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.

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 ...

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 ...

It’s an educated guess, not a proof. But a good conjecture will guide math forward, pointing the way into the mathematical unknown. Mountain climbing is a beloved metaphor for mathematical research.

Abstract: I will introduce the notion of cobordism and explain how Morse functions extend to them. I will then give a brief sketch of the h-cobordism theorem, its limitations and explain how the ...

Alexandru Ghitza receives funding from the Australian Research Council. A couple of months ago, Japanese mathematician Shinichi Mochizuki posted the latest in a series of four papers claiming the ...

Abstract: We prove some structure results for transversely reducible Sasaki manifolds. In particular, we show a Sasaki manifold with positive Ricci curvature is transversely irreducible, and so join ...