In this note we discuss a recent proof of the formula for the worldsheet instanton prepotential predicted by Mirror Symmetry for the quintics in <b>P</b>⁴. One of the key ingredients in the proof is the equivariant cohomology groups on the so-called linear sigma model moduli spaces. We introduce the notion of admissible data on the equivariant cohomology groups of the linear sigma model. An admissible data may be thought of as a sequence of equivariant classes satisfying certain algebraic conditions. It arises naturally from Kontsevich's stable map compactification of moduli spaces of maps from curves into projective manifolds. The structures of admissible data help reduce many counting problems to checking certain combinatorial structure of a compactification. The mirror transformation of Candelas et al turns out to be a transformation between two admissible data associated respectively to the linear and the non-linear

This paper provides a rigorous proof of the existence of an infinite number of black hole solutions to the Einstein-Yang/Mills equations with gauge group<i>SU</i>(2), for any event horizon. It is also demonstrated that the ADM mass of each solutions is finite, and that the corresponding Einstein metric tends to the associated Schwarzschild metric at a rate 1/<i>r</i> ², as<i>r</i> tends to infinity.

Based on examples from superstring/D-brane theory since the work of Douglas and Moore on resolution of singularities of a superstring target-space Y via a D-brane probe, the richness and the complexity of the stack of punctual D0-branes on a variety, and as a guiding question, we lay down a conjecture that any resolution Y of a variety Y over Y can be factored through an embedding of Y into the stack Y of punctual D0-branes of rank Y on Y for Y in Y , where Y depends on the germ of singularities of Y . We prove that this conjecture holds for the resolution Y of a reduced singular curve Y over Y . In string-theoretical language, this says that the resolution Y of a singular curve Y always arises from an appropriate D0-brane aggregation on Y and that the rank of the Chan-Paton module of the D0-branes involved can be chosen to be arbitrarily large.

This article is concerned with the dynamics of a mixture of gases. Under the assumption that all the gases are isothermal and inviscid, we show that the governing equations have an elegant conservation-dissipation structure. With the help of this structure, a multicomponent diffusion law is derived mathematically rigorously. This clarifies a long-standing non-uniqueness issue in the field for the first time. The multicomponent diffusion law derived here takes the spatial gradient of an entropic variable as the thermodynamic forces and satisfies a nonlinear version of the Onsager reciprocal relations.

In this sequel to [L-Y1],[LLSY], and [L-Y2](respectively arXiv: 0709.1515 [math. AG], arXiv: 0809.2121 [math. AG], and arXiv: 0901.0342 [math. AG]), we study a D-brane probe on a conifold from the viewpoint of the Azumaya structure on D-branes and toric geometry. The details of how deformations and resolutions of the standard toric conifold Y can be obtained via morphisms from Azumaya points are given. This should be compared with the quantum-field-theoretic/D-brany picture of deformations and resolutions of a conifold via a D-brane probe sitting at the conifold singularity in the work of Klebanov and Witten [KW](arXiv: hep-th/9807080) and Klebanov and Strasser [KS](arXiv: hep-th/0007191). A comparison with resolutions via noncommutative desingularizations is given in the end.