Math
149 articles
Alan Turing and the Power of Negative Thinking
Mathematical proofs based on a technique called diagonalization can be relentlessly contrarian, but they help reveal the limits of algorithms.

Risky Giant Steps Can Solve Optimization Problems Faster
New results break with decades of conventional wisdom for the gradient descent algorithm.

Math Proof Draws New Boundaries Around Black Hole Formation
For a half century, mathematicians have tried to define the exact circumstances under which a black hole is destined to exist.

The Lawlessness of Large Numbers
Mathematicians can often figure out what happens as quantities grow infinitely large. What about when they are just a little big?

The Unbelievable Untangling Worms
Solving nature’s greatest Gordian knot.

Are All Brains Good at Math?
Math provokes dread in so many people—yet we are all born with a sense for numbers.

A Numerical Mystery From the 19th Century Finally Gets Solved
Two mathematicians have proven Patterson’s conjecture, which was designed to explain a strange pattern in sums involving prime numbers.

Math’s “Oldest Problem Ever” Gets a New Answer
A new proof significantly strengthens a decadesold result about the ubiquity of ways to represent whole numbers as sums of fractions.

Imaginary Numbers Are Reality
How the modern world arose from imaginary numbers.

An Ancient Geometry Problem Falls to New Mathematical Techniques
Three mathematicians show, for the first time, how to form a square with the same area as a circle by cutting them into interchangeable pieces that can be visualized.

Mathematicians Find Structure in Biased Polynomials
New work establishes a tighter connection between the rank of a polynomial and the extent to which it favors particular outputs.

In Topology, When Are Two Shapes the Same?
As topologists seek to classify shapes, the effort hinges on how to define a manifold and what it means for two of them to be equivalent.

Computer Scientists Discover Limits of Major Research Algorithm
The most widely used technique for finding the largest or smallest values of a math function turns out to be a fundamentally difficult computational problem.

Why You May Have More Friends Than Your Friends Do
There’s a rude charm to the title, “Why Your Friends Have More Friends Than You.” It’s catchy, like the title of an antagonistic explainer: Here are the causes of your lackluster social life. It sounds more like a New York Times oped than an academic paper. But in fact, “Why Your Friends Have More Friends […]

Pandemic Puts Mathematical Modeling Through Its Paces
Mathematical tools that proved essential during the pandemic were in many cases invented by mathematicians who had no particular goal in mind.

Mathematicians Answer Old Question About Odd Graphs
A pair of mathematicians solved a legendary question about the proportion of vertices in a graph with an odd number of connections.

How Mathematicians Use Homology to Make Sense of Topology
Originally devised as a rigorous means of counting holes, homology provides a scaffolding for mathematical ideas, allowing for a new way to analyze the shapes within data.

New Quantum Algorithms Finally Crack Nonlinear Equations
Two teams found different ways for quantum computers to process nonlinear systems by first disguising them as linear ones.

How the Slowest Computer Programs Illuminate Math’s Fundamental Limits
The goal of the “busy beaver” game is to find the longestrunning computer program. Its pursuit has surprising connections to some of the most profound questions and concepts in mathematics.

Undergraduate Math Student Pushes Frontier of Graph Theory
At 21, Ashwin Sah has produced a body of work that senior mathematicians say is nearly unprecedented for a college student.