding almost-periodic functions.

(www.quantamagazine.org)

Tech almost periodic functions

The Busy Beaver Game: A Race to the Universe's Edge

2025-08-25

The Busy Beaver Game: A Race to the Universe's Edge

Mathematician Tibor Radó's Busy Beaver game challenges finding the longest-running Turing machine for a given number of rules. Recent years have seen a thrilling competition between Shawn Ligocki and Pavel Kropitz in the BB(6) challenge, pushing the boundaries of computation. Their discoveries resulted in runtimes exceeding the number of atoms in the universe, showcasing both the incredible advancements in computing power and the ingenuity of algorithms.

(www.quantamagazine.org)

Tech Computation

How Neural Networks Recognize Cats: From Simple Classifiers to Complex Models

2025-08-24

How Neural Networks Recognize Cats: From Simple Classifiers to Complex Models

Teaching a computer to recognize a cat in a photo isn't straightforward. However, neural networks now easily accomplish this by learning from millions or billions of examples. This article uses cat photo recognition as an example to explain the basic principles of neural networks: building a simple classifier that uses mathematical functions (neurons) to process input data and ultimately find the optimal boundary to distinguish between categories. The article explains the workings of neural networks in an accessible way, understandable even without a programming background.

(www.quantamagazine.org)

AI

The Surprising Power of Randomness in Algorithms

2025-08-16

The Surprising Power of Randomness in Algorithms

From simulating nuclear processes to primality testing, randomness plays a surprisingly crucial role in computer science. While seemingly paradoxical, pure randomness helps uncover the structure that solves a problem. For instance, Fermat's Little Theorem, combined with random numbers, provides an efficient way to test if a large number is prime. Although deterministic equivalents exist in theory, randomized algorithms often prove more efficient in practice. In some cases, like finding shortest paths in graphs with negative edge weights, randomized algorithms are the only known efficient approach. Randomness offers a clever strategy to tackle complex computational problems.

(www.quantamagazine.org)

Development randomized algorithms primality testing

The Mystery of Thirst: How the Brain Senses Dehydration

2025-08-12

The Mystery of Thirst: How the Brain Senses Dehydration

New research reveals the mechanism by which the brain senses thirst. Instead of directly detecting water deficiency, the brain monitors blood salt concentration through circumventricular organs near the hypothalamus, such as the OVLT and SFO. When salt concentration is too high or the water-salt ratio is imbalanced, these organs signal the brain, triggering thirst. Interestingly, the brain doesn't wait for water absorption to determine hydration; it uses sensors in the mouth and gut to quickly estimate water intake, shutting off the thirst signal promptly. This suggests thirst isn't simply a water deficiency signal, but rather the brain's 'educated guess' about the body's internal environment.

(www.quantamagazine.org)

Tech thirst mechanism brain sensing

Breaking the Sorting Barrier: A New Algorithm Speeds Up Shortest-Path Finding

2025-08-07

Breaking the Sorting Barrier: A New Algorithm Speeds Up Shortest-Path Finding

For decades, a classic problem in computer science—finding the shortest path from a specific starting point in a network to every other point—has been limited by a 'sorting barrier'. Recently, Ran Duan and his team at Tsinghua University have broken this barrier, devising a new algorithm that surpasses all sorting-based algorithms in speed. The algorithm cleverly uses clustering strategies and the Bellman-Ford algorithm, avoiding point-by-point sorting and achieving significant performance improvements, opening a new chapter in shortest-path problem research.

(www.quantamagazine.org)

Development shortest path sorting barrier

Cells Remember Too: Challenging the Definition of Memory

2025-08-05

Cells Remember Too: Challenging the Definition of Memory

Neuroscientist Nikolay Kukushkin at NYU has found that both nerve and kidney cells can differentiate patterns of neurotransmitter bursts and form memories lasting up to a day. This suggests that even non-neural cells can perform pattern recognition and memory, challenging the traditional neuroscientific definition of memory. The research indicates that the formation of cellular memory is related to the spacing of stimuli; spaced stimuli more easily form lasting memories, similar to the mechanisms of memory formation in animals. The study also reveals long-standing biases in the scientific community, limiting memory to observable behavioral changes and ignoring cellular-level memory mechanisms.

(www.quantamagazine.org)

Tech cellular memory

Undergrad Cracks a Math Conjecture: Tackling the Mizohata-Takeuchi Problem

2025-08-02

Undergrad Cracks a Math Conjecture: Tackling the Mizohata-Takeuchi Problem

Hannah Cairo, an undergraduate at UC Berkeley, unexpectedly made significant headway on a simplified version of the Mizohata-Takeuchi conjecture while taking a graduate course in Fourier restriction theory. Initially a homework problem, Cairo became captivated by it, extending the work to more complex formulations. Her advisor, Professor Ruixiang Zhang, was impressed by her passion and focus. This story highlights the potential of young scholars and the dedication to intellectual exploration.

