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Biography
Rami Grossberg is a full professor of mathematics at Carnegie Mellon University and works in model theory.
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" (a version of Shelah's categoricity conjecture) for Abstract Elementary Classes with the amalgamation property, that are tame. 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
L
ω
1
,
ω
{\displaystyle {\mathit {L}}_{{\omega _{1}},\omega }}
). Let
ψ
{\displaystyle \psi }
be a sentence. If
ψ
{\displaystyle \psi }
is categorical in a cardinal
>
ℶ
ω
1
{\displaystyle \;>\beth _{\omega _{1}}}
then
ψ
{\displaystyle \psi }
is categorical in all cardinals
>
ℶ
ω
1
{\displaystyle \;>\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.
Other examples of his results in pure model theory include: generalizing the Keisler–Shelah omitting types theorem for
L
(
Q
)
{\displaystyle {\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
L
ω
1
,
ω
{\displaystyle {\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
L
ω
1
,
ω
{\displaystyle {\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.