The major obstacle in mathematical modeling in science that is also responsible for the
variety of different phenomena appearing is its nonlinear nature.
Richard Kollar's field of research is nonlinear analysis that includes mathematical modeling and the study of existence and stability of coherent structures
as nonlinear waves, vortices, and defects, appearing in models ranging from nonlinear optics, or condensed matter physics to chemical processes in human brain. The value of these problems lies not only in their far-reaching consequences
for applications, but also in the interesting mathematics underlying them.
The goal of the three projects in this proposal is to gain insight by studying interesting particular applied problems, and apply it to build and simplify the general theory.
The goal of the first project is to study heat damage of cells, particularly during burn injuries and hyperthermic cancer treatments. Based on his current research, Kollar proposes to extend his mathematical model to include important effects as increased vascular permeability or three-dimensional nonhomogeneous environment.
In the second project Kollar, in collaboration with R. Pego, B. Deconinck and N. Kutz, studies stability of certain nonlinear waves. Besides other investigations it requires an extension of the Evans function technique for detection of unstable eigenvalues to three-dimensional and non-local problems.
The third project, in collaboration with P. Miller, proposes to use Krein signature and Pontryagin spaces in the study of inverse scattering-spectral problems. The idea discussed in the proposal is to use Krein signature to restrict the position of spectra for potentials satisfying a single-lobe condition introduced by Klaus and Shaw.
A prominent common feature of this proposal is a very novel approach to classical problems and the unification of different theories.
Fields of science
Call for proposal
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