My research concerns asymptotic analysis and homogenization of partial differential operators. The problems originate often in mathematical physics and describe phenomena like electromagnetic wave scattering on small particles and heat transfer.
In the classical homogenization theory, one studies mixtures of materials with different properties. Such mixtures might have two scales: a microscopic scale describing the microstructure of the heterogeneous material and a macroscopic scale representing the size of the material. The homogenization theory provides tools to rigorously substitute the heterogeneous material with microstructure by a new homogeneous material. The properties of the new homogeneous material are called effective (homogenized) and the resulting equations are often easier to solve than the original equations. One well-known example is a derivation of the Darcy law for transport in porous media from Navier-Stokes equations.
Now I am interested in signal propagation in neurons, and I am working on the derivation of a 1D nonlinear cable equation from a 3D-model based on the Hodgkin-Huxley model of current flow through ionic channels in neural membrane.
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