Problems with physical constraints, such as the incompressibility constraint for mass conservation in fluids or Gauss's laws for electric and magnetic fields, result in generalized saddle point systems. So-called structure-preserving finite elements respect the constraints pointwise, resulting in more physically accurate solutions that are typically robust with respect to some problem parameters. However, constructing these finite elements may involve complicated spaces for the Lagrange multiplier variables. Augmented Lagrangian methods (ALMs) provide one process to compute the solution without the need for an explicit basis for the Lagrange multiplier space. In this talk, we present new convergence estimates for a standard ALM method, sometimes called the iterated penalty method, applied to structure-preserving discretizations of linear saddle point systems.

Deformation theory studies how varieties and other algebro-geometric objects vary in families. A central part of the subject is formal deformation theory, where one deforms over an Artinian base; such deformation problems are governed by Lie algebraic models. 

We pose the question of deforming varieties over nilpotent but not necessarily Artinian bases. These turn out to be classified by the same Lie algebraic models plus some topological structure. More precisely, we will consider partition Lie algebras in the category of ultrasolid modules, a variation of the solid modules of Clausen and Scholze that give a well-behaved category akin to topological modules.

To approach this result, we decompose deformation problems into n-nilpotent layers. Each of these layers is individually easier to understand, and is classified by simpler variants of partition Lie algebras.