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James Cummings, Professor Ph.D., Cambridge University Email: jcumming AT andrew DOT cmu DOT edu Office: Wean Hall 7101 Phone: 4122682551 Personal web siteResearch: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, 865896

Cummings, James and Foreman, Matthew and Magidor, Menachem
Canonical structure in the universe of set theory. II.
Ann. Pure Appl. Logic 142 (2006), no. 13, 5575

Cummings, James and Foreman, Matthew and Magidor, Menachem
Canonical structure in the universe of set theory. I.
Ann. Pure Appl. Logic 129 (2004), no. 13, 211243

Cummings, James and Foreman, Matthew and Magidor, Menachem
The noncompactness of square.
J. Symbolic Logic 68 (2003), no. 2, 637643.

Cummings, James and Schimmerling, Ernest
Indexed squares.
Israel J. Math. 131 (2002), 6199.

Cummings, James and Foreman, Matthew and Magidor, Menachem
Squares, scales and stationary reflection,
Journal of Mathematical Logic 1 (2001), no 1, 35–98

Apter, Arthur and Cummings, James
A global version of a theorem of BenDavid and Magidor.
Ann. Pure Appl. Logic 102 (2000), no. 3, 199–222.

Cummings, James and Shelah, Saharon
Some independence results on reflection.
J. London Math. Soc. (2) 59 (1999), no. 1, 37–49.

Cummings, James & Foreman, Matthew
The tree property.
Adv. Math. 133 (1998), no. 1, 1–32.

Cummings, James
Souslin trees which are hard to specialise.
Proc. Amer. Math. Soc. 125 (1997), no. 8, 2435–2441.

Cummings, James
Collapsing successors of singulars
Proc. Amer. Math. Soc. 125 (1997), no. 9, 2703–2709.

Cummings, James & Shelah, Saharon
Cardinal invariants above the continuum
Ann. Pure Appl. Logic 75 (1995), no. 3, 251–268.

Cummings, James
Strong ultrapowers and long core models
J. Symbolic Logic 58 (1993), no. 1, 240–248.

Cummings, James
A model in which GCH holds at successors but fails at limits
Trans. Amer. Math. Soc. 329 (1992), no. 1, 1–39.
