Across many mathematical disciplines one encounters projective systems of algebraic varieties indexed by a combinatorial datum, such as a natural number, a finite graph, or a lattice polytope. As the datum grows, the algebraic complexity of the corresponding variety (measured, for instance, in terms of its defining equations or higher-order syzygies) typically increases. But in good cases it eventually stabilises in a well-defined manner, especially when the family admits a direct system of sufficiently large symmetry groups. Exactly when this stabilisation phenomenon can be expected is still poorly understood, and this question motivates much current research activity in algebraic geometry and adjacent branches. After a brief general setup, which involves passing to the projective limit of the varieties and the direct limit of their symmetry groups, I will discuss a number of concrete instances where stabilisation occurs, both from classical algebraic geometry and from other areas of mathematics.[-]

We study two types of actions on King's moduli spaces of quiver representations over a field k, and we decompose their fixed loci using group cohomology in order to give modular interpretations of the components. The first type of action arises by considering finite groups of quiver automorphisms. The second is the absolute Galois group of a perfect field k acting on the points of this quiver moduli space valued in an algebraic closure of k; the fixed locus is the set of k-rational points, which we decompose using the Brauer group of k, and we describe the rational points as quiver representations over central division algebras over k. Over the field of complex numbers, we describe the symplectic and holomorphic geometry of these fixed loci in hyperkaehler quiver varieties using the language of branes. Over the reals, the rational points of these quiver moduli spaces come from either real or quaternionic quiver representations, and we compute the Poincaré polynomials of both components.
This is joint work with Florent Schaffhauser.[-]

Character varieties of closed surfaces have a natural Poisson structure whose quantization may be constructed in terms of the corresponding quantum group. When the quantum parameter is a root of unity, this quantization carries a central subalgebra isomorphic to the algebra of functions on the classical character variety. In this talk I will describe a procedure which allows one to obtain Azumaya algebras via quantum Hamiltonian reduction. As an application, I will show that quantizations of character varieties at roots of unity are Azumaya over the corresponding classical character varieties.
This is a report on joint work with Iordan Ganev and David Jordan.[-]