We review first Azumaya geometry and D-branes in the realm of algebraic geometry along the line of Polchinski-Grothendieck Ansatz from our earlier work and then use it as background to introduce Azumaya C^{\infty} -manifolds with a fundamental module and morphisms therefrom to a projective complex manifold. This gives us a description of D-branes of A-type. Donaldson's picture of Lagrangian and special Lagrangian submanifolds as selected from the zero-locus of a moment map on a related space of maps can be merged into the setting. As a pedagogical toy model, we study D-branes of A-type in a Calabi-Yau torus. Simple as it is, it reveals several features of D-branes, including their assembling/disassembling. The 4th theme of Sec. 2.4, the 2nd theme of Sec. 4.2, and Sec. 4.3 are to be read respectively with Gmez-Sharpe (arXiv: hep-th/0008150), Donagi-Katz-Sharpe (arXiv: hep-th/0309270), and Denef (arXiv: hep-th/0107152). Some string-theoretical remarks are given at the end of each section.

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.

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.

We study SYZ mirror symmetry in the context of non-Khler CalabiYau manifolds. In particular, we study the six-dimensional Type II supersymmetric <i>SU</i>(3) systems with RamondRamond fluxes, and generalize them to higher dimensions. We show that FourierMukai transform provides the mirror map between these Type IIA and Type IIB supersymmetric systems in the semi-flat setting. This is concretely exhibited by nilmanifolds.