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polynomials

3D illustration of Gardam's crystallographic group
group theory

Mathematician Disproves 80-Year-Old Algebra Conjecture

April 12, 2021

Inside the symmetries of a crystal shape, a postdoctoral researcher has unearthed a counterexample to a basic conjecture about multiplicative inverses.

An illustration of a doughnut-shaped elliptic curve intertwined with the Julia set.
number theory

Mathematicians Set Numbers in Motion to Unlock Their Secrets

February 22, 2021

A new proof demonstrates the power of arithmetic dynamics, an emerging discipline that combines insights from number theory and dynamical systems.

Abstractions blog

Undergraduates Hunt for Special Tetrahedra That Fit Together

February 9, 2021

A group of MIT undergraduates is searching for tetrahedra that tile space, the latest effort in a millennia-long inquiry. They’ve already made a new discovery.

Colorful isualization of the 59 isolated examples of tetrahedra that have rational dihedral angles
geometry

Tetrahedron Solutions Finally Proved Decades After Computer Search

February 2, 2021

Four mathematicians have cataloged all the tetrahedra with rational angles, resolving a question about basic geometric shapes using techniques from number theory.

Illustration of mathematicians descending through a cavern whose walls are lined with polynomials.
polynomials

Mathematicians Resurrect Hilbert’s 13th Problem

January 14, 2021

Long considered solved, David Hilbert’s question about seventh-degree polynomials is leading researchers to a new web of mathematical connections.

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

An Infinite Universe of Number Systems

October 19, 2020

The p-adics form an infinite collection of number systems based on prime numbers. They’re at the heart of modern number theory.

An illustration of computer scientists looking down a new road that breaks open a traveling salesperson route.
algorithms

Computer Scientists Break Traveling Salesperson Record

October 8, 2020

After 44 years, there’s finally a better way to find approximate solutions to the notoriously difficult traveling salesperson problem.

Photo of iron filings spread along magnetic field lines on a green and yellow background
number theory

Mathematician Measures the Repulsive Force Within Polynomials

May 14, 2020

Vesselin Dimitrov’s proof of the Schinzel-Zassenhaus conjecture quantifies the way special values of polynomials push each other apart.

Quantized Academy

To Win This Numbers Game, Learn to Avoid Math Patterns

May 7, 2020

Sizing up patternless sets is hard, so mathematicians rely on simple bounds to help answer their questions.