I am interested in all areas of applied mathematics, but my current research is
in numerical methods and mathematical modeling of biological systems. A list of publications with links can be found here.
dissertation research involved a generalized version of the
model, a highly nonlinear system of partial differential equations that
describes the fascinating
biological phenomenon of chemotaxis
with my advisors Jay Gopalakrishnan and Patrick De Leenheer, I have
developed theoretical results on a generalized Keller-Segel
model that suggest chemotaxis as a means of initiation of pattern
formation. A paper on the results can be found here
I also programmed finite element method solutions (both standard and
non-standard discretizations) to find numerical stationary solutions
for the Keller-Segel model. Time simulations indicate that
these stationary solutions are stable, and therefore would be expressed
in nature. Some steady state solutions and time simulations are shown
below, and results have been published here
result of my dissertation research is the development of a novel
computational algorithm for identifying potential patterns that may be
seen in solutions to nonlinear partial differential equations.
Some of the patterns visualized using our method of spectral bands can
be seen below.
Growth and Remodeling
The bone in our bodies is constantly
being destroyed and rebuilt in order to repair microfractures and keep
our bones strong. Some of my more recent research has involved
questions surrounding the mathematical modeling of this process. One
question I have been investigating is pattern formation in the bones of
patients with the rare bone disorder idiopathic hyperphosphatasia.
Patients with this disorder have bones which take on a paralell plate
pattern, as opposed to healthy children which have more of a honey comb
pattern (see Pivonka
). Preliminary results of the application of the method
of spectral bands indicate the the production level of the protein
osteoprotegerin (OPG) may affect the patterning of bone tissue in this
manner as seen below. Each row shows the patterning of the bone at a
different level of OPG production, and each column corresponds to a different point in time.