(joint with Lipnowski, Sawin) The class group is a natural abelian group one can associated to a number field, and it is natural to ask how it varies in families. Cohen and Lenstra famously proposed a model for families of quadratic fields based on random matrices of large rank, and this was later generalized by Cohen-Martinet. However, their model was observed by Malle to have issues when the base field contains roots of unity.
(joint with Bhargava,Shankar,Taniguchi,Thorne, and Zhao) Zhang’s conjecture asserts that for fixed positive integers m, n, the size of the m-torsion in the class group of a degree n number field is smaller than any power of the discriminant. In all but a handful of cases, the best known result towards this conjecture is the ”convex” bound given by the Brauer-Siegel Theorem. We make progress on this conjecture by giving a”subconvex” bound on the size of the 2-torsion of the class group of a number field in terms of its discriminant, for any value of n.
Number fields are fields which are finite extensions of Q. They come with a canonical invariant called the discriminant, which can be thought of as the volume of a certain canonically associated lattice. While these objects are central to modern number theory, it turns out that counting them is extremely difficult. More precisely, what is the asymptotic behavior of N (n,X) the number of degree n field extensions of Q with discriminant at most X as X grows, while n remains fixed?
A tropical variety is a piecewise linear object that (sometimes) appears as a shadow of an algebraic variety. In this talk, a gentle introduction to the Chow homology and cohomology of tropical varieties will be given. Two family of examples of combinatorial nature will be emphasized: the Stanley-Reisner ring of the boundary of a simplicial polytope, modulo linear system of parameters, and the Chow ring of a matroid.