The complexity of the normal surface solution space

Normal surface theory is a central tool in algorithmic three-dimensional topology, and the enumeration of vertex normal surfaces is the computational bottleneck in many important algorithms. However, it is not well understood how the number of such surfaces grows in relation to the size of the underlying triangulation. Here we address this problem in both theory and practice. In theory, we tighten the exponential upper bound substantially; furthermore, we construct pathological triangulations that prove an exponential bound to be unavoidable. In practice, we undertake a comprehensive analysis of millions of triangulations and find that in general the number of vertex normal surfaces is remarkably small, with strong evidence that our pathological triangulations may in fact be the worst case scenarios. This analysis is the first of its kind, and the striking behaviour that we observe has important implications for the feasibility of topological algorithms in three dimensions.Comment: Extended abstract (i.e., conference-style), 14 pages, 8 figures, 2 tables; v2: added minor clarification

Investigating the Role of FGF8 Signaling in Neurogenesis of the Developing Zebrafish Eye

In the embryonic zebrafish, the fibroblast growth factor 8a (FGF8) signaling network is essential for proper development and maintenance of retinal ganglion cells (RGCs) as well as the hyaloid vasculature, the vessels that supply the eye with nutrients during development. Disruption of FGF8 signaling via knock down of FGF8 or pharmacologic inhibition of FGF receptors (FGFRs) results in extensive abnormalities throughout the developing eye. Our preliminary data indicated that in developing zebrafish, mRNA expression of fgf8a is present exclusively in the RGCs, while the fibroblast growth factor receptor 1 (fgfr1b) is expressed exclusively in the area of the hyaloid vasculature. These results led us to hypothesize that FGF8 signals from the RGCs to the vasculature of the developing eye, and that this signaling network is essential for proper eye development. In order to test this hypothesis, we demonstrated the ability to detect downstream phosphorylation events in response to acute FGF8 stimulation in cells that expressed FGFR1 using Western blot and immunofluorescence (IF). Next, we established a zebrafish eye explant culture system to treat the cells of the developing zebrafish eye in vitro. Using transgenic zebrafish lines expressing green fluorescent protein (GFP) tags in either the differentiating RGCs or the vascular cells of the eye, we attempted to identify the specific cells capable of responding to FGF8. Our data indicate that recombinant FGF8 is capable of activating detectable intracellular signaling cascades, such as ERK phosphorylation, in cultured endothelial cells. Furthermore, FGF8 is capable of inducing signaling in some of the cells from the developing zebrafish eye, but not in the RGCs. These findings support our proposed model in which FGF8 signals from the RGCs to the hyaloid vasculature, resulting in the activation of signaling pathways that are necessary for proper development of the hyaloid vasculature and RGCs

A duplicate pair in the SnapPea census

We identify a duplicate pair in the well-known Callahan-Hildebrand-Weeks census of cusped finite-volume hyperbolic 3-manifolds. Specifically, the six-tetrahedron non-orientable manifolds x101 and x103 are homeomorphic.Comment: 5 pages, 3 figures; v2: minor edits. To appear in Experimental Mathematic

Optimizing the double description method for normal surface enumeration

Many key algorithms in 3-manifold topology involve the enumeration of normal surfaces, which is based upon the double description method for finding the vertices of a convex polytope. Typically we are only interested in a small subset of these vertices, thus opening the way for substantial optimization. Here we give an account of the vertex enumeration problem as it applies to normal surfaces, and present new optimizations that yield strong improvements in both running time and memory consumption. The resulting algorithms are tested using the freely available software package Regina.Comment: 27 pages, 12 figures; v2: Removed the 3^n bound from Section 3.3, fixed the projective equation in Lemma 4.4, clarified "most triangulations" in the introduction to section 5; v3: replace -ise with -ize for Mathematics of Computation (note that this changes the title of the paper

The Anatomy of Start-Stop Growth

This paper investigates the remarkable extremes of growth experiences within countries and examines the changes that occur when growth starts and stops. We find three main results. First, all but the very richest countries experience both growth miracles and failures over substantial periods. Second, growth accounting reveals that physical capital accumulation plays a negligible role in growth take-offs and a larger but still modest role in growth collapses. The implied role of productivity in these shifts is also directly reflected in employment reallocations and changes in trade. Third, growth accelerations and collapses are asymmetric phenomena. Collapses typically feature reduced manufacturing and investment amidst increasing price instability, whereas growth takeoffs are primarily associated with large and steady expansions in international trade. This asymmetry suggests that the roads into and out of rapid growth expansions may not be the same. The results stand in contrast to much growth theory and conventional wisdom: despite much talk of poverty traps, even very poor countries regularly grow rapidly, and the role of aggregate investment in growth accelerations is negligible.

Hit or Miss? The Effect of Assassinations on Institutions and War

Assassinations are a persistent feature of the political landscape. Using a new data set of assassination attempts on all world leaders from 1875 to 2004, we exploit inherent randomness in the success or failure of assassination attempts to identify assassination's effects. We find that, on average, successful assassinations of autocrats produce sustained moves toward democracy. We also find that assassinations affect the intensity of small-scale conflicts. The results document a contemporary source of institutional change, inform theories of conflict, and show that small sources of randomness can have a pronounced effect on history.