We define a notion of stability for chiral ring of four dimensional N= 1 theory by introducing test chiral rings and generalized a maximization. We conjecture that a chiral ring is the chiral ring of a superconformal field theory if and only if it is stable. We then study N= 1 field theory derived from D3 branes probing a three-fold singularity X, and show that the K stability which implies the existence of Ricci-flat conic metric on X is equivalent to the stability of chiral ring of the corresponding field theory.

In this note, we prove the sharp DaviesGaffneyGrigoryan Lemma for minimal heat kernels on graphs.

In this paper, we provide existence and estimates of the equation du A v^du i= i

We consider a static, spherically symmetric system of a Dirac particle in a classical gravitational and SU (2) Yang-Mills field. We prove that the only black-hole solutions of the corresponding Einstein-Dirac-Yang/Mills equations are the Bartnik-McKinnon black-hole solutions of the SU (2) Einstein-Yang/Mills equations; thus the spinors must vanish identically. This indicates that the Dirac particles must either disappear into the black-hole or escape to infinity.

The original motivation for solving of the Einstein equation is to understand space-time in the absence of matter; see the essay by Tod in this volume for an overview of the mathematics of general relativity. The equation governs the manner in which space-time is inuenced by the sole force of gravity. As is well known, singularities such as black holes can occur. The study of the vacuum Einstein equations is a difficult problem in nonlinear hyperbolic systems; see the essay by Christodoulou in this volume. Complete space-times with nontrivial gravitational radiation are not well understood.