Faculty 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:

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.