The Navier-Stokes Equations, Explained

This video explores why the Navier-Stokes equations are so difficult, what it means for their solutions to blow up, and how a new AI-assisted proof is reshaping one of mathematics’ most famous unsolved problems.

When Hannah Cairo was a 17-year-old high school student, she did something mathematicians had been unable to do for four decades: she overturned the Mizohata-Takeuchi conjecture, a famous prediction about how wave energy behaves. In this video, Cairo and the mathematicians she worked with explain how she found the counterexample and what it means for the field.

Biologist Michael Levin argues that intelligence is a continuum of goal-directed problem-solving that extends far beyond brains, from cells to potentially even nonliving systems.

For thousands of years, mathematicians avoided the concept of infinity. Then, in 1874, Georg Cantor published a four-page paper that changed mathematics forever. Hidden inside was a shocking claim: some infinities are larger than others. Today, Cantor is celebrated as the father of set theory and one of the great mathematical revolutionaries. But newly uncovered letters reveal a far more complicated story — one involving collaboration, secrecy, and plagiarism.

2025’s most surprising breakthroughs in biology included a finding that a father’s environmental exposures can impact the development of their offspring, research confirming that intelligence evolved independently in birds and mammals, and a new mathematical model reveals that evolution happens in explosive bursts.

2025’s most surprising computational revelations included a new fundamental relationship between time and space, an undergraduate who overthrew a 40-year-old conjecture, and an important milestone in quantum computing.

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.

In 2025, physicists spotted a paradigm-shifting new black hole, doubled down on weakening dark energy, and discovered that the Earth’s deep interior may be more connected than previously thought, reshaping our picture of the planet’s engine.

Carlo Rovelli discusses his research on time and his view that it should not appear in the quantum theory of gravity.