Past

Over the last several decades, many mesh generation methods and a plethora of adaptive methods for solving differential equations have been developed.  In this talk, we take a general approach for describing the mesh generation problem, which can be considered as being in some sense equivalent to determining a coordinate transformation between physical space and a computational space.  Our description provides some new theoretical insights into precisely what is accomplished from mesh equidistribution (which is a standard adaptivity tool used in practice) and mesh alignment.  We show how variational mesh generation algorithms, which have historically been the most common and important ones, can generally be compared using these mesh generation principles.  Lastly, we relate these to a variety of moving mesh methods for solving time-dependent PDEs.

This is joint work with Weizhang Huang, Kansas University

We consider the problem of cavitation in nonlinear elasticity, or the formation of macroscopic cavities in elastic materials from microscopic defects, when subjected to large tension at the boundary.

The main goal is to determine the optimal locations where the body prefers the cavities to open, the preferred number of cavities, their optimal sizes, and their optimal shapes. To this aim it is necessary to analyze the elastic energy of an incompressible deformation creating multiple cavities, in a way that accounts for the interaction between the cavitation singularities. Based on the quantitative version of the isoperimetric inequality, as well as on new explicit constructions of incompressible deformations creating cavities of different shapes and sizes, we provide energy estimates showing that, for certain loading conditions, there are only the following possibilities:

• only one cavity is created, and if the loading is isotropic then it is created at the centre
• multiple cavities are created, they are spherical, and the singularities are well separated
• there are multiple cavities, but they act as a single spherical cavity, they are considerably distorted, and the distance between the cavitation singularities must be of the same order as the size of the initial defects contained in the domain.

In the latter case, the formation of thin structures between the cavities is observed, reminiscent of the initiation of ductile fracture by void coalesence.

This is joint work with Sylvia Serfaty (LJLL, Univ. Paris VI).

Postponed until May

I'll describe an approach to perturbative quantum field theory
which is philosophically similar to the deformation quantization approach
to quantum mechanics. The algebraic objects which appear in our approach --
factorization algebras -- also play an important role in some recent work
in topology (by Francis, Lurie and others).  This is joint work with Owen
Gwilliam.

We review the relation between the geometry of Kleinian singularities and Dynkin diagrams of types ADE, recalling in particular the construction of a braid group action of type A, D, or E on the derived category of coherent sheaves on the minimal resolution of a Kleinian singularity. By work of Seidel-Thomas, this action was known to be faithful in type A. We extend this faithfulness result to types ADE, which provides the missing ingredient for completing Bridgeland's description of spaces of stability conditions for certain triangulated categories associated to Kleinian singularities. Our main tool is the Garside normal form for braid group elements. This project is joint work with Hugh Thomas from the University of New Brunswick.

I will consider the physical background, and general thinking behind, the recent programme aimed at applying topos theory to the foundations of physics.