Grant: Discovery Launch Supplement

Roughly speaking, number theory is the branch of mathematics concerned with the study of whole numbers. It is one of the oldest branches of mathematics, dating back at least to around 1800 BC. Algebraic geometry is a field of mathematics that is in the intersection of algebra and geometry, another two ancient branches of mathematics. Roughly speaking, algebraic geometry is concerned with geometric objects described by polynomial equations. Modern algebraic geometry has a reputation of being one of the most abstract fields of mathematics. In modern times, number theory and algebraic geometry have both found applications outside mathematics. Among other applications, number theory lies at the foundation of cryptography and cybersecurity, and algebraic geometry has deep connections to theoretical physics. Number theory and algebraic geometry have, in fact, a great deal of intersection. Not only that, concrete objects and phenomena of great interest in number theory seem to be mysteriously linked to highly abstract objects and notions in algebraic geometry. This link is the subject of some of the deepest open problems in mathematics. These include, for instance, the "Grothendieck Period conjecture", the "Hodge-Nori conjecture", and the "Beilinson-Bloch conjectures". All these problems are considered unreachable with the current state of knowledge and any progress towards them would be invaluable to the field of mathematics. Broadly speaking, the objective of this research proposal is to further advance our knowledge of the mysterious link between number theory and algebraic geometry, and develop techniques that shed light on aspects of this link. We will investigate several problems that are motivated by and take important steps towards the deep conjectures such as the ones mentioned above. The problems we will work on will be of various difficulty levels. They will be used to train researchers in different career stages, from the postdoctoral to the undergraduate level. The Highly Qualified Personnel that will be trained can pursue a vast range of careers, from academic jobs to jobs in finance, cryptography and cybersecurity (among other options).