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p-adic numbers

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A deltoid and other mathematical shapes.
geometry

A Question About a Rotating Line Helps Reveal What Makes Real Numbers Special

By Kevin Hartnett
July 26, 2022
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The Kakeya conjecture predicts how much room you need to point a line in every direction. In one number system after another — with one important exception — mathematicians have been proving it true.

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.

Illustration of figures building a large structure, with a new material being used for the latest additions
polynomials

Mathematicians Find Long-Sought Building Blocks for Special Polynomials

By Kelsey Houston-Edwards
May 25, 2021
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Hilbert’s 12th problem asked for novel analogues of the roots of unity, the building blocks for certain number systems. Now, over 100 years later, two mathematicians have produced them.

A visualization of the tower formed by the 3-adic numbers
Abstractions blog

An Infinite Universe of Number Systems

By Kelsey Houston-Edwards
October 19, 2020
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The p-adics form an infinite collection of number systems based on prime numbers. They’re at the heart of modern number theory.

Photo of Dr. Peter Scholze
2018 Fields Medal and Nevanlinna Prize

A Master of Numbers and Shapes Who Is Rewriting Arithmetic

By Erica Klarreich
August 1, 2018
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The 30-year-old math sensation Peter Scholze is now one of the youngest Fields medalists for “the revolution that he launched in arithmetic geometry.”

Minhyong Kim in front of a whiteboard
number theory

Secret Link Uncovered Between Pure Math and Physics

By Kevin Hartnett
December 1, 2017
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An eminent mathematician reveals that his advances in the study of millennia-old mathematical questions owe to concepts derived from physics.

profiles

The Oracle of Arithmetic

By Erica Klarreich
June 28, 2016
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At 28, Peter Scholze is uncovering deep connections between number theory and geometry.

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