David Holcman
Incumbent of the Madeleine Haas Russel Career Development Chair
My research is divided into two different directions. The first direction
of research concerns linear and nonlinear analysis on Riemannian manifolds.
The purpose is to study how the geometry of the manifold is involved in the
behavior of solutions of partial differential equations. The class of
problems includes Yamabe's type equations, semi-classical limit of the
stationary Schrodinger equation and the influence of certain class vector
fields on the concentration phenomena.
The second direction I'm involved in concerns modeling microstructures in biological systems. Examples are trafficking of receptors in the membrane of a cell: how long it takes for a protein to travel to a specific location. Another class of problems is how to predict the function of microstructures like dendritic spines, located on the surface of the dendrite. The approach consists in deriving coarse grained equations at a cellular level starting from a molecular level. In particular I'm interested in constructing a model of synaptic plasticity, process that underlie learning and memory. The last project, I'm involved concerns retinal photoreceptors such as cones and rods: why a rod photoreceptor can detect a single photon, whereas cones cannot? In general the mathematical approach to address the previous questions strongly uses analysis of partial differential equations, asymptotic analysis, stochastic processes and differential geometry and sometimes computer simulations.
Recent Publications
- [with I. Kupka] Perturbation Methods and First Order Differential Equations on Riemannian Manifolds. Quarterly J. of Mathematics, to appear.
- [with Z. Schuss] Escape through a small opening: receptor trafficking in a synaptic membrane. J. of Statistical Physics 117 (2004).
- [with Z. Schuss] Modeling Calcium Dynamics in Dendritic Spines. SIAM of Applied Math. (in press).