Randomly triangulated surfaces are versatile models for lipid bilayer vesicles, cell membranes, viral capsids, colloid-covered oil-water interfaces, etc. . Polymerized membranes are described by triangulated surfaces of fixed connectivity. Fluid membranes are characterized by
diffusion and flow within the membrane plane, and therefore have to be modelled by dynamically triangulated surfaces. Similarly, the description of crystal defects induced by thermal fluctuations or external forces requires dynamical triangulations. Two applications of triangulated surfaces are discussed in detail.
Motivated by the vesicle formation at the plasma membrane of cells by clathrin adsorption, we have studied budding processes of two-component membranes composed of crystalline and fluid states with two different spontaneous curvatures . Before a bud appears from the crystalline part of the membrane, several defects - dislocations
and disclinations - must be created and spread in the bud area. We have investigated the phase behavior, as well as the distribution and dynamics
of defects in the crystalline domains.
The flow behavior of cells and vesicles is important in many applications in biology and medicine. For example, the flow properties of blood in micro-vessels is determined by the rheological properties of the red blood cells. Furthermore, microfluidic devices have been developed recently, which allow the manipulation of small amounts of suspensions of particles
or cells. A mesoscale simulations technique for solvent hydrodynamics (multi-particle collision dynamics ) is employed to study the dynamical
bevavior of fluid vesicles and model red blood cells both in shear and capillary flows [4,5]. Several types of dynamical behaviors as well as shape transformations occur as a function
of shear rate (or flow velocity), membrane viscosity and internal viscosity.
 G. Gompper and D.~M. Kroll, in Statistical Mechanics of Membranes and Surfaces (2nd edition), p.359-426, edited by D.~R. Nelson, T. Piran and S. Weinberg (World Scientific, Singapore, 2004).
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 H. Noguchi and G. Gompper, Proc. Natl. Acad. Sci. USA 102, 14159 (2005).