My research is multidisciplinary and has found its way as publications into journals ranging from those for biologists to nanotechnologists to mathematicians to computer scientists. I model organisms as 3D surfaces, whether extant or extinct, based on geometry, using deterministic mathematics involving parametric equations, implicit functions, differential geometry, and vector and tensor calculus. Numerical representation of 3D geometry is achieved, and these results are used in, for example, multivariate morphospace analysis. I approach taxonomy and systematics from the perspective of algebraic and differential topology with the infusion of fuzzy set and fuzzy measure theories. I have developed applications for use in ecology, ecological informatics, and more broadly, environmental sciences as well. In collaboration with Dan Miller (Invertebrate fossils Collections Manager), we are developing applications and analyses of statistical representation of 3D surfaces of organisms ranging in size from diatoms to ostracods to mollusks and beyond. Museum collections are highly useful in modeling whether deterministic or statistical analyses are used.
With reference to biology and paleobiology, my research has been centered on diatoms which I have studied for over 20 years. Many of my studies involve Great Lakes diatoms. In addition, I have collected diatoms from Neogene deposits in Heraklion, Crete for study. Other microfossils in these deposits are of interest including silicoflagellates and radiolarians as well as the presence of sponge spicules. Middle Devonian ostracods from the Silica Formation in the Michigan Basin are another source of interest. We have extensive museum collections at U-M that facilitate my research in this area. I have developed 2D and am in the process of developing 3D Zernike polynomials to model ostracod carapaces in studying their morphological evolution. In general, I approach the biological and paleobiological sciences from a mathematical perspective with the intention of developing new ways to acquire and analyze data with respect to evolutionary patterns exhibited by past and present life on Earth.
I am also interested in studying whether diatoms have undergone stasis since the Late Cretaceous. Heavily silicified, extravagantly ornamented centric diatoms became extinct just after the Albian as less silicified, less ornamented centric and pennate diatoms diversified in the Early Cretaceous. Many of the same forms continue to flourish in the Recent. Rapid speciation in marine taxa occurred as a result of a cooling trend during the Cenozoic, but subsequently, diversification declined and remains at the same level today. Stasis may have occurred only with regard to particular taxa or among a group of taxa during distinct time intervals. Morphological variation via modeled 3D diatom forms will be used in simulations to assess diatom evolution and stasis.
Ongoing studies on microfossils include Middle Devonian ostracods from the Silica Formation in the Michigan Basin. Ostracods are used as indicator organisms for reconstructing paleoenvironmental and paleoclimatic conditions, since they have a calcified carapace, are well-preserved, and provide an almost continuous fossil record throughout the Phanerozoic. The Late Devonian extinction event at the Frasnian-Famennian boundary can be studied in terms of morphological evolution of ostracods and the environmental conditions. Ostracod carapace 3D surfaces are modeled using deterministic methods involving parametric equations and solutions to the geometric representation of the ostracod forms. 3D digital point clouds of ostracod carapaces are used to represent surface morphology and analyzed statistically. 3D digital methods and statistical analysis are conducted with Dan Miller, invertebrate fossils collections manager, based on his research in this area. In both cases, morphospace analysis can be used to assess disparity before and after the extinction event either by using simulations with ostracod models or with statistical analysis of point cloud data.