(www.quantamagazine.org)

Development conjecture undergraduate

Yellowstone Bacteria Defies Textbook Biology: Simultaneous Aerobic and Anaerobic Respiration

2025-07-29

Yellowstone Bacteria Defies Textbook Biology: Simultaneous Aerobic and Anaerobic Respiration

A groundbreaking discovery challenges our understanding of cellular respiration. Scientists have found a bacterium in a Yellowstone National Park hot spring capable of simultaneously performing both aerobic and anaerobic respiration—a feat previously thought impossible. This bacterium's unique metabolic pathway offers new insights into how life transitioned from anaerobic to aerobic respiration after the appearance of oxygen. It also highlights the astonishing diversity and adaptability of the microbial world. Published in Nature Communications, this research provides a new perspective on how life adapts to extreme environments.

(www.quantamagazine.org)

Tech cellular respiration

Knowledge Distillation: How Small AI Models Can Challenge the Giants

2025-07-24

Knowledge Distillation: How Small AI Models Can Challenge the Giants

DeepSeek's R1 chatbot, released earlier this year, caused a stir by rivaling the performance of leading AI models from major companies, but at a fraction of the cost and computing power. This led to accusations that DeepSeek used knowledge distillation, a technique potentially involving unauthorized access to OpenAI's o1 model. However, knowledge distillation is a well-established AI technique, dating back to a 2015 Google paper. It involves transferring knowledge from a large 'teacher' model to a smaller 'student' model, significantly reducing costs and size with minimal performance loss. This method has become ubiquitous, powering improvements to models like BERT, and continues to show immense potential across various AI applications. The controversy highlights the power and established nature of this technique, not its novelty.

(www.quantamagazine.org)

AI Knowledge Distillation AI Model Compression

AI Revolutionizes Physics: From LIGO to Novel Quantum Entanglement Experiments

2025-07-22

AI Revolutionizes Physics: From LIGO to Novel Quantum Entanglement Experiments

Artificial intelligence is revolutionizing physics research. This article details AI's application in enhancing LIGO's sensitivity, discovering symmetries in Einstein's relativity from Large Hadron Collider data, and even finding a new equation for dark matter clumping. Most impressively, AI-designed quantum entanglement experiments, surpassing previous designs in simplicity and efficiency, have been successfully validated in China, showcasing AI's immense potential in experimental design and data analysis.

(www.quantamagazine.org)

Tech

Singularity Theorems Proven in Non-Smooth Spacetimes

2025-07-19

Singularity Theorems Proven in Non-Smooth Spacetimes

Mathematicians have long sought to prove singularity theorems in general relativity, such as Hawking's singularity theorem, but these theorems rely on the assumption of smooth spacetime. Recently, researchers cleverly used a 'triangle comparison method' and 'optimal transport theory' to prove special cases of these singularity theorems in non-smooth spacetimes, even extending to more general spacetime models. This breakthrough not only strengthens the mathematical foundation of the Big Bang singularity theory but also provides new mathematical tools for quantum gravity research, paving the way for unifying general relativity and quantum physics.

(www.quantamagazine.org)

Tech singularity theorems non-smooth spacetime

Cryptographic Security Shaken: Attack on Fiat-Shamir Transformation

2025-07-10

Cryptographic Security Shaken: Attack on Fiat-Shamir Transformation

New research has challenged the long-held assumption of the random oracle model in cryptography. Researchers demonstrated a method to trick proof systems using the widely adopted Fiat-Shamir transformation, enabling them to certify false statements. This transformation is crucial in systems like blockchains for verifying computations from external servers, relying on the random oracle model's assumption. The research shows that even under this assumption, attacks are possible. This finding necessitates a re-evaluation of the random oracle model and its implications for numerous cryptographic applications, raising concerns about blockchain security and the potential for cryptocurrency theft.

(www.quantamagazine.org)

Tech

Convex Geometry Cracks Decades-Old Sphere Packing Problem

2025-07-08

Convex Geometry Cracks Decades-Old Sphere Packing Problem

A decades-old problem in mathematics, the efficient packing of spheres in high-dimensional space, has been significantly advanced. Professor Boaz Klartag, using a novel approach from convex geometry, cleverly improved an existing method, achieving a substantial increase in packing efficiency. By using a random process to adjust an ellipsoid, he found a more efficient way to pack spheres than any previous method, improving efficiency by hundreds or even millions of times in high dimensions. This breakthrough not only sets a new record for sphere packing but also reignites the debate on the optimal sphere packing in high-dimensional space, offering new insights for cryptography and communications.

