The lecture will be about complete embedded minimal surfaces in R^3 (some immersed ones help the explanations). Before 1978
very little was known and after 1984 progress became rapid. I will
illustrate with pictures, how observed features of known surfaces
led to new, increasingly abstract, constructions. I will not assume
that the audience consists of minimal surface experts, the lecture
is intended to be understandable by graduate students.

In this talk I present my recent results on the regularity conditions for a solution to the 3D Navier-Stokes equations with powers of the Laplacian, which incorporates the vorticity direction and its magnitude simultaneously. For the proof of the we exploit geometric properties of the vortex stretching term as well as the estimate using the Triebel-Lizorkin type of norms.

Biochemical reaction networks provide a paradigm for many dynamical systems in biology. The paradigm can be generalized to describe "variable-structure systems" in which objects larger than molecules (such as cells) also change in number and in their relationships over time. In the course of building mathematical and software tools for understanding such networks and dynamical systems, we have identified some interesting applied mathematical problems whose reformulation and solution would be very useful in current computational biology. For example, we can identify partial differential equations whose solution would be especially instructive for enzyme kinetics. These problems arise at the level of small reaction networks, multimolecular complexes, and the development of multicellular tissues. Developmental examples include modeling the shoot meristem of a plant.

A common mathematical framework for models at these different spatial scales can be given in terms of "dynamical grammars". In a dynamical grammar, an input/output syntax for an elementary chemical or biological processes is mapped to an operator algebra expression for the generator of the temporal dynamics associated with that process. Many processes act simultaneously (in parallel) if their generators are summed. Contingent spatial relationships are expressed in terms of dynamical graph grammars, whose formulation could perhaps be improved by use of ideas from topology and differential geometry. By solving such problems, we may hope to construct a useful modeling language of sufficient generality to describe multiscale, variable-structure dynamical systems that arise naturally in biology.

We consider the problem of quenching the flame in a framework of passive reaction-diffusion model. We ask which flows are more efficient in supressing reaction, and prove bounds on the relationship between flow strength and the initial flame size for different classes of flows. The estimates we prove agree very well with numerical experiments carried out in collaboration with astrophysics ASC group at the University of Chicago. The problem is closely related to proving norm bounds for the evloution semigroup corresponding to the passive scalar model. The techniques involve PDE and probability tools, and further natural questions indicate interesting links with spectral theory of elliptic operators and dynamical systems.