We present single-cell clustering using bifurcation analysis (SCUBA), a novel computational method for extracting lineage relationships from single-cell gene expression data and modeling the dynamic changes associated with cell differentiation. SCUBA draws techniques from nonlinear dynamics and stochastic differential equation theories, providing a systematic framework for modeling complex processes involving multilineage specifications. By applying SCUBA to analyze two complementary, publicly available datasets we successfully reconstructed the cellular hierarchy during early development of mouse embryos, modeled the dynamic changes in gene expression patterns, and predicted the effects of perturbing key transcriptional regulators on inducing lineage biases. The results were robust with respect to experimental platform differences between RT-PCR and RNA sequencing. We selectively tested our predictions in Nanog mutants and found good agreement between SCUBA predictions and the experimental data. We further extended the utility of SCUBA by developing a method to reconstruct missing temporal-order information from a typical single-cell dataset. Analysis of a hematopoietic dataset suggests that our method is effective for reconstructing gene expression dynamics during human B-cell development. In summary, SCUBA provides a useful single-cell data analysis tool that is well-suited for the investigation of developmental processes.single cell | gene expression | bifurcation | differentiation

We develop numerical methods for approximating Ricci flat metrics on CalabiYau hypersurfaces in projective spaces. Our approach is based on finding balanced metrics, and builds on recent theoretical work by Donaldson. We illustrate our methods in detail for a one parameter family of quintics. We also suggest several ways to extend our results.

We develop an iterative method for finding solutions to the hermitian Yang-Mills equation on stable holomorphic vector bundles, following ideas recently developed by Donaldson. As illustrations, we construct numerically the hermitian Einstein metrics on the tangent bundle and a rank three vector bundle on P 2 . In addition, we find a hermitian Yang-Mills connection on a stable rank three vector bundle on the Fermat quintic.

We analyze the link between the occurrence of massless B-type D-branes for specific values of moduli and monodromy around such points in the moduli space. This allows us to propose a classification of all massless B-type D-branes at any point in the moduli space of Calabi-Yau's. This classification then justifies a previous conjecture due to Horja for the general form of monodromy. Our analysis is based on using monodromies around points in moduli space where a single D-brane becomes massless to generate monodromies around points where an infinite number become massless. We discuss the various possibilities within the classification.