(www.quantamagazine.org)

Misc

Quantum Paradox Shakes Foundations of Physics

2025-07-07

Quantum Paradox Shakes Foundations of Physics

A new thought experiment challenges the foundations of quantum mechanics. The experiment, involving four agents and intricate quantum measurements, leads to contradictory results: two observers reach opposite conclusions about the same event. This suggests at least one of three fundamental assumptions is false: quantum mechanics is universal; measurements have single outcomes; and different observers' quantum predictions aren't contradictory. The experiment forces a re-evaluation of quantum interpretations like many-worlds and spontaneous collapse theories, potentially hinting at a novel understanding of reality.

(www.quantamagazine.org)

Tech quantum paradox reality

Unlocking the Universe's Elemental Origins: Scientists Crack the i-Process Mystery Using FRIB

2025-07-03

Unlocking the Universe's Elemental Origins: Scientists Crack the i-Process Mystery Using FRIB

Scientists at Michigan State University's Facility for Rare Isotope Beams (FRIB) have successfully observed the decay of key isotopes in the i-process, precisely measuring their neutron capture rates. This provides crucial evidence to explain the unusual abundance of heavy elements in some metal-poor, carbon-enhanced stars and offers a new perspective on the origin of heavy elements in the universe. The team plans to apply this technique to the r-process to further unravel the mystery of the origin of heavier elements like gold, silver, and platinum.

(www.quantamagazine.org)

Tech elemental origins

Harvard Professor Unravels the Math Behind Möbius Strips, Brain Folds, and Termite Mounds

2025-06-30

Harvard University professor L. Mahadevan uses mathematics and physics to explore the form and function of everyday phenomena. From the equilibrium shape of a Möbius strip to the complex factors driving biological systems like morphogenesis and social insect colonies, his curiosity knows no bounds. In this podcast episode, he shares his research inspirations, explaining how gels, gypsum, and LED lights can help uncover form and function in biological systems, and how noisy random processes might underlie our intuitions about geometry. He explores brain folds, simulating the folding process with gel experiments, and reveals how termites build massive mounds to regulate temperature and ventilation.

(www.quantamagazine.org)

Tech biological morphogenesis

The Impossible Tetrahedron: From Math Problem to Real-World Object

2025-06-26

The Impossible Tetrahedron: From Math Problem to Real-World Object

Mathematicians have long studied the 'monostable tetrahedron' – a unique shape stable on only one side. Theoretically, this shape is achievable through clever mass distribution, but building one proved incredibly challenging. Gergő Almádi and his team, after complex calculations and multiple failed attempts, finally constructed a monostable tetrahedron model using a carbon fiber frame and tungsten carbide components. This successful model not only validates mathematical theory but also offers new avenues for future engineering designs, such as lunar landers.

(www.quantamagazine.org)

Tech

Challenging Infinity: An Expedition to the Edge of the Mathematical Universe

2025-06-24

Challenging Infinity: An Expedition to the Edge of the Mathematical Universe

A group of mathematicians, meeting in the Finnish Arctic Circle, explored the mysteries of infinity within the mathematical universe. They discovered two new cardinal numbers that defy the established hierarchy, instead 'exploding' into a new class of infinities, challenging the known order of the mathematical universe. This discovery sparked a heated debate about the structure of the mathematical universe, with some arguing it represents substantial progress, while others question its validity. The core of the debate lies in the understanding of mathematical axiom systems and the exploration of the nature of infinity.

(www.quantamagazine.org)

Misc mathematical universe

Microbial Ecosystems: Phase Transitions and the Surprising Vulnerability of Diverse Communities

2025-06-20

Microbial Ecosystems: Phase Transitions and the Surprising Vulnerability of Diverse Communities

MIT researchers discovered that microbial ecosystems undergo phase transitions, similar to those in physics, progressing through stable, partially extinct, and wildly fluctuating states. Surprisingly, diverse, fluctuating ecosystems were more susceptible to invasion by new species, contradicting established ecological theory. The study reveals that a higher survival fraction of initial species increases vulnerability to invasion. The Lotka-Volterra model confirmed these results, suggesting this is an emergent property of complex dynamic systems.

(www.quantamagazine.org)

Tech microbes

Entropic Gravity: Is Gravity Not a Fundamental Force?

2025-06-16

Entropic Gravity: Is Gravity Not a Fundamental Force?

For centuries, physicists have grappled with understanding the nature of gravity. Newton's law of universal gravitation, while effective, left its mechanism of action at a distance unexplained. Einstein's general relativity offered a deeper explanation, but it also has limitations. Recently, entropic gravity proposes that gravity isn't a fundamental force, but rather a collective effect of deeper, more microscopic physics, similar to 17th-century mechanical models. New research models this effect using quantum bits, suggesting that gravity arises from the interaction of these qubits with massive objects, resulting in an apparent attractive force due to entropy increase. While still in its early stages, this model opens up new experimental avenues for gravity research, such as testing gravitational effects in quantum superpositions, potentially shedding light on fundamental issues like wave function collapse.

