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Behold Modular Forms, the ‘Fifth Fundamental Operation’ of Math
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
On its surface, the Kakeya conjecture is a simple statement about rotating needles. But it underlies a wealth of mathematics.
Why Mathematical Proof Is a Social Compact
Number theorist Andrew Granville on what mathematics really is — and why objectivity is never quite within reach.
Two Students Unravel a Widely Believed Math Conjecture
Mathematicians thought they were on the cusp of proving a conjecture about the ancient structures known as Apollonian circles. But a summer project would lead to its downfall.
Elliptic Curves Yield Their Secrets in a New Number System
Ana Caraiani and James Newton have extended an important result in number theory to the imaginary realm.
Why Mathematicians Re-Prove What They Already Know
It’s been known for thousands of years that the primes go on forever, but new proofs give fresh insights into how theorems depend on one another.
New Proof Distinguishes Mysterious and Powerful ‘Modular Forms’
Using “refreshingly old” tools, mathematicians resolved a 50-year-old conjecture about how to categorize important functions called modular forms, with consequences for number theory and theoretical physics.
Probability and Number Theory Collide — in a Moment
Mathematicians are taking ideas developed to study random numbers and applying them to a broad range of categories.
The Year in Math
Four Fields Medals were awarded for major breakthroughs in geometry, combinatorics, statistical physics and number theory, even as mathematicians continued to wrestle with how computers are changing the discipline.