Departmental Colloquium

Studies of thin liquid films give rise to a wealth of mathematical problems in nonlinear PDE and provide the opportunity for mathematicians to become involved in interdisciplinary research. I will review several recent problems in this area including the formation and stability of undercompressive shocks in Marangoni-gravity driven coating flows, modeling of `spinodal dewetting' and bifurcation of patterns, finite time singularities and self-similarity, and numerical schemes based on Lyapunov functions and multiplicative splitting. All problems will illustrate the interaction between science, analysis, and methods from modern applied mathematics.

The zero set V(f) of any homogeneous polynomial f of degree d in (n+1) variables can be viewed as a degree d hypersurface in projective n -space. Fixing the degree d and n, all such hypersurfaces are parametrized by the coefficients of the corresponding polynomial up to a constant factor. It turns out that V(f) is singular precisely when the coefficients of f satisfy a polynomial equation \Delta=0 depending on d and n. this generalizes the well known relation b^2-4ac=0 for n=1, d=2 and therefore is called "discriminant relation". The hypersurface V(\Delta) in the corresponding projective space is called "discriminant hypersurface". It is known to be a highly singular object whose invariants (like homology groups etc.) are difficult to calculate. Some years ago Vassiliev outlined a method to approach this problem. Steenbrink has shown that this method gives much more information, namely that ALL of the mixed-Hodge theory is hidden in this approach. Together with Steenbrink I realized that this method allows to describe the mixed Hodge theory of the moduli space of SMOOTH hypersurfaces of fixed degree in a fixed projective space. In the talk I will address the more elementary aspects of this approach.

Crystalline microstructures made from high-contrast dielectric materials have the interesting property that for certain material arrangements, the frequency spectrum for electromagnetic wave propagation admits a complete band gap. Such structures are relatively new and are expected to have a large impact in optoelectronics. We consider the problem of determining material arrangements which result in maximal band gaps. This can be formulated as an extremization problem, which turns out to have a nonsmooth but Lipschitz continuous objective. Well-posedness of the problem is studied, and a generalized gradient ascent maximization algorithm is proposed. Computational experiments are presented in which several novel structures with large band gaps are obtained.

Loewner's equation provides a powerful analytic tool and unified approach to the study of various aggregation processes in the complex plane (such as Laplacian random walks, Hele-Shaw and diffusion limited aggregation models). I will discuss some open problems and recent developments in this area.

Toric varieties are special algebraic varieties defined by some combinatorial data. It turns out that these data also define certain modular forms on the upper half plane, which we call toric forms. Linear span of toric forms has non-trivial number-theoretic properties, related to the conjecture of Birch and Swinnerton-Dyer.

Gromov's recent work on hyperbolic groups showed that one can study finitely generated groups via the 'large scale geometry' of their Cayley graphs. The appropriate geometric notion in this context is that of a quasi-isometry rather than isometry. In my talk I will focus on the decomposition theory of groups. It turns out that the 'asymptotic topology' of the Cayley graph determines in many cases whether a group admits a splitting as an amalgamated product or HNN-extension. In the case of splittings over infinite cyclic groups the proof of this fact is based on a new topological characterization of the plane.