- Rami Grossberg
Rami Grossberg is an associate professor of
mathematics atCarnegie Mellon University and works inmodel theory . Grossberg's recent work has revolved around theclassification theory of non-elementary classes, and is part of the active effort to prove two ofSaharon Shelah 's outstanding categoricityconjecture s:Conjecture 1. (Categoricity for mathit{L}_omega_1},omega}). Let psi be a sentence. If psi is categorical in a cardinal eth_{omega_{1 then psi is categorical in all cardinals eth_{omega_{1. See
Infinitary logic andBeth number .Conjecture 2. (Categoricity for AECs) See [http://www.math.cmu.edu/~rami/Rami-NBilgi.pdf] and [http://www2.math.uic.edu/~jbaldwin/pub/turino2.pdf] . 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"). Furthemore, 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.External links
* [http://www.math.cmu.edu/~rami/#papers A list of Rami Grossberg's publications]
* [http://www.math.cmu.edu/~rami/Rami-NBilgi.pdf Some of the basics of classification theory for AECs]
* [http://www2.math.uic.edu/~jbaldwin/pub/turino2.pdf A survey of recent work on AECs]
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