The affine Plateau problem

In this paper, we study a second order variational problem for locally convex hypersurfaces, which is the affine invariant analogue of the classical Plateau problem for minimal surfaces. We prove existence, regularity and uniqueness results for hypersurfaces maximizing affine area under appropriate boundary conditions.

     

Minimal volume with length bounded below

We study a broad class of problems where volume is minimized among metrics on a smooth, compact Riemannian manifold that keep the length of a fixed set of curves bounded below. They can be seen as a generalization of isosystolic inequalities. Necessary and sufficient conditions are given for continuous minima in a conformal class and necessary conditions are given for local minima.     

The research of thirteen students at Howard University

Results from, and problems related to, the dissertations of my Ph.D. students are discussed.     

Introduction of bis-discotic and bis-calamitic mesogenic addends to C 60

The synthesis and characterization of four C 60 Bingel cyclopropanation adducts incorporating bis-biphenylene (three adducts) and bis-triphenylene (one adduct) moieties are described. The thermal analysis (POM and DSC) of these materials reveals that they are not liquid crystalline. However, two of the precursor bis-biphenylene malonate esters possess monotropic mesophases. Furthermore, each of the corresponding C 60 adducts is miscible in the melts of the precursor malonate ester, and at low dopings, retains the liquid crystalline monotropic mesophases of the precursor.     

Synthesis and thiol–ene photopolymerization of allyl‐ether functionalized dendrimers

Well‐defined, allyl‐ether functional, first‐generation dendrimers have been synthesized. The convergent growth approach was utilized, using the anhydride of the allyl‐ether terminated building block. Three different core moieties were used: trimethylolpropane, trisphenol, and ditrimethylolpropane. The coupling reactions proceeded in good yields and all compounds were characterized by NMR, MALDI‐TOF, and SEC. The allyl‐terminated dendrimers were crosslinked by thiol–ene chemistry, using a multifunctional thiol, TriThiol, to give clear and smooth films. The photopolymerization was conducted in the presence of a photoinitiator, Irgacure 651, and no traces of either allyl‐ether groups or thiols were observed by FT‐Raman after cure. All crosslinked films were characterized with respect to mechanical (DMA) and thermal (DSC) properties. It was found that homogeneous networks were formed and that the core functionality and structure had little effect on the network properties. © 2008 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 46: 1339–1348, 2008     

A type 2 Ferrier rearrangement-based synthesis of d-myo-inositol 1,4,5-trisphosphate

The synthesis of d-myo-inositol 1,4,5-trisphosphate (InsP₃) from methyl α-d-glucopyranose, via a type 2 Ferrier rearrangement is reported. A key intermediate in this synthesis possesses orthogonal protecting groups at the 1-, 4- and 5-position, making it a versatile starting point for the synthesis of unnatural InsP₃ derivatives. Biological evaluation of the synthetic InsP₃ demonstrates that this compound evokes selective Ca²⁺ release via activation of InsP₃ receptors.     

An exact solution of Einstein's equation with a dilaton field

An exact solution of Einstein gravity coupled to a dilaton field is found. The solution is conformally flat and is invariant under Lorentz transformations. The singularities and conformal structure of the metric are examined.