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The Biggest Breakthroughs in Mathematics: 2025

2025 marked a historic year in mathematics. Researchers solved a major case of Hilbert’s ambitious sixth problem, proved a sweeping new theorem about hyperbolic surfaces, and settled the longstanding three-dimensional Kakeya conjecture.

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Lisa Manning on the Dynamics of Glasses and Embryos

June 11, 2018

Lisa Manning, a physicist at Syracuse University, describes how the physics of glassy materials helps to explain how some organs assume their correct shape during embryonic development.

Michela Massimi: Defending the Philosophy of Science

May 24, 2018

Michela Massimi argues that the philosophy of science doesn’t have to be useful to scientists for it to be useful to humanity.

Günter Ziegler Seeks God’s Perfect Math Proofs

March 19, 2018

Günter Ziegler describes one of the most famous and beautiful proofs in Proofs From THE BOOK, a book he co-authored with Martin Aigner.

Barbara Engelhardt on How to Improve Statistical Analyses of Genomes

February 27, 2018

Barbara Engelhardt, a computer scientist at Princeton University, explains why traditional machine-learning techniques have often fallen short for genomic analysis, and how researchers are overcoming that challenge.

Daniel Goldman and His Smart Robots

February 14, 2018

Goldman explains how “smarticles” work together to demonstrate collective behavior.

Gil Kalai: Why Quantum Computers Won’t Work

February 7, 2018

Kalai argues that limiting the noise in a quantum computer will also limit the computational power of the system.

Erich Jarvis on Theories About the Origin of Vocal Learning

January 30, 2018

Neuroscientist Erich Jarvis discusses how the brain circuitry for vocal learning in songbirds and humans evolved from systems for controlling body movements and why so few species have this ability.

Ed Boyden on the Promise of Expansion Microscopy

January 18, 2018

Ed Boyden explains how expansion microscopy could illuminate deep mysteries about how the brain works and improve cancer diagnosis, among other advances.