Formal definition of the Hodge decomposition of a smooth projective variety $X$. Definition of an algebraic cycle $Z$ and the correspondence to its cohomology class ...

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.

Research on Riemann surfaces in the proof of the BSD conjecture. Properties of the motive of the Jacobian of algebraic curves in the proof of the BSD conjecture. Geometry of the moduli space of ...

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 ...

Since the conjecture was already known to be true in one and two dimensions, they sought to prove it in three: to show that if you can shift copies of one shape to tile all of three-dimensional space, ...

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 ...

We classify simply connected compact Sasaki manifolds of dimension 2n+1 with positive transverse bisectional curvature. In particular, the Kähler cone corresponding to such manifolds must be ...