(www.quantamagazine.org)

Tech entropic gravity

50-Year-Old Conjecture on Space vs. Time in Computation Cracked

2025-06-07

50-Year-Old Conjecture on Space vs. Time in Computation Cracked

A central question in complexity theory is the relationship between P and PSPACE, classes encompassing problems solvable in reasonable time and space, respectively. Intuitively, space is a more powerful resource than time because it's reusable. For 50 years, researchers aimed to prove PSPACE is larger than P, meaning some problems are impossible to solve quickly but solvable with limited space. Hopcroft, Paul, and Valiant made a breakthrough in 1975, showing space is slightly more powerful than time. However, this progress was limited by the 'simulation' approach. Ryan Williams finally broke the deadlock with a novel approach, solving the long-standing problem.

(www.quantamagazine.org)

Development

The Brain's Energy Budget: Why Focus Leads to Fatigue

2025-06-06

The Brain's Energy Budget: Why Focus Leads to Fatigue

New research unveils the secrets of the brain's energy efficiency. The brain operates far more efficiently than previously thought, a legacy of our ancestors' evolution in energy-scarce environments. Even at rest, the brain performs extensive background tasks, including prediction and maintaining homeostasis. Intense mental activity significantly increases energy consumption, explaining why prolonged focus leads to fatigue. The brain has evolved mechanisms to limit energy expenditure, such as reducing neuronal firing rates and synaptic transmission efficiency, maximizing information transmission efficiency per energy unit. This research provides insights into the brain's mechanisms and the limits of human cognitive capacity.

(www.quantamagazine.org)

Tech

Reversible Computing: A Low-Energy Revolution for AI?

2025-06-02

Reversible Computing: A Low-Energy Revolution for AI?

The inherent energy loss in computer computation, like Hansel and Gretel's discarded breadcrumbs, has long been a challenge. Landauer pioneered reversible computing, but it was initially deemed a dead end. Bennett's 'uncomputation' offered a new path, cleverly avoiding data deletion to reduce energy waste, but speed remained an issue. MIT engineers attempted low-loss chip designs, but progress was slow. Recently, as computer circuits approach physical limits and the demand for parallel AI computation rises, reversible computing has gained renewed interest. Earley's research precisely quantifies the energy savings, paving the way for commercial applications. The founding of Vaire Computing marks a milestone in the transition from theory to reality.

(www.quantamagazine.org)

Tech

Geometry: From Land Measurement to Understanding the Universe

2025-05-30

This episode of the podcast 'The Joy of Why' features theoretical physicist Yang-Hui He discussing the evolution of geometry. From its ancient roots in land measurement and pyramid construction to its pivotal role in Einstein's general relativity, geometry's influence is explored. He argues that geometry serves as a unifying language for modern physics and speculates on AI's potential to revolutionize the field. The hosts also discuss the tension between formal mathematics and intuition-driven insight, and the two types of mathematicians: 'birds' and 'hedgehogs'.

(www.quantamagazine.org)

Tech

Singularities: Physics' Unbreakable Dead Ends?

2025-05-28

Singularities: Physics' Unbreakable Dead Ends?

The birth of the universe and the center of a black hole both point to singularities—points where the fabric of spacetime breaks down. Einstein's general relativity predicts singularities, but it fails there. Recent research shows that singularities persist even when considering quantum effects, challenging physicists' efforts to build a complete theory of quantum gravity. This suggests that our universe may contain regions where spacetime structure completely disintegrates, time stops, and everything becomes unpredictable. Future quantum gravity theories might explain singularities, but the concept of spacetime may need redefinition.

(www.quantamagazine.org)

Tech

Solved: The Sum-Free Sets Conjecture

2025-05-25

Solved: The Sum-Free Sets Conjecture

A seemingly simple mathematical problem—the sum-free sets conjecture—has baffled mathematicians for decades. The conjecture explores whether, within any set of integers, there exists a large subset where the sum of any two numbers in the subset is not also in the subset. In 1965, the renowned mathematician Paul Erdős posed the question, providing a lower bound. Despite many attempts to improve upon it, progress remained stagnant until February of this year, when Oxford graduate student Benjamin Bedert finally solved the problem, demonstrating that any set of integers contains a large sum-free subset, significantly larger than previously estimated. Bedert's proof cleverly combines techniques from diverse mathematical fields, offering new approaches to similar problems. This achievement is hailed as a major breakthrough in mathematics.

(www.quantamagazine.org)

Misc sum-free sets

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