The original version of this story appeared in Quanta Magazine. Large language models work well because they’re so large. The latest models from OpenAI, Meta, and DeepSeek use hundreds of billions of “parameters”—the adjustable knobs that determine connections among data and get tweaked during the training process. With more parameters,Continue Reading
Moore designed his pinball machine to complete the analogy to the Turing machine. The starting position of the pinball represents the data on the tape being fed into the Turing machine. Crucially (and unrealistically), the player must be able to adjust the ball’s starting location with infinite precision, meaning thatContinue Reading
Those are pretty stringent constraints, so it wasn’t obvious that the extra memory could ever prove useful. But to their surprise, Buhrman and Cleve showed that if you tweak bits in just the right way, you really can get extra computational oomph out of a full memory. “That was aContinue Reading
But one of Malle’s graduate students was on the case. Britta Späth. “Our Obsession” In 2003, Späth arrived at the University of Kassel to start her doctorate with Malle. She was almost perfectly suited for working on the McKay conjecture: Even in high school, she could spend days or weeksContinue Reading
In a 1985 paper, the computer scientist Andrew Yao, who would go on to win the A.M. Turing Award, asserted that among hash tables with a specific set of properties, the best way to find an individual element or an empty spot is to just go through potential spots randomly—anContinue Reading
In other words, Hilbert’s 10th problem is undecidable. Mathematicians hoped to follow the same approach to prove the extended, rings-of-integers version of the problem—but they hit a snag. Gumming Up the Works The useful correspondence between Turing machines and Diophantine equations falls apart when the equations are allowed to haveContinue Reading
The original version of this story appeared in Quanta Magazine. Most of life’s engines run on sunlight. Photons filter down through the atmosphere and are eagerly absorbed by light-powered organisms such as plants and algae. Through photosynthesis, the particles of light power a cellular reaction that manufactures chemical energy (inContinue Reading
The original version of this story appeared in Quanta Magazine. Calculus is a powerful mathematical tool. But for hundreds of years after its invention in the 17th century, it stood on a shaky foundation. Its core concepts were rooted in intuition and informal arguments, rather than precise, formal definitions. TwoContinue Reading
The original version of this story appeared in Quanta Magazine. Computer scientists often deal with abstract problems that are hard to comprehend, but an exciting new algorithm matters to anyone who owns books and at least one shelf. The algorithm addresses something called the library sorting problem (more formally, theContinue Reading
The original version of this story appeared in Quanta Magazine. Prochlorococcus bacteria are so small that you’d have to line up around a thousand of them to match the thickness of a human thumbnail. The ocean seethes with them: The microbes are likely the most abundant photosynthetic organism on theContinue Reading
