Project Details
Projekt
GRK 1294: Analysis, Simulation and Design of Nanotechnological Processes
Subject Area
Mathematics
Term
from 2006 to 2015
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 12406856
Nanotechnology is a key technology of the 21st century. It deals with material structures of the size of one up to hundred nanometers, that is, sizes of large virus, small bacteria or the wavelength of light. These structures become especially important when they show properties not known from usual materials.
The aim of this Research Training Group is the analysis, simulation and design of nanotechnological processes from a mathematical point of view. Every project will concentrate on a mathematical aspect of a problem in nanotechnology in cooperation with a scientist in physics or engineering. In this way it is guaranteed that the projects are relevant for applications.
The projects in the first period are mainly motivated by problems in nonlinear optics. In particular, photonic crystals and ultra-short pulses will be investigated. For example, photonic crystals, which are used for the transmission of light, exhibit bandgaps. Bandgaps are also found in semi-conductors, in which charges are transported. Hopefully, this will result in applications of similar importance. In the analysis and simulation of the partial differential equations that describe nanostructures several mathematical difficulties arise, e.g., the treatment of nonlinearities, simultaneous occurrence of continuous and discrete spectrum, problems with multiple scales, and the ill-posedness of these problems. In our examples we hope to find Maxwell equations and Nonlinear Schrödinger equations in different geometries. The techniques come from analysis, scientific computing, the theory of inverse problems, for example, multiscale-analysis, Fourier- and Blochwave-analysis, adaptive finite elements, wavelet-collocation and direct and inverse scattering theory.
The aim of this Research Training Group is the analysis, simulation and design of nanotechnological processes from a mathematical point of view. Every project will concentrate on a mathematical aspect of a problem in nanotechnology in cooperation with a scientist in physics or engineering. In this way it is guaranteed that the projects are relevant for applications.
The projects in the first period are mainly motivated by problems in nonlinear optics. In particular, photonic crystals and ultra-short pulses will be investigated. For example, photonic crystals, which are used for the transmission of light, exhibit bandgaps. Bandgaps are also found in semi-conductors, in which charges are transported. Hopefully, this will result in applications of similar importance. In the analysis and simulation of the partial differential equations that describe nanostructures several mathematical difficulties arise, e.g., the treatment of nonlinearities, simultaneous occurrence of continuous and discrete spectrum, problems with multiple scales, and the ill-posedness of these problems. In our examples we hope to find Maxwell equations and Nonlinear Schrödinger equations in different geometries. The techniques come from analysis, scientific computing, the theory of inverse problems, for example, multiscale-analysis, Fourier- and Blochwave-analysis, adaptive finite elements, wavelet-collocation and direct and inverse scattering theory.
DFG Programme
Research Training Groups
Applicant Institution
Karlsruher Institut für Technologie
Spokespersons
Professor Dr. Willy Dörfler, from 4/2006 until 2/2011; Professorin Dr. Marlis Hochbruck, since 2/2011
