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# Mathematics

## Latest Articles

### Why Mathematical Proof Is a Social Compact

Number theorist Andrew Granville on what mathematics really is — and why objectivity is never quite within reach.

### An Old Conjecture Falls, Making Spheres a Lot More Complicated

The telescope conjecture gave mathematicians a handle on ways to map one sphere to another. Now that it has been disproved, the universe of shapes has exploded.

### Math Proof Draws New Boundaries Around Black Hole Formation

For a half century, mathematicians have tried to define the exact circumstances under which a black hole is destined to exist. A new proof shows how a cube can help answer the question.

### 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.

### Ninth Dedekind Number Found by Two Independent Groups

The numbers count a variety of seemingly unrelated mathematical structures.

### New Proof Shows That ‘Expander’ Graphs Synchronize

The proof establishes new conditions that cause connected oscillators to sway in sync.

### Math That Lets You Think Locally but Act Globally

Knowing a little about the local connections on flight maps and other networks can reveal a lot about a system’s global structure.

### Mathematicians Solve Long-Standing Coloring Problem

A new result shows how much of the plane can be colored by points that are never exactly one unit apart.

### New Proof Threads the Needle on a Sticky Geometry Problem

A new proof marks major progress toward solving the Kakeya conjecture, a deceptively simple question that underpins a tower of conjectures.