SMS 2014 - Overview
Counting objects of arithmetic interest (such as quadratic forms, number fields, elliptic curves, curves of a given genus, ...) in order of increasing arithmetic complexity, is among the most fundamental enterprises in number theory, going back (at least) to the fundamental work of Gauss on composition of binary quadratic forms and class groups of quadratic fields.
In the past decade tremendous progress has been achieved, notably through Bhargava's revolutionary program blending elegant algebraic techniques with powerful analytic ideas. It suffices to mention the striking upper bounds on the size of Selmer groups (and therefore ranks) of elliptic curves and even Jacobians of hyperelliptic curves of higher genus, among the many other breakthroughs that have grown out of this remarkable circle of ideas.
The 2014 Summer School will be devoted to covering these recent developments, with the objective of attracting researchers who are in the early stages of their career into this active and rapidly developing part of number theory.