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algebraic geometry

foundations of mathematics

With Category Theory, Mathematics Escapes From Equality

October 10, 2019

Two monumental works have led many mathematicians to avoid the equal sign. The process has not always gone smoothly.

Colored spheres arranged in pairs.
prime numbers

Big Question About Primes Proved in Small Number Systems

September 26, 2019

The twin primes conjecture is one of the most important and difficult questions in mathematics. Two mathematicians have solved a parallel version of the problem for small number systems.

Portrait of Carolina Araujo at IMPA in Brazil
Q&A

A Movement to Close the Gender Gap in Mathematics

January 22, 2019

The Brazilian mathematician Carolina Araujo, who calls herself “a bit of an anarchist,” is organizing meetings and building a support network to study and solve the problems women face in mathematics.

abstractions blog

New Proof Shows Infinite Curves Come in Two Types

November 7, 2018

Alexander Smith’s work on the Goldfeld conjecture reveals fundamental characteristics of elliptic curves.

number theory

Without a Proof, Mathematicians Wonder How Much Evidence Is Enough

October 31, 2018

A new statistical model appears to undermine long-held assumptions in number theory. How much should it be trusted when all that really matters is proof?

Art for "Titans of Mathematics Clash Over Epic Proof of ABC Conjecture"
number theory

Titans of Mathematics Clash Over Epic Proof of ABC Conjecture

September 20, 2018

Two mathematicians have found what they say is a hole at the heart of a proof that has convulsed the mathematics community for nearly six years.

Art for "Why Mathematicians Can’t Find the Hay in a Haystack"
Abstractions blog

Why Mathematicians Can’t Find the Hay in a Haystack

September 17, 2018

In math, sometimes the most common things are the hardest to find.

Illustration of a complex shape casting a shadow
geometry

Tinkertoy Models Produce New Geometric Insights

September 5, 2018

An upstart field that simplifies complex shapes is letting mathematicians understand how those shapes depend on the space in which you visualize them.

Photo illustration of Caucher Birkar
2018 Fields Medal and Nevanlinna Prize

An Innovator Who Brings Order to an Infinitude of Equations

August 1, 2018

The mathematician Caucher Birkar was born on a subsistence farm and raised in the middle of the brutal war between Iran and Iraq. After fleeing to England, he has gone on to impose order on a wild landscape of mathematical equations.