James Cummings

Professor
Ph.D., Cambridge University

Office: Wean Hall 7101
Phone: (412) 268-2551
E-mail: jcumming@andrew.cmu.edu
Personal web site

Research

My research interests lie in set theory, with a focus on large cardinals, forcing and inner models. A major theme in modern set theory is the determination of the "consistency strength" of combinatorial statements; typically this strength is measured by a large cardinal axiom, with forcing used to show the axiom is sufficient and inner models used to show that it is necessary. I am also interested in purely combinatorial questions, and in applications of set theory to areas such as algebra and analysis.

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, 35–98

Apter, Arthur and Cummings, James
A global version of a theorem of Ben-David 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.