Rami Grossberg is a full professor of
mathematics at
Carnegie Mellon University and works in
model theory.
Work
Grossberg's work in the past few years has revolved around the
classification theory of non-elementary classes. In particular, he has provided, in joint work with
Monica VanDieren, a proof of an upward "
Morley's Categoricity Theorem
In mathematical logic, a theory is categorical if it has exactly one model ( up to isomorphism). Such a theory can be viewed as ''defining'' its model, uniquely characterizing the model's structure.
In first-order logic, only theories with a ...
" (a version of Shelah's categoricity conjecture) for
Abstract Elementary Classes with the amalgamation property, that are
tame
Tame may refer to:
*Taming, the act of training wild animals
*River Tame, Greater Manchester
*River Tame, West Midlands and the Tame Valley
* Tame, Arauca, a Colombian town and municipality
* "Tame" (song), a song by the Pixies from their 1989 al ...
. In another work with VanDieren, they also initiated the study of ''tame'' Abstract Elementary Classes. Tameness is both a crucial technical property in categoricity transfer proofs and an independent notion of interest in the area – it has been studied by Baldwin, Hyttinen, Lessmann, Kesälä, Kolesnikov, Kueker among others.
Other results include a best approximation to the main gap conjecture for AECs (with Olivier Lessmann), identifying AECs with JEP, AP, no maximal models and tameness as the uncountable analog to Fraïssé's constructions (with VanDieren), a stability spectrum theorem and the existence of Morley sequences for those classes (also with VanDieren).
In addition to this work on the Categoricity Conjecture, more recently, with Boney and Vasey, new understanding of
frames in AECs and
forking (in the abstract elementary class setting) has been obtained.
Some of Grossberg's work may be understood as part of the big project on
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 An infinitary logic is a logic that allows infinitely long statements and/or infinitely long proofs. Some infinitary logics may have different properties from those of standard first-order logic. In particular, infinitary logics may fail to be co ...
and
Beth number
In mathematics, particularly in set theory, the beth numbers are a certain sequence of infinite cardinal numbers (also known as transfinite numbers), conventionally written \beth_0,\ \beth_1,\ \beth_2,\ \beth_3,\ \dots, where \beth is the second H ...
.
''Conjecture 2.'' (Categoricity for AECs) Se
an
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''.
Other 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
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
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
Extp(''G'', ''Z'') at the first singular strong limit cardinal.
Personal life
Grossberg married his former doctoral student and frequent collaborator,
Monica VanDieren.
References
External links
Rami Grossberg*
*
A survey of recent work on AECs
{{DEFAULTSORT:Grossberg, Rami
Year of birth missing (living people)
Living people
Israeli mathematicians
20th-century American mathematicians
21st-century American mathematicians
Carnegie Mellon University faculty
Model theorists