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A Fight to Fix Geometry’s Foundations

When two mathematicians raised pointed questions about a classic proof that no one really understood, they ignited a years-long debate about how much could be trusted in a new kind of geometry.

Abstractions blog

Nobel Prize Awarded for Quantum Topology

Three physicists have been awarded the Nobel Prize in Physics for rewriting our understanding of exotic quantum states on the surfaces of materials.

Abstractions blog

Air Traffic Control for Random Surfaces

Mathematicians have had a hard time finding commonalities in large groups of random shapes — until recently.


A Unified Theory of Randomness

Researchers have uncovered deep connections among different types of random objects, illuminating hidden geometric structures.


The Oracle of Arithmetic

At 28, Peter Scholze is uncovering deep connections between number theory and geometry.


How Strange Twists in DNA Orchestrate Life

Coils and twirls in DNA’s double-helix change how the molecule behaves, opening a new role for topology in the study of life.

foundations of mathematics

Will Computers Redefine the Roots of Math?

The Fields medalist Vladimir Voevodsky has died at 51. This 2015 article describes his computer-aided quest to eliminate human error and rewrite the century-old rules underlying all of mathematics.


A Proof That Some Spaces Can’t Be Cut

Mathematicians have solved the century-old triangulation conjecture, a major problem in topology that asks whether all spaces can be subdivided into smaller units.


Scientists Conjure Curves From Flatness

Researchers have found a set of rules for imbuing flat surfaces with curvature, enabling them to form a virtually unlimited range of three-dimensional structures.