Moduli Spaces in Algebraic Geometry

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Arithmetic geometry is an important branch of mathematics that lies at the intersection of algebra, geometry, and number theory. The applications of the field are numerous, ranging from coding theory to cryptography to string theory. My research program is centered around several long-standing conjectures.***The first of these is the Batyrev-Manin conjecture. It predicts in a precise way "how many" rational solutions there are to certain systems of polynomial equations. Although the conjecture is known in only a handful of cases, this far reaching statement greatly guides our thinking as to how rational solutions should behave in a wide level of generality. My program aims to extend the cases where this conjecture is known.***A second conjecture concerns Hilbert schemes. These objects play a fundamental role in algebraic geometry, as they are often used to show the existence of other nice spaces, called moduli spaces. A longstanding problem within the field is to give a "rule" (moduli interpretation) that governs which algebras are on the main component of the Hilbert scheme of points. One of the biggest anticipated impacts of my research program for the mathematical community is the introduction of new tools with which to study the Hilbert scheme of points.***The two main objects featured in my research program are stacks and Galois closures of ring extensions, as introduced by myself and Manjul Bhargava. Stacks are generalizations of the orbifolds showing up in theoretical physics. They are spaces equipped with additional structure encoding symmetries. One aspect of my research is the use of stacks in answering questions about ordinary spaces (ones with no additional stacky structure). Galois closures of ring extensions are highly accessible objects, and so lend themselves to projects at all levels. Accordingly, my research program will serve Canada through the training of highly qualified personnel.**