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cummings
James Cummings, Professor
Ph.D., Cambridge University
E-mail: jcumming AT andrew DOT cmu DOT edu
Office: Wean Hall 7101
Phone: 412-268-2551
Personal web site

Research:

I work mostly in the area of combinatorial set theory, which is the study of combinatorial objects (eg graphs, posets, colourings) in a setting where the underlying set is infinite. I am particularly interested in "Singular cardinal combinatorics". My work has touched on a number of areas in set theory including forcing, large cardinals, Ramsey theory and PCF theory. Recent research topics have included the structure theory of linear orderings and posets, rainbow Ramsey theory, strong forcing axioms, and new forcing techniques for obtaining consistency results in singular cardinal combinatorics. I have also done some work in finite combinatorics, using tools including algebraic constructions and the flag algebra method to prove asymptotic extremal results about colourings of finite complete graphs.

Selected Publications:

  • Abraham, Uri and Cummings, James and Smyth, Clifford Some results in polychromatic Ramsey theory. J. Symbolic Logic 72 (2007), no. 3, 865--896
  • Cummings, James and Foreman, Matthew and Magidor, Menachem Canonical structure in the universe of set theory. II. Ann. Pure Appl. Logic 142 (2006), no. 1-3, 55--75
  • Cummings, James and Foreman, Matthew and Magidor, Menachem Canonical structure in the universe of set theory. I. Ann. Pure Appl. Logic 129 (2004), no. 1-3, 211--243
  • Cummings, James and Foreman, Matthew and Magidor, Menachem The non-compactness of square. J. Symbolic Logic 68 (2003), no. 2, 637--643.
  • Cummings, James and Schimmerling, Ernest Indexed squares. Israel J. Math. 131 (2002), 61--99.
  • Cummings, James and Foreman, Matthew and Magidor, Menachem Squares, scales and stationary reflection, Journal of Mathematical Logic 1 (2001), no 1, 3598
  • Apter, Arthur and Cummings, James A global version of a theorem of Ben-David and Magidor. Ann. Pure Appl. Logic 102 (2000), no. 3, 199222.
  • Cummings, James and Shelah, Saharon Some independence results on reflection. J. London Math. Soc. (2) 59 (1999), no. 1, 3749.
  • Cummings, James & Foreman, Matthew The tree property. Adv. Math. 133 (1998), no. 1, 132.
  • Cummings, James Souslin trees which are hard to specialise. Proc. Amer. Math. Soc. 125 (1997), no. 8, 24352441.
  • Cummings, James Collapsing successors of singulars Proc. Amer. Math. Soc. 125 (1997), no. 9, 27032709.
  • Cummings, James & Shelah, Saharon Cardinal invariants above the continuum Ann. Pure Appl. Logic 75 (1995), no. 3, 251268.
  • Cummings, James Strong ultrapowers and long core models J. Symbolic Logic 58 (1993), no. 1, 240248.
  • Cummings, James A model in which GCH holds at successors but fails at limits Trans. Amer. Math. Soc. 329 (1992), no. 1, 139.