We study viscosity solutions of the partial differential equation [special characters omitted] where U ⊆ [special characters omitted] is bounded and open, f ∈ C( U) ∩ L^∞ (U), and [special characters omitted] is the infinity Laplacian.

Our first result is the Max-Ball Theorem, which states that if u ∈ USC(U) is a viscosity subsolution of [special characters omitted] and ϵ > 0, then the function v( x) := max_{B¯( x,ϵ)} u satisfies [special characters omitted] for all x ∈ U _2ϵ := {x ∈ U : dist(x, ∂ U) > 2ϵ}. The left-hand side of this latter inequality is a monotone finite difference scheme that is comparatively easy to analyze. The Max-Ball Theorem allows us to lift results for this finite difference scheme to the continuum equation. In particular, we obtain a new proof of uniqueness of viscosity solutions to the Dirichlet problem when f ≡ 0, inf f > 0, or sup f < 0. The results mentioned so far are joint work with S. Armstrong.

The Max-Ball Theorem is also useful in the analysis of numerical methods for the infinity Laplacian. We obtain a rate of convergence for the numerical method of Oberman [32]. We also present a new adaptive finite difference scheme.

We also prove some results in Model Theory. We study rank-preserving interpretations of theories of finite Morley rank in strongly minimal sets. In particular, we partially answer a question posed by Hasson [20], showing that definable degree is not necessary for such interpretations. We generalize Ziegler's fusion of structures of finite Morley rank [38] to a class of theories without definable degree. Our main combinatorial lemma also allows us to repair a mistake in [23].