The Gross-Kudla-Schoen modified diagonal cycle on the triple product of the modular curve with itself provides a wealth of arithmetic information about modular forms, including derivatives of complex L-functions, special values of p-adic L-functions, and points on elliptic curves, known as Chow-Heegner points. In this talk, I will discuss a formula expressing the p-adic logarithm of a Chow-Heegner point in terms of the coefficients of the ordinary projection of certain p-adic modular forms.

We prove a B-SD conjecture for elliptic curves (for the p^infinity Selmer groups with arbitrary rank) a la Mazur-Tate and Darmon in anti-cyclotomic setting, for certain primes p. This is done, among other things, by proving a conjecture of Kolyvagin in 1991 on p-indivisibility of (derived) Heegner points over ring class fields. Some applications follow, for example, the p-part of the refined B-SD conjecture in the rank one case.

Mandatory for 298 Grad Seminar students who have been a TA for less than 2 quarters at UCI

Mandatory for 298 Grad Seminar students who have been a TA for less than 2 quarters at UCI

The L2 norm of the Riemannian curvature tensor is a natural energy to associate to a Riemannian manifold, especially in dimension 4.  A natural path for understanding the structure of this functional and its minimizers is via its gradient flow, the "L2 flow."  This is a quasi-linear fourth order parabolic equation for a Riemannian metric, which one might hope shares behavior in common with the Yang-Mills flow.  We verify this idea by exhibiting structural results for finite time singularities of this flow resembling results on Yang-Mills flow.  We also exhibit a new short-time existence statement for the flow exhibiting a lower bound for the existence time purely in terms of a measure of the volume growth of the initial data.  As corollaries we establish new compactness and diffeomorphism finiteness theorems for four-manifolds generalizing known results to ones with have effectively minimal hypotheses/dependencies.  These results all rely on a new technique for controlling the growth of distances along a geometric flow, which is especially well-suited to the L2 flow.