Geometry of Morphogenesis

We introduce a formalism for the geometry of eukaryotic cells and organisms.Cells are taken to be star-convex with good biological reason. This allows for a convenient description of their extent in space as well as all manner of cell surface gradients. We assume that a spectrum of such cell surface markers determines an epigenetic code for organism shape. The union of cells in space at a moment in time is by definition the organism taken as a metric subspace of Euclidean space, which can be further equipped with an arbitrary measure. Each cell determines a point in space thus assigning a finite configuration of distinct points in space to an organism, and a bundle over this configuration space is introduced with fiber a Hilbert space recording specific epigenetic data. On this bundle, a Lagrangian formulation of morphogenetic dynamics is proposed based on Gromov-Hausdorff distance which at once describes both embryo development and regenerative growth

Exploring the Potential Diversity of Early Type Ia Supernova Light Curves

During the first several days after explosion, Type Ia supernova light curves probe the outer layers of the exploding star and therefore provide important clues for identifying their progenitors. We investigate how both the shallow $^{56}$Ni distribution and the presence of circumstellar material shape these early light curves. This is performed using a series of numerical experiments with parameterized properties for systematic exploration. Although not all of the considered models may be realized in nature (and indeed there are arguments why some of them should not occur), the spirit of this work is to provide a broader exploration of the diversity of possibilities. We find that shallower $^{56}$Ni leads to steeper, bluer light curves. Differences in the shape of the rise can introduce errors in estimating the explosion time and thus impact efforts to infer upper limits on the progenitor or companion radius from a lack of observed shock cooling emission. Circumstellar material can lead to significant luminosity during the first few days, but its presence can be difficult to identify depending on the degree of nickel mixing. In some cases, the hot emission of circumstellar material may even lead to a signature similar to interaction with a companion, and thus in the future additional diagnostics should be gathered for properly assessing early light curves.Comment: Revised version with additional figures and discussions. 8 pages, 15 figures, accepted for publication in The Astrophysical Journa