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 Rami Grossberg
Associate Professor
Ph.D., The Hebrew University of Jerusalem
Office: Wean Hall 7204
Phone: (412) 268-8482
E-mail: rami@andrew.cmu.edu
Personal web site
Research
I'm interested in model theory (a branch of logic), especially in the
classification theory of infinitary logics, set theory, and the
applications of these areas to algebra. In the last decade my research
focuses in classification theory of Abstract Elementary Classes (AEC),
which are a semantic generalization of Lw1,w. The main test question is
Shelah's categoricity conjecture from the mid seventies. My recent results
on AECs are currently the best approximations to Shelah's main-gap and
categoricity conjectures.
Examples of my results in pure model theory include: generalizing the
Keisler-Shelah omitting types theorem for L(Q) to successors of singular
cardinals; with Shelah, introducing the notion of unsuper-stability for
infinitary logics, and proving a non-structure theorem, which is used to
resolve a problem of Fuchs and Salce in the theory of modules; with Hart,
proving a structure theorem for Lw1,w, which resolves an instance of
Morley's conjecture for excellent classes; and I've studied the notion of
relative saturation and its connection to Shelah's conjecture for Lw1,w.
With Lessmann I proved the main-gap theorem for AECs with a nicely behaved
dependence relation (including \aleph_0-stable homogenous models). With
VanDieren I initiated the study of tame AECs, proved existence of Morley
sequences and a stability spectrum theorem. Lately with VanDieren I proved
Shelah's categoricity conjecture for tame AECs with amalgamation from
categoricity in one successor.
Examples of my results in applications to algebra include: 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); and with Shelah, showing that there is a jump in
cardinality of the abelian group Extp(G,Z) at the first singular strong
limit cardinal.
Publications
R. Grossberg and M. VanDieren. Categoricity from one successor cardinal in
Tame Abstract Elementary Classes. Journal of Mathematical Logic, to appear.
R. Grossberg and M. VanDieren. Galois-stability for Tame Abstract
Elementary Classes, Journal of Mathematical Logic. 6, No. 1 (2006) 25 - 49.
R. Grossberg and M. VanDieren. Shelah's Categoricity Conjecture from a
successor for Tame Abstract Elementary Classes, Journal of Symbolic Logic.
71, (2006) 2, 553 - 568.
R. Grossberg and O. Lessmann. Abstract decomposition theorem and
applications. Contemporary Mathematics, 380, (2005), AMS, pp. 73 - 108.
R. Grossberg, A. Kolesnikov, I. Tomasic, and M. VanDieren. The equality
S1 = D = R, Mathematical Logic Quarterly, 49, (2003), pp. 115 - 128.
R. Grossberg. Classification theory for non-elementary classes.
Contemporary Mathematics, 302, (2002), AMS, pp. 165 - 204.
R. Grossberg, J. Iovino and O. Lessmann. A primer of Simple theories, Arch.
Math. Logic, 41, (2002), 541 - 580.
R. Grossberg and O. Lessmann. Shelah's stability spectrum and homogeneity
spectrum in finite diagrams, Archive for mathematical Logic, 41, (2002) 1,
1 - 31.
R. Grossberg and O. Lessmann. Dependence Relation in Pregeometries. Algebra
Universalis, 44, (2000) pp 199 - 216.
A list of all publications is available.
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