Stability conditions on derived categories of algebraic varieties and their wall-crossings have given algebraic geometers a number of new tools to study the geometry of moduli spaces of stable sheaves. In work in progress with Macri, Lahoz, Nuer, Perry and Stellari, we are extending this toolkit to a the "relative" setting, i.e. for a family of varieties. Our construction comes with relative moduli spaces of stable objects; this gives additional ways of constructing new families of varieties from a given family, thereby potentially relating different moduli spaces of varieties.

A C^infinity scheme is a version of a scheme that uses a maximal spectrum. The category of C^infinity schemes contains the category of Manifolds as a full subcategory, as well as being closed under fibre products. In other words, this category is equipped to handle intersection singularities of smooth spaces.

While originally defined in the set up of Synthetic Differential Geometry, C^infinity schemes have more recently been used to describe derived manifolds, for example, the d-manifolds of Joyce. There are applications of this in Symplectic Geometry, such as the describing the moduli space of J-holomorphic forms.

In this talk, I will describe the category of C^infinity schemes, and how this idea can be extended to manifolds with corners. If time, I will mention the applications of this in derived geometry.

Symplectic reduction is the natural quotient construction for symplectic manifolds. Given a free and proper action of a Lie group G on a symplectic manifold M, this process produces a new symplectic manifold of dimension dim(M) - 2 dim(G). For non-free actions, however, the result is usually fairly singular. But Sjamaar-Lerman (1991) showed that the singularities can be understood quite precisely: symplectic reductions by non-free actions are partitioned into smooth symplectic manifolds, and these manifolds fit nicely together in the sense that they form a stratification.

Symplectic reduction has an analogue in hyperkähler geometry, which has been a very important tool for constructing new examples of these special manifolds. In this talk, I will explain how Sjamaar-Lerman’s results can be extended to this setting, namely, hyperkähler quotients by non-free actions are stratified
spaces whose strata are hyperkähler.

For many moduli problems, in order to construct a moduli space as a geometric invariant theory quotient, one needs to impose a notion of (semi)stability. Using recent results in non-reductive geometric invariant theory, we explain how to stratify certain moduli stacks in such a way that each stratum admits a coarse moduli space which is constructed as a geometric quotient of an action of a linear algebraic group with internally graded unipotent radical. As many stacks are
naturally filtered by quotient stacks, this involves describing how to stratify certain quotient stacks. Even for quotient stacks for reductive group actions, we see that non-reductive GIT is required to construct the coarse moduli spaces of the higher strata. We illustrate this point by studying the example of the moduli stack of coherent sheaves over a projective scheme. This is joint work with G. Berczi, J. Jackson and F. Kirwan.