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group theory

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A modular form is shown as a distorted, rainbow-hued dome atop concentric circles.
number theory

New Proof Distinguishes Mysterious and Powerful ‘Modular Forms’

By Jordana Cepelewicz
March 9, 2023
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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.

number theory

Probability and Number Theory Collide — in a Moment

By Leila Sloman
January 12, 2023
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Mathematicians are taking ideas developed to study random numbers and applying them to a broad range of categories.

An illustration of a large, 3D equation that almost looks like a sculpture (that is being inspected by different people, and is surrounded by different numbers and measuring equipment such as rulers and protractors. The equation is x2 − dy2 = –1).
number theory

Mathematicians Crack a Simple but Stubborn Class of Equations

By Jordana Cepelewicz
August 10, 2022
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Ever since Archimedes, mathematicians have been fascinated by equations that involve a difference between squares. Now two mathematicians have proven how often these equations have solutions, concluding a decades-old quest.

Quantized Academy

The Simple Math Behind the Mighty Roots of Unity

By Patrick Honner
September 23, 2021
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Solutions to the simplest polynomial equations — called “roots of unity” — have an elegant structure that mathematicians still use to study some of math’s greatest open questions.

Animated illustration showing multiple permutations of colorful letters
group theory

Galois Groups and the Symmetries of Polynomials

By Allison Whitten
August 3, 2021
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By focusing on relationships between solutions to polynomial equations, rather than the exact solutions themselves, Évariste Galois changed the course of modern mathematics.

Illustration showing geometric figures at left connected via wormhole to numbers at right.
Langlands program

New Shape Opens ‘Wormhole’ Between Numbers and Geometry

By Kevin Hartnett
July 19, 2021
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Laurent Fargues and Peter Scholze have found a new, more powerful way of connecting number theory and geometry as part of the sweeping Langlands program.

3D illustration of Gardam's crystallographic group
group theory

Mathematician Disproves 80-Year-Old Algebra Conjecture

By Erica Klarreich
April 12, 2021
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Inside the symmetries of a crystal shape, a postdoctoral researcher has unearthed a counterexample to a basic conjecture about multiplicative inverses.

Diagram showing a colorful visualization of the simple Lie group
Abstractions blog

The ‘Useless’ Perspective That Transformed Mathematics

By Kevin Hartnett
June 9, 2020
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Representation theory was initially dismissed. Today, it’s central to much of mathematics.

Multimedia

The Map of Mathematics

By Kevin Hartnett
February 13, 2020
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Explore our surprisingly simple, absurdly ambitious and necessarily incomplete guide to the boundless mathematical universe.


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