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Mathematics

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Mathematicians Cross the Line to Get to the Point

September 25, 2023

A new paper establishes a long-conjectured bound about the size of the overlap between sets of lines and points.

Behold Modular Forms, the ‘Fifth Fundamental Operation’ of Math

September 21, 2023

Modular forms are one of the most beautiful and mysterious objects in mathematics. What are they?

A Tower of Conjectures That Rests Upon a Needle

September 12, 2023

On its surface, the Kakeya conjecture is a simple statement about rotating needles. But it underlies a wealth of mathematics.

The Biggest Smallest Triangle Just Got Smaller

September 8, 2023

A new proof breaks a decades-long drought of progress on the problem of estimating the size of triangles created by cramming points into a square.

Alan Turing and the Power of Negative Thinking

September 5, 2023

Mathematical proofs based on a technique called diagonalization can be relentlessly contrarian, but they help reveal the limits of algorithms.

Q&A

Why Mathematical Proof Is a Social Compact

August 31, 2023

Number theorist Andrew Granville on what mathematics really is — and why objectivity is never quite within reach.

An Old Conjecture Falls, Making Spheres a Lot More Complicated

August 22, 2023

The telescope conjecture gave mathematicians a handle on ways to map one sphere to another. Now that it has been disproved, the universe of shapes has exploded.

Complexity Theory’s 50-Year Journey to the Limits of Knowledge

August 17, 2023

How hard is it to prove that problems are hard to solve? Meta-complexity theorists have been asking questions like this for decades. A string of recent results has started to deliver answers.

Math Proof Draws New Boundaries Around Black Hole Formation

August 16, 2023

For a half century, mathematicians have tried to define the exact circumstances under which a black hole is destined to exist. A new proof shows how a cube can help answer the question.

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