For homogeneous, defect-free TQFTs, (1) n+\epsilon-dimensional versions of the theories are relatively easy to construct; (2) an n+\epsilon-dimensional theory can be extended to n+1-dimensional (i.e. the top-dimensional path integral can be defined) if certain more restrictive conditions related to handle cancellation are satisfied; and (3) applying this path integral construction to a handle decomposition of an n+1-manifold yields a state sum description of the path integral.  In this talk, I'll show that the same pattern holds for defect TQFTs.  The adaptation of homogeneous results to the defect setting is mostly straightforward, with the only slight difficulty being the purely topological problem of generalizing handle theory to manifolds with defects.  If time allows, I'll describe two applications: a Verlinde-like dimension formula for the dimension of the ground state of fracton systems, and a generalization, to arbitrary dimension, of Ostrik's theorem relating algebra objects to modules (gapped boundaries).

In this talk, we'll consider the numerical approximation of singularly perturbed reaction-diffusion partial differential equations, by finite element methods (FEMs).

Solutions to such problems feature boundary layers, the width of which depends on the magnitude of the perturbation parameter. For many hears, some numerical analysts have been preoccupied with constructing methods that can resolve any layers present, and for which one can establish an error estimate that is  independent of the perturbation parameter. Such methods are called "parameter robust", or (in some norms) "uniformly convergent".

In this talk we'll begin with the simplest possible parameter robust FEM: a standard Galerkin finite element method (FEM) applied on a suitably constructed  mesh using a priori information. However, from a practical point of view, not very scalable. To resolve this issue we consider the application of sparse grid techniques. These methods have many variants, two of which we'll consider: the hierarchical basis approach (e.g., Zenger, 1991) and the
two-scale method (e.g., many papers by Aihui Zhou and co-authors). The former can be more efficient, while the latter is considered simpler in both theory and practice.

Our goal is to try to unify these two approaches (at least in two dimensions), and then extend to three-dimensional problems, and, moreover, to other FEMs.