Bridges Between Subriemannian Geometry and Algebraic Geometry

We consider how the problem of determining normal forms for a specific class of nonholonomic systems leads to various interesting and concrete bridges between two apparently unrelated themes. Various ideas that traditionally pertain to the field of algebraic geometry emerge here organically in an attempt to elucidate the geometric structures underlying a large class of nonholonomic distributions known as Goursat constraints. Among our new results is a regularization theorem for curves stated and proved using tools exclusively from nonholonomic geometry, and a computation of topological invariants that answer a question on the global topology of our classifying space. Last but not least we present for the first time some experimental results connecting the discrete invariants of nonholonomic plane fields such as the RVT code and the Milnor number of complex plane algebraic curves.Comment: 10 pages, 2 figures, Proceedings of 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Madrid 201

Complete spelling rules for the Monster tower over three-space

The Monster tower, also known as the Semple tower, is a sequence of manifolds with distributions of interest to both differential and algebraic geometers. Each manifold is a projective bundle over the previous. Moreover, each level is a fiber compactified jet bundle equipped with an action of finite jets of the diffeomorphism group. There is a correspondence between points in the tower and curves in the base manifold. These points admit a stratification which can be encoded by a word called the RVT code. Here, we derive the spelling rules for these words in the case of a three dimensional base. That is, we determine precisely which words are realized by points in the tower. To this end, we study the incidence relations between certain subtowers, called Baby Monsters, and present a general method for determining the level at which each Baby Monster is born. Here, we focus on the case where the base manifold is three dimensional, but all the methods presented generalize to bases of arbitrary dimension.Comment: 14 pages, 4 figures; new titl

A Monster Tower Approach to Goursat Multi-Flags

We consider here the problem of classifying orbits of an action of the dif- feomorphism group of 3-space on a tower of fibrations with P2-fibers that generalize the Monster Tower due to Montgomery and Zhitomirskii. As a corollary we give the first steps towards the problem of classifying Goursat 2-flags of small length. In short, we classify the orbits within the first four levels of the Monster Tower and show that there is a total of 34 orbits at the fourth level in the tower.Comment: Documents has 30 pages, and it contains 7 figures. We have also included two appendices detailing some computations which appeared in the main tex

TOPkit: An Online Faculty Development Resource

Faculty preparation is a critical success factor for any program that involves online teaching and learning (Kane, Shaw, & Pang, 2016; Merillat & Scheibmeir, 2016; Parker, Maor, & Herrington, 2013). Although experts in their respective fields, faculty may not have studied online pedagogy or instructional design before stepping into the onsite or online classroom. While few educators would argue against offering faculty development, institutions grapple with limited funding and campus resources (e.g., limited space, scheduling constraints) for faculty training. To pool resources for this purpose, the State University System of Florida Board of Governors committed to sponsoring a program to develop an online toolkit and annual workshop to support faculty development efforts statewide. The result was The Teaching Online Preparation Toolkit (TOPkit), an open educational resource (website and two free, customizable online faculty development courses) that faculty developers anywhere can use. This article describes the online toolkit and specifically highlights the toolkit’s pair of sample courses that may be downloaded and customized

Facilitating Resolution of Land Disputes: Applying Information Technology to Store and Access Historical Land Assignment Records

The lack of a reliable system to store and easily access historical documents regarding land assignments has led to many land disputes within the Jemez Pueblo of New Mexico. In this project we designed and developed an automated system, sensitive to the unique culture of the Pueblo, to facilitate resolution of these land disputes

Neuregulin-4 contributes to the establishment of cutaneous sensory innervation

Recent work has shown that neuregulin‐4 (NRG4) is a physiological regulator of the growth of sympathetic axons and CNS dendrites in the developing nervous system. Here, we have investigated whether NRG4 plays a role in sensory axon growth and the establishment of cutaneous sensory innervation. Imaging early nerve fibers in the well‐characterized cutaneous trigeminal territory, the brachial plexus, and thorax revealed very marked and highly significant decreases in nerve fiber length and branching density in Nrg4−/− embryos compared with Nrg4+/+ littermates. NRG4 promoted neurotrophin‐independent sensory axon growth from correspondingly early trigeminal ganglion and DRG neurons in culture but not from enteroceptive nodose ganglion neurons. High levels of Nrg4 mRNA were detected in cutaneous tissues but not in sensory ganglia. Our findings suggest that NRG4 is an important target‐derived factor that participates in the establishment of early cutaneous sensory innervation