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.

We consider sequences of metrics, $g_j$, on a Riemannian manifold, $M$, which converge smoothly on compact sets away from a singular set $ S \subset M$, to a metric, $g_\infty$, on $M ∖setminus S$. We prove theorems which describe when $M_j=(M,g_j) $converge in the Gromov-Hausdorff sense to the metric completion, $(M_\infty,d_\infty), of $(M∖setminus S,g_\infty)$. To obtain these theorems, we study the intrinsic flat limits of the sequences. A new method, we call hemispherical embedding, is applied to obtain explicit estimates on the Gromov-Hausdorff and Intrinsic Flat distances between Riemannian manifolds with diffeomorphic subdomains.