In the first (theoretical) part of this paper, we prove a number of constraints on hypothetical counterexamples to the Casas-Alvero conjecture, building on ideas of Graf von Bothmer, Labs, Schicho and ...

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.

Proceedings of the American Mathematical Society, Vol. 81, No. 2 (Feb., 1981), pp. 193-194 (2 pages) In [5] Swarup states a group theoretic conjecture P2 and shows that $\mathrm{P1} \Rightarrow ...

One of the oldest and simplest problems in geometry has caught mathematicians off guard—and not for the first time. Since antiquity, artists and geometers have wondered how shapes can tile the entire ...

The Kaplansky conjectures are three long-standing open problems on the group rings of torsion-free groups. Last week, Dr. Giles Gardam, postdoc in Mathematics Münster's topology group, announced that ...

The grandest project in mathematics has received a rare gift, in the form of a mammoth 350-page paper posted in February that will change the way researchers around the world investigate some of the ...

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 imaging ...

To the surprise of experts in the field, a postdoctoral statistician has solved one of the most important problems in high-dimensional convex geometry. In the mid-1980s, the mathematician Jean ...

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. “At some ...