Rami Grossberg

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Rami Grossberg is an associate professor of mathematics at Carnegie Mellon University and works in model theory. Grossberg's recent work has revolved around the classification theory of non-elementary classes, and is part of the active effort to prove two of Saharon Shelah's outstanding categoricity conjectures:

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.

Conjecture 2. (Categoricity for AECs) See [1] and [2]. Let K be an AEC. There exists a cardinal μ(K) such that categoricity in a cardinal greater than μ(K) implies categoricity in all cardinals greater than μ(K). Furthermore, μ(K) is the Hanf number of K.

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}.

Examples of his results in applications to algebra include the finding that under the weak continuum hypothesis there is no universal object in the class of uncountable locally finite groups (answering a question of Macintyre and Shelah); with Shelah, showing that there is a jump in cardinality of the abelian group Extp(G, Z) at the first singular strong limit cardinal; and, with Shelah, eliminating the use of the diamond in the proof of existence theorem for complete universal locally finite groups in several cardinalities.

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