Rami Grossberg

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.