Bounds on 2-torsion in class groups of number fields

In this talk, we will discuss a proof of the first known nontrivial bounds on the sizes of the 2-torsion subgroups of the class group of number fields, for all degrees.  We will also discuss applications of this result on bounds of rank of elliptic curves, bounds on the number of integral points on elliptic curves and on the number of A_4 quartic fields of bounded discriminant. This is joint work with M. Bhargava, A. Shankar, T. Taniguchi, F. Thorne and J. Tsimerman.