Arithmetic geometry is a vibrant field at the intersection of number theory and algebraic geometry, focussing on the study of polynomial equations and the distribution of their rational solutions ...

They made some progress, re-proving the conjecture in two dimensions using different techniques—ones they hoped would be applicable to the three-dimensional case. But then they hit a wall.

Numbers From Shapes Beginning in the early 1980s Vladimir Drinfeld and later Alexander Beilinson proposed that there should be a way to interpret Langlands’ conjectures in geometric terms. The ...

D. B. A. Epstein, V. Markovic, The Logarithmic Spiral: A Counterexample to the K = 2 Conjecture, Annals of Mathematics, Second Series, Vol. 161, No. 2 (Mar., 2005 ...

Algebraic geometry explores the geometric objects that are sets of solutions to algebraic equations — for example, a circle of radius r can be described by x 2+ y 2= r 2.

Chinese mathematician Wang Hong has solved an “infamous” geometry problem called the Kakeya conjecture within three dimensions. It is considered a breakthrough that could have implications for ...