We consider a mutation-finite quiver Q. Associated to Q are two interesting complex manifolds: (1) the space of stability conditions for the derived CY3 category of the Ginzburg algebra associated to the quiver, (2) the complex cluster Poisson variety. Each of these manifolds is constructed from the combinatorics of the quiver, though in very different ways. Tom Bridgeland and I introduce an "interpolation" of these complex manifolds: we give a construction of a complex manifold together with a submersion to the complex plane which we call the stability twistor space. The fiber over the origin is a finite quotient of the space of stability conditions whereas every other fiber is an etale cover of the cluster Poisson variety associated to the quiver. This etale map is a generalization of Thurston's grafting map. Constructing sections of the twistor family is closely related to solving the Riemann-Hilbert problem that encodes the Donaldson-Thomas invariants.

A generically log smooth (gls) family is a degeneration f: X → 𝔸¹ of algebraic varieties which is log smooth in relative codimension one; as such, they are the logarithmic analog of families of normal varieties. Gls families play a fundamental role in smoothing reducible schemes because most of them do not admit a global log smooth structure. With a few exceptions, the log singularities of gls families are still poorly understood; it is therefore useful to study explicit examples. In this talk, I present the first general algorithmic tool to show that a family f: X → 𝔸¹ given by explicit generators and relations X = Spec 𝕜[t,x₁,...,xₙ]/(f₁,...,fᵣ) is indeed log smooth in relative codimension one, hence a gls family.

Given a Laurent polynomial with reflexive Newton polytope, we give a criterion ensuring that it gives rise to a deformation family with smooth general fibre. One-parameter deformations in this family correspond to mutations of the Laurent polynomial.

In dimension three, the smooth fibres arising from this construction include all 98 deformation families of very ample smooth Fano threefolds. This is an upcoming result that generalises the published work Laurent Polynomials and Deformations of Non-Isolated Gorenstein Toric Singularities.

Building on that framework, I will also outline evidence towards a conjecture of Corti–Filip–Petracci predicting a one-to-one correspondence between 0-mutable Laurent polynomials and the smoothing deformation components of three-dimensional affine Gorenstein toric varieties.