In the early 2000s, Daan Krammer and Stephen Bigelow independently proved that braid groups are linear. They used the Lawrence-Krammer-Bigelow (LKB) representation for generic values of its variables q and t. The $t$ variable is related to the Garside structure of the braid group used in Krammer's algebraic proof. The q variable, associated with the dual Garside structure

of the braid group, has received less attention.

In this dissertation we give a geometric interpretation of the q portion

of the LKB representation in terms of an action of the braid group on

the space of non-degenerate euclidean simplices. In our

interpretation, braid group elements act by systematically reshaping

(and relabeling) euclidean simplices. The reshapings associated to

the simple elements in the dual Garside structure of the braid group

are of an especially elementary type that we call relabeling and

rescaling.

This paper explores the commercial potential of indigenous African crops and models a supply chain that integrates small-scale cooperative farms with large-scale cultivation for agricultural export. The research addresses the existing systems' deficiencies while analyzing the prospect for global supply chain expansion for food products sourced from Africa to China. The model seeks to overcome current challenges and enable competitive advantage, as well as infrastructure and economic development.

Spon-1 and pxn-2 are essential basement membrane genes crucial to the elongation stage of C. elegans morphogenesis. However, the interactions of these proteins with other components of the basement membrane and the extracellular matrix are not very well known. In attempts to find factors that interact with spon-1 and pxn-2, EMS screens of both mutants were performed. A suppressor of spon-1 mutants, ju1185, was isolated from the spon-1 screen. We mapped this suppressor to chromosome II between -0.46 and +6.80 using two-point mapping with SNPs. Our rescue experiments suggests that this is a missense mutation in srap-1. We found that overexpression of this gene also suppresses spon-1 suggesting that this may be a gain-of-function mutation. In the pxn-2 mutant screen, a previously isolated suppressor ju1123 rescued embryonic and larval lethality, but has reduced locomotion due to defects in the synaptic release of acetylcholine. In initial observations of the mutant and suppressor nerve cords of C. elegans, we observed an abnormal separation of the dorsal nerve cord axon bundle and GABAergic synapses in commissures. With these observations, we hope to find the relationship between the separations of the nerve cord to pxn-2’s role in synaptic vesicle release. These results can potentially reveal additional roles played by these basement membrane genes, as well as discover additional genes essential in morphogenesis, applicable to the study of the epithelia in vertebrates.

The microtubule (MT) arrays in mature polarized cells comprise mostly of non-centrosomal MTs; their regulation is poorly understood. Recently, the MT minus end binding proteins CAMSAPs have been shown to be important regulators of non-centrosomal microtubule organization and dynamics. In this dissertation I describe my work on the C. elegans member of the CAMSAP family, PTRN-1, and its roles in the injury responses of two types of tissues, the neurons and the epidermis. First, I found that PTRN-1 is required for axon regeneration. Though PTRN-1 is largely dispensable for normal neuronal development, its proper regulation of neuronal microtubule dynamics is required for axons to regrow after injury. The second part of my work focuses on PTRN-1’s role in epidermal development and wound response by regulating epidermal microtubule stability. Mutations in the gene death associated protein kinase 1 (DAPK1)/dapk-1 results in both hyperactive innate immune response and enhanced wound closure in the epidermis. A genetic suppressor screen of the dapk-1 mutant identified PTRN-1 as a key player in dapk-1 mediated wound phenotypes. Further work uncovered DAPK-1 as a potential novel regulator of PTRN-1’s function in epidermal MT architecture and dynamics. My thesis work reveals new in vivo roles of PTRN-1 in regulating non-centrosomal MTs for normal tissue damage response.