# Rami Grossberg

Conjecture 1. (Categoricity for $\mathit{L}_{{\omega_1},\omega}$). Let $\psi$ be a sentence. If $\psi$ is categorical in a cardinal $\; >\beth_{\omega_{1}}$ then $\psi$ is categorical in all cardinals $\; >\beth_{\omega_{1}}$. See Infinitary logic and Beth number.
Examples of his results in pure model theory include: generalizing the Keisler–Shelah omitting types theorem for $\mathit{L(Q)}$ to successors of singular cardinals; with Shelah, introducing the notion of unsuper-stability for infinitary logics, and proving a nonstructure theorem, which is used to resolve a problem of Fuchs and Salce in the theory of modules; with Hart, proving a structure theorem for $\mathit{L}_{{\omega_1},\omega}$, which resolves Morley's conjecture for excellent classes; and the notion of relative saturation and its connection to Shelah's conjecture for $\mathit{L}_{{\omega_1},\omega}$.