Towards a Nano Geometry? Geometry and Dynamics on Nano Scale

This paper applies I.M. Gelfand’s distinction between adequate and non-adequate use of mathematical language in different contexts to the newly opened window of model-based measurements of intracellular dynamics. The specifics of geometry and dynamics on the mesoscale of cell physiology are elaborated in contrast to the familiar Newtonian mechanics and the more recent, but by now also rather well established quantum field theories. Examples are given originating from the systems biology of insulin secreting pancreatic beta-cells and the mathematical challenges of an envisioned non-invasive control of magnetic nanoparticles. 1. The Challenge of Nano Structures There are many different geometries around. Do we really need new kinds of geometries? Why and how? 1.1. Nanoparticle Based Transducer for Intracellular Structures. These days, we witness a dramatic progress in various technologies devoted to capturing intracellular dynamics of highly differentiated animal cells like the delicate insulin secreting pancreatic β-cell with its thousands of free moving insulin granules, rails of microtubuli, fences of actin filaments, zoos of organelles, proteins, genes, ion channels, electro-static and electrodynamic phenomena. A radically new world of geometry and dynamics is evolving before our eyes. The most decisive technological advances are in the following domains: • Life imaging, for instance confocal multi-beam laser microscopy, admitting up to 40 frames per second for tracking position and movement of suitably prepared nanoparticles within the cell and without overheating the tissue; • Magnetic nanoparticle design and coating, admitting electromagnetic manipulation, docking to selected organelles and tracking their movement; • Computer supported collection and administration of huge databases. This raises the question: Are we on the verge of a need and, then, of the emergence of radically different new mathematical concepts, as predicted by the late I.M. Gelfand in 2003 at a conference (see [Eti]) on The Unity of Mathematics, held in honor of his 90th birthday. He said: Date: January 31, 2012. 2000 Mathematics Subject Classification. Primary 92C37; Secondary 35R35, 53A55.