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Rami Grossberg, Professor
Ph.D., The Hebrew University of Jerusalem
E-mail: rami@andrew.cmu.edu
Office: Wean Hall 7204
Phone: 412-268-8482
Personal web site

Research:

My primary area of interest is model theory which is one of the major fields of mathematical logic. My secondary interest is in related combinatorial, set-theoretic problems and applications to algebra.

I am focusing in the development of a model theory for non first-order axiomaziable classes, primarily in the context of Abstract Elementary Classes. The program of developing classification theory for AECs was established by Shelah more than 35 years ago, it is his main focus. The goal is to discover structural properties and concepts that will bring model of AECs to a stage of maturity in par with much older areas of pure mathematics like commutative algebra and algebraic geometry. Shelah proposed several test problems to measure progress with the oldest dates back to 1976, the categoricity conjecture. Shelah alone published more than 1,000 pages of difficult mathematics for various partial approximations, but the conjecture is still open. Another far reaching conjecture of Shelah is an extension of what is known as the “Main Gap Theorem”. With Bradd Hart we proved a case of Shelah’s main gap conjecture for atomic models of a first-order theory (known as excellent classes), later this was extended in my work with my former student Olivier Lessmann for the broader class of homogenous models. At present this is the best known approximation to Shelah’s main gap conjecture. With my former student Monica VanDieren we identified a subclass called tame AECs and we established for tame AECs the strongest known case of Shelah’s catgeoricity conjecture.

Recently with my current student Will Boney we managed to discover the illusive notion of forking for AECs and established its basic properties as an abstract independence relation (generalizing pre-geometries or matroids). I expect this to have a major effect on future developments.


Selected 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.