# James Cummings

## Professor

**Address:**

6130 Wean Hall

Department of Mathematical Sciences

Carnegie Mellon University

5000 Forbes Avenue

Pittsburgh, PA 15213

**P:** 412-268-2551

# Education

Ph.D. in Mathematics, Cambridge University

Postdoctoral Appointments:

- Mathematical Sciences Research Institute
- Massachusetts Institute of Technology
- Dartmouth College
- Hebrew University of Jerusalem

# Research

I work mostly in the area of combinatorial set theory, which is the study of combinatorial objects (e.g. 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.

# Select Publications

Cummings, J., and Morgan, C. (2017). Dowker and super-Dowker filters. *Proceedings of the American Mathematical Society*, 145, 5381–5390.

Cummings, J., Džamonja, M., Magidor, M., Morgan, C., Shelah, S. (2017). A framework for forcing constructions at successors of singular cardinals. *Transactions of the American Mathematical Society,* 369, 7405–7441.

Cummings, J., and Lambie-Hanson,C., (2016). Bounded stationary reflection. *Proceedings of the American Mathematical Society,* 144, 861–873.

Cummings, J., Friedman,S. D., and M. Golshani, (2015). Collapsing the cardinals of HOD. *Journal of Mathematical Logic,* 15, 1550007.

Cummings, J., Král',D., Pfender,F., Sperfeld,K., Treglown, A., and Young, M. (2013). Monochromatic triangles in three-coloured graphs. *Journal of Combinatorial Theory Series B,* 103, 489–503.

Abraham,U., Bonnet, R., Cummings, J., Džamonja, M., and Thompson, K., (2012). A scattering of orders. *Transactions of the American Mathematical Societ*y, 364, 6259–6278.