Ilia Smilga
Margulis spacetimes and crooked planes
We are interested in the following problem: which groups can act
properly on R^n by affine transformations, or in other terms, can occur
as a symmetry group of a "regular affine tiling"? If we additionally
require that they preserve a Euclidean metric (i.e. act by affine
isometries), then these groups are well-known: they all contain a
finite-index abelian subgroup. If we remove this requirement, a
surprising result due to Margulis is that the free group can act
properly on R^3. I shall explain how to construct such an action.
Charles Parker
Unexpected Behavior in Finite Elements for Linear Elasticity
One of the first problems that finite elements were designed to approximate is the small deformations of a linear elastic body; i.e. the 2D/3D version of Hooke's law for springs from elementary physics. However, for nearly incompressible materials, such as rubber, certain finite elements seemingly lose their approximation power. After briefly reviewing the equations of linear elasticity and the basics of finite element methods, we will spend most of the time looking at a few examples that highlight this unexpected behavior. We conclude with a theoretical result that (mostly) explains these findings.