In this note, we give a list of Calabi-Yau hypersurfaces in weighted projective 4-spaces with the property that a hypersurface contains naturally a pencil of K3 variety. For completeness we also obtain a similar list in the case K3 hypersurfaces in weighted projective 3-spaces. The first list significantly enlarges the list of K3-fibrations of\KlemmLercheMayr~ which has been obtained on some assumptions on the weights. Our lists are expected to correspond to examples of the so-called heterotic-type II duality\KachruVafa\KachruSilverstein.

We explain how Polchinski's work on D-branes re-read from a noncommutative version of Grothendieck's equivalence of local geometries and function rings gives rise to an intrinsic prototype definition of D-branes (of B-type) as an Azumaya-type noncommutative space. Several originally open-string induced properties of D-branes can be reproduced solely by this intrinsic definition. We study also the moduli space of D0-branes on a commutative target space in this setup. Some of its features resembles gas of D0-branes in Vafa's work.

This paper gives a survey of wave dynamics in the Kerr space-time geometry, the mathematical model of a rotating black hole in equilibrium. After a brief introduction to the Kerr metric, we review the separability properties of linear wave equations for fields of general spin s= 0,rac {1}{2}, 1, 2, corresponding to scalar, Dirac, electromagnetic fields and linearized gravitational waves. We give results on the long-time dynamics of Dirac and scalar waves, including decay rates for massive Dirac fields. For scalar waves, we give a rigorous treatment of superradiance and describe rigorously a mechanism of energy extraction from a rotating black hole. Finally, we discuss the open problem of linear stability of the Kerr metric and present partial results.

String theory has had a profound influence on research in Calabi-Yau spaces over the past twenty-five years. We first briefly mention some of the work in Khler Calabi-Yau manifolds that was influenced by the discovery of mirror symmetry in the late 1980s. We then discuss some of the mathematical motivations behind the recent work on non-Khler Calabi-Yau manifolds, which arise in string compactifications with fluxes. After extending mirror symmetry to non-Khler Calabi-Yau manifolds, we show how this leads to new cohomologies and invariants of non-Khler symplectic manifolds.

One great challenge of genomic research is to efficiently and accurately identify complex gene regulatory networks. The development of high-throughput technologies provides numerous experimental data such as DNA sequences, protein sequence, and RNA expression profiles makes it possible to study interactions and regulations among genes or other substance in an organism. However, it is crucial to make inference of genetic regulatory networks from gene expression profiles and protein interaction data for systems biology. This study will develop a new approach to reconstruct time delay Boolean networks as a tool for exploring biological pathways. In the inference strategy, we will compare all pairs of input genes in those basic relationships by their corresponding -scores for every output gene. Then, we will combine those consistent relationships to reveal the most probable relationship and reconstruct the genetic network. Specifically, we will prove that state transition pairs are sufficient and necessary to reconstruct the time delay Boolean network of nodes with high accuracy if the number of input genes to each gene is bounded. We also have implemented this method on simulated and empirical yeast gene expression data sets. The test results show that this proposed method is extensible for realistic networks.