Counting problems, from the perspective of moments

Abstract: Many questions in number theory can be phrased as counting problems. How many number fields are there? How many elliptic curves are there? How many integral solutions to this system of Diophantine equations are there? If the answer is “infinitely many,” we want to understand the order of growth for the number of objects we are counting in the “family." But in many settings we are also interested in finer-grained questions, like: how many number fields are there, with fixed degree and fixed discriminant? We know the answer is “finitely many,” but it would have important consequences if we could show the answer is always “very few indeed.” In this talk, we will describe a way that these finer-grained questions can be related to the bigger infinite-family questions. Then we will use this perspective to survey interconnections between several big open conjectures in number theory, related in particular to class groups and number fields.

number theory

Audience: researchers in the topic

London number theory seminar

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For a record of talks predating this website see: wwwf.imperial.ac.uk/~buzzard/LNTS/numbtheo_past.html