The classification of countable models of set theory

We study the complexity of the classification problem for countable models of set theory ($\mathsf{ZFC}$). We prove that the classification of arbitrary countable models of $\mathsf{ZFC}$ is Borel complete, meaning that it is as complex as it can conceivably be. We then give partial results concerning the classification of countable well-founded models of $\mathsf{ZFC}$.     

Turing invariant sets and the perfect set property

We show that $\text{◂...▸}\mathsf{ZF}+\mathsf{DC}+\text{“all}\text{Turing}\text{invariant}\text{sets}\text{of}\text{reals}\text{have}\text{the}\text{perfect}\text{set}\text{property”}$ implies that all sets of reals have the perfect set property. We also show that this result generalizes to all countable analytic equivalence relations.     

On the evolution equation of compressible vortex sheets

We are concerned with supersonic vortex sheets for the Euler equations of compressible inviscid fluids in two space dimensions. For the problem with constant coefficients we derive an evolution equation for the discontinuity front of the vortex sheet. This is a pseudo-differential equation of order two. In agreement with the classical stability analysis, if the Mach number $\mathsf{M}$ satisfies , the symbol is elliptic and the problem is ill-posed. On the contrary, if then the problem is weakly stable, and we are able to derive a wave-type a priori energy estimate for the solution, with no loss of regularity with respect to the data. Then we prove the well-posedness of the problem, by showing the existence of the solution in weighted Sobolev spaces.     

Computability of graphs

We consider topological pairs $(A,B)$, $B\subseteq A$, which have computable type, which means that they have the following property: if X is a computable topological space and $f:A\to X$ a topological imbedding such that $f(A)$ and $f(B)$ are semicomputable sets in X, then $f(A)$ is a computable set in X. It is known, e.g., that $(M,\partial M)$ has computable type if M is a compact manifold with boundary. In this paper we examine topological spaces called graphs and we show that we can in a natural way associate to each graph G a discrete subspace E so that $(G,E)$ has computable type. Furthermore, we use this result to conclude that certain noncompact semicomputable graphs in computable metric spaces are computable.     

Graph-like compacta: Characterizations and Eulerian loops

A compact graph-like space is a triple $\left(X,V,E\right)$, where $X$ is a compact, metrizable space, $V\subseteq X$ is a closed zero-dimensional subset, and $E$ is an index set such that $X⧹V\cong \text{◂+▸}E\times \left(0,1\right)$. New characterizations of compact graph-like spaces are given, connecting them to certain classes of continua, and to standard subspaces of Freudenthal compactifications of locally finite graphs. These are applied to characterize Eulerian graph-like compacta.