Godunov's theorem: Difference between revisions
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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 [[Stable theory|classification theory]] of non-elementary classes, and is part of the active effort to prove two of [[Saharon Shelah]]'s outstanding categoricity [[conjecture]]s: | |||
'''Conjecture 1.''' (Categoricity for <math>\mathit{L}_{{\omega_1},\omega}</math>). Let <math>\psi</math> be a [[sentence (mathematical logic)|sentence]]. If <math>\psi</math> is categorical in a cardinal <math>\; >\beth_{\omega_{1}}</math> then <math>\psi</math> is categorical in all cardinals <math>\; >\beth_{\omega_{1}}</math>. See [[Infinitary logic]] and [[Beth 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 <math>\mathit{L(Q)}</math> 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 <math>\mathit{L}_{{\omega_1},\omega}</math>, which resolves Morley's conjecture for excellent classes; and the notion of relative saturation and its connection to Shelah's conjecture for <math>\mathit{L}_{{\omega_1},\omega}</math>. | |||
Examples of his results in applications to algebra include the finding that under the [[continuum hypothesis|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] | |||
{{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. --> | |||
| NAME = Grossberg, Rami | |||
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| SHORT DESCRIPTION = American mathematician | |||
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| DATE OF DEATH = | |||
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}} | |||
{{DEFAULTSORT:Grossberg, Rami}} | |||
[[Category:Year of birth missing (living people)]] | |||
[[Category:Living people]] | |||
[[Category:Israeli mathematicians]] | |||
[[Category:American mathematicians]] | |||
[[Category:Carnegie Mellon University faculty]] | |||
[[Category:Model theorists]] | |||
Revision as of 14:36, 16 January 2013
Template:Notability 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 ). Let be a sentence. If is categorical in a cardinal then is categorical in all cardinals . 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). 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 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 , which resolves Morley's conjecture for excellent classes; and the notion of relative saturation and its connection to Shelah's conjecture for .